reflection physics problem solving

Physics Problems with Solutions

Reflection of light rays, examples and solutions, reflection of light rays on a reflecting surface.

Laws of reflection

(1) The incident light ray, the reflected light ray and the normal to the interface at the point of incidence make a plane called the plane of incidence (2) The angle of incidence i and the angle of reflection r have the same size.

A light ray strikes a reflective plane surface at an angle of 56° with the surface. a) Find the angle of incidence. b) Find the angle of reflection. c) Find the angle made by the reflected ray and the surface. d) Find the angle made by the incident and reflected rays.

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1.2 The Law of Reflection

Learning objectives.

By the end of this section, you will be able to:

• Explain the reflection of light from polished and rough surfaces
• Describe the principle and applications of corner reflectors

Whenever we look into a mirror, or squint at sunlight glinting from a lake, we are seeing a reflection. When you look at a piece of white paper, you are seeing light scattered from it. Large telescopes use reflection to form an image of stars and other astronomical objects.

The law of reflection states that the angle of reflection equals the angle of incidence, or

The law of reflection is illustrated in Figure 1.5 , which also shows how the angle of incidence and angle of reflection are measured relative to the perpendicular to the surface at the point where the light ray strikes.

We expect to see reflections from smooth surfaces, but Figure 1.6 illustrates how a rough surface reflects light. Since the light strikes different parts of the surface at different angles, it is reflected in many different directions, or diffused. Diffused light is what allows us to see a sheet of paper from any angle, as shown in Figure 1.7 (a). People, clothing, leaves, and walls all have rough surfaces and can be seen from all sides. A mirror, on the other hand, has a smooth surface (compared with the wavelength of light) and reflects light at specific angles, as illustrated in Figure 1.7 (b). When the Moon reflects from a lake, as shown in Figure 1.7 (c), a combination of these effects takes place.

When you see yourself in a mirror, it appears that the image is actually behind the mirror ( Figure 1.8 ). We see the light coming from a direction determined by the law of reflection. The angles are such that the image is exactly the same distance behind the mirror as you stand in front of the mirror. If the mirror is on the wall of a room, the images in it are all behind the mirror, which can make the room seem bigger. Although these mirror images make objects appear to be where they cannot be (like behind a solid wall), the images are not figments of your imagination. Mirror images can be photographed and videotaped by instruments and look just as they do with our eyes (which are optical instruments themselves). The precise manner in which images are formed by mirrors and lenses is discussed in an upcoming chapter on Geometric Optics and Image Formation .

Corner Reflectors (Retroreflectors)

A light ray that strikes an object consisting of two mutually perpendicular reflecting surfaces is reflected back exactly parallel to the direction from which it came ( Figure 1.9 ). This is true whenever the reflecting surfaces are perpendicular, and it is independent of the angle of incidence. (For proof, see [link] at the end of this section.) Such an object is called a corner reflector , since the light bounces from its inside corner. Corner reflectors are a subclass of retroreflectors, which all reflect rays back in the directions from which they came. Although the geometry of the proof is much more complex, corner reflectors can also be built with three mutually perpendicular reflecting surfaces and are useful in three-dimensional applications.

Many inexpensive reflector buttons on bicycles, cars, and warning signs have corner reflectors designed to return light in the direction from which it originated. Rather than simply reflecting light over a wide angle, retroreflection ensures high visibility if the observer and the light source are located together, such as a car’s driver and headlights. The Apollo astronauts placed a true corner reflector on the Moon ( Figure 1.10 ). Laser signals from Earth can be bounced from that corner reflector to measure the gradually increasing distance to the Moon of a few centimeters per year.

Working on the same principle as these optical reflectors, corner reflectors are routinely used as radar reflectors ( Figure 1.11 ) for radio-frequency applications. Under most circumstances, small boats made of fiberglass or wood do not strongly reflect radio waves emitted by radar systems. To make these boats visible to radar (to avoid collisions, for example), radar reflectors are attached to boats, usually in high places.

As a counterexample, if you are interested in building a stealth airplane, radar reflections should be minimized to evade detection. One of the design considerations would then be to avoid building 90 ° 90 ° corners into the airframe.

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24.2: Reflection, Refraction, and Dispersion

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learning objectives

• Formulate the relationship between the angle of reflection and the angle of incidence

Whenever you look into a mirror or squint at sunlight glinting off a lake, you are seeing a reflection. When you look at the text in a book, you are actually seeing the light that is reflected from it. Large telescopes use reflections to form images of stars and other astronomical objects. In fact, the only way we can see an object that does not itself emit light is if that object reflects light.

The law of reflection is illustrated in, which also shows how the angles are measured relative to the perpendicular to the surface at the point where the light ray strikes. The law of reflection is very simple: The angle of reflection equals the angle of incidence. When we see our reflection in a mirror, it appears that our image is actually behind the mirror — we see the light coming from a direction determined by the law of reflection. The angles are such that our image appears exactly the same distance behind the mirror as we stand away from the mirror.

Mirror Reflection : An image in a mirror appears as though it is behind the mirror. The two rays shown are those that strike the mirror at just the correct angles to be reflected into the eyes of the viewer. The image appears to come from the direction the rays are coming from when they enter the viewer’s eyes.

Law of Reflection : The law of reflection states that the angle of reflection equals the angle of incidence: θr = θi. The angles are measured relative to the perpendicular to the surface at the point where the ray strikes the surface.

We expect to see reflections off a smooth surface. However, light strikes different parts of a rough surface at different angles, and it is reflected in many different directions (“diffused”). Diffused light is what allows us to see a sheet of paper from any angle. Many objects, such as people, clothing, leaves, and walls, have rough surfaces and can be seen from all sides. A mirror, on the other hand, has a smooth surface (compared with the wavelength of light) and reflects light at specific angles. When the moon reflects off the surface of a lake, a combination of these effects takes place.

Reflection : A brief overview of reflection and the law of reflection.

The Law of Refraction: Snell’s Law and the Index of Refraction

The amount that a light ray changes its direction depends both on the incident angle and the amount that the speed changes.

• Formulate the relationship between the index of refraction and the speed of light

It is easy to notice some odd things when looking into a fish tank. For example, you may see the same fish appearing to be in two different places. This is because light coming from the fish to us changes direction when it leaves the tank, and in this case, it can travel two different paths to get to our eyes. The changing of a light ray’s direction (loosely called bending) when it passes through variations in matter is called refraction. Refraction is responsible for a tremendous range of optical phenomena, from the action of lenses to voice transmission through optical fibers.

Law of Refraction : Looking at the fish tank as shown, we can see the same fish in two different locations, because light changes directions when it passes from water to air. In this case, the light can reach the observer by two different paths, and so the fish seems to be in two different places. This bending of light is called refraction and is responsible for many optical phenomena.

Refraction: The changing of a light ray’s direction (loosely called bending) when it passes through variations in matter is called refraction.

Speed of Light

The speed of light c not only affects refraction, it is one of the central concepts of Einstein’s theory of relativity. The speed of light varies in a precise manner with the material it traverses. It makes connections between space and time and alters our expectations that all observers measure the same time for the same event, for example. The speed of light is so important that its value in a vacuum is one of the most fundamental constants in nature as well as being one of the four fundamental SI units.

Why does light change direction when passing from one material ( medium ) to another? It is because light changes speed when going from one material to another.

Law of Refraction

A ray of light changes direction when it passes from one medium to another. As before, the angles are measured relative to a perpendicular to the surface at the point where the light ray crosses it. The change in direction of the light ray depends on how the speed of light changes. The change in the speed of light is related to the indices of refraction of the media involved. In mediums that have a greater index of refraction the speed of light is less. Imagine moving your hand through the air and then moving it through a body of water. It is more difficult to move your hand through the water, and thus your hand slows down if you are applying the same amount of force. Similarly, light travels slower when moving through mediums that have higher indices of refraction.

The amount that a light ray changes its direction depends both on the incident angle and the amount that the speed changes. For a ray at a given incident angle, a large change in speed causes a large change in direction, and thus a large change in angle. The exact mathematical relationship is the law of refraction, or “Snell’s Law,” which is stated in equation form as:

\[\mathrm{n_1 \sin \theta _1 = n_2 \sin \theta _2}\]

Here n 1 and n 2 are the indices of refraction for medium 1 and 2, and θ 1 and θ 2 are the angles between the rays and the perpendicular in medium 1 and 2. The incoming ray is called the incident ray and the outgoing ray the refracted ray, and the associated angles the incident angle and the refracted angle. The law of refraction is also called Snell’s law after the Dutch mathematician Willebrord Snell, who discovered it in 1621. Snell’s experiments showed that the law of refraction was obeyed and that a characteristic index of refraction n could be assigned to a given medium.

Understanding Snell’s Law with the Index of Refraction : This video introduces refraction with Snell’s Law and the index of refraction.The second video discusses total internal reflection (TIR) in detail. http://www.youtube.com/watch?v=fvrvqm3Erzk

Total Internal Reflection and Fiber Optics

Total internal reflection happens when a propagating wave strikes a medium boundary at an angle larger than a particular critical angle.

• Formulate conditions required for the total internal reflection

Total internal reflection is a phenomenon that happens when a propagating wave strikes a medium boundary at an angle larger than a particular critical angle with respect to the normal to the surface. If the refractive index is lower on the other side of the boundary and the incident angle is greater than the critical angle, the wave cannot pass through and is entirely reflected. The critical angle is the angle of incidence above which the total internal reflectance occurs.

What is Total Internal Reflection? : Describes the concept of total internal reflection, derives the equation for the critical angle and shows one example.

Critical angle

The critical angle is the angle of incidence above which total internal reflection occurs. The angle of incidence is measured with respect to the normal at the refractive boundary (see diagram illustrating Snell’s law ). Consider a light ray passing from glass into air. The light emanating from the interface is bent towards the glass. When the incident angle is increased sufficiently, the transmitted angle (in air) reaches 90 degrees. It is at this point no light is transmitted into air. The critical angle θcθc is given by Snell’s law, n1sinθ1=n2sinθ2n1sin⁡θ1=n2sin⁡θ2. Here, n 1 and n 2 are refractive indices of the media, and θ1θ1 and θ2θ2are angles of incidence and refraction, respectively. To find the critical angle, we find the value for θ1θ1 when θ2θ2= 90° and thus sinθ2=1sin⁡θ2=1. The resulting value of θ1θ1 is equal to the critical angle θc=θ1=arcsin(n2n1)θc=θ1=arcsin⁡(n2n1). So the critical angle is only defined when n 2 /n 1 is less than 1.

Fig 1 : Refraction of light at the interface between two media, including total internal reflection.

Optical Fiber

Total internal reflection is a powerful tool since it can be used to confine light. One of the most common applications of total internal reflection is in fibre optics. An optical fibre is a thin, transparent fibre, usually made of glass or plastic, for transmitting light. The construction of a single optical fibre is shown in.

Fig 2 : Fibers in bundles are clad by a material that has a lower index of refraction than the core to ensure total internal reflection, even when fibers are in contact with one another. This shows a single fiber with its cladding.

The basic functional structure of an optical fiber consists of an outer protective cladding and an inner core through which light pulses travel. The overall diameter of the fiber is about 125 μm and that of the core is just about 50 μm. The difference in refractive index of the cladding and the core allows total internal reflection in the same way as happens at an air-water surface show in. If light is incident on a cable end with an angle of incidence greater than the critical angle then the light will remain trapped inside the glass strand. In this way, light travels very quickly down the length of the cable over a very long distance (tens of kilometers). Optical fibers are commonly used in telecommunications, because information can be transported over long distances, with minimal loss of data. Another common use can be found in medicine in endoscopes. The field of applied science and engineering concerned with the design and application of optical fibers are called fiber optics.

Total Polarization

Brewster’s angle is an angle of incidence at which light with a particular polarization is perfectly transmitted through a surface.

• Calculate the Brewster’s angle from the indices of refraction and discuss its physical mechanism

Brewster’s angle (also known as the polarization angle) is an angle of incidence at which light with a particular polarization is perfectly transmitted through a transparent dielectric surface, with no reflection. When unpolarized light is incident at this angle, the light that is reflected from the surface is therefore perfectly polarized. This special angle of incidence is named after the Scottish physicist Sir David Brewster (1781–1868).

The physical mechanism for this can be qualitatively understood from the manner in which electric dipoles in the media respond to p-polarized light (whose electric field is polarized in the same plane as the incident ray and the surface normal). One can imagine that light incident on the surface is absorbed, and then re-radiated by oscillating electric dipoles at the interface between the two media. The refracted light is emitted perpendicular to the direction of the dipole moment; no energy can be radiated in the direction of the dipole moment. Thus, if the angle of reflection θ 1 (angle of reflection) is equal to the alignment of the dipoles (90 – θ 2 ), where θ 2 is angle of refraction, no light is reflected.

Fig 1 : An illustration of the polarization of light that is incident on an interface at Brewster’s angle.

This geometric condition can be expressed as \(\theta _ { 1 } + \theta _ { 2 } = 90 ^ { \circ }\), where θ 1 is the angle of incidence and θ 2 is the angle of refraction. Using Snell’s law (n 1 sinθ1 = n 2 sinθ 2 ), one can calculate the incident angle θ 1 = B at which no light is reflected: \(\mathrm { n } _ { 1 } \sin \left( \theta _ { \mathrm { B } } \right) = \mathrm { n } _ { 2 } \sin \left( 90 ^ { \circ } - \theta _ { \mathrm { B } } \right)\) Solving for θ B gives \(\theta _ { \mathrm { B } } = \arctan \left( \frac { \mathrm { n } _ { 2 } } { \mathrm { n } _ { 1 } } \right)\).

When light hits a surface at a Brewster angle, reflected beam is linearly polarized. shows an example, where the reflected beam was nearly perfectly polarized and hence, blocked by a polarizer on the right picture. Polarized sunglasses use the same principle to reduce glare from the sun reflecting off horizontal surfaces such as water or road.

Fig 2 : Photograph taken of a window with a camera polarizer filter rotated to two different angles. In the picture at left, the polarizer is aligned with the polarization angle of the window reflection. In the picture at right, the polarizer has been rotated 90° eliminating the heavily polarized reflected sunlight.

Polarization Experience : A polarizing filter allows light of a particular plane of polarization to pass, but scatters the rest of the light. When two polarizing filters are crossed, almost no light gets through. Some materials have molecules that rotate the plane of polarization of light. When one of these materials is placed between crossed polarizing filters, more light is allowed to pass through.

Dispersion: Rainbows and Prisims

Dispersion is defined as the spreading of white light into its full spectrum of wavelengths.

• Describe production of rainbows by a combination of refraction and reflection processes

We see about six colors in a rainbow—red, orange, yellow, green, blue, and violet; sometimes indigo is listed, too. These colors are associated with different wavelengths of light. White light, in particular, is a fairly uniform mixture of all visible wavelengths. Sunlight, considered to be white, actually appears to be a bit yellow because of its mixture of wavelengths, but it does contain all visible wavelengths. The sequence of colors in rainbows is the same sequence as the colors plotted versus wavelength. What this implies is that white light is spread out according to wavelength in a rainbow. Dispersion is defined as the spreading of white light into its full spectrum of wavelengths. More technically, dispersion occurs whenever there is a process that changes the direction of light in a manner that depends on wavelength. Dispersion, as a general phenomenon, can occur for any type of wave and always involves wavelength-dependent processes.

Colors of a Rainbow : Even though rainbows are associated with seven colors, the rainbow is a continuous distribution of colors according to wavelengths.

Refraction is responsible for dispersion in rainbows and many other situations. The angle of refraction depends on the index of refraction, as we saw in the Law of Refraction. We know that the index of refraction n depends on the medium. But for a given medium, n also depends on wavelength. Note that, for a given medium, n increases as wavelength decreases and is greatest for violet light. Thus violet light is bent more than red light and the light is dispersed into the same sequence of wavelengths.

Pure Light and Light Dispersion : (a) A pure wavelength of light falls onto a prism and is refracted at both surfaces. (b) White light is dispersed by the prism (shown exaggerated). Since the index of refraction varies with wavelength, the angles of refraction vary with wavelength. A sequence of red to violet is produced, because the index of refraction increases steadily with decreasing wavelength.

Rainbows are produced by a combination of refraction and reflection. You may have noticed that you see a rainbow only when you look away from the sun. Light enters a drop of water and is reflected from the back of the drop. The light is refracted both as it enters and as it leaves the drop. Since the index of refraction of water varies with wavelength, the light is dispersed, and a rainbow is observed. (There is no dispersion caused by reflection at the back surface, since the law of reflection does not depend on wavelength. ) The actual rainbow of colors seen by an observer depends on the myriad of rays being refracted and reflected toward the observer’s eyes from numerous drops of water. The arc of a rainbow comes from the need to be looking at a specific angle relative to the direction of the sun.

Light Reflecting on Water Droplet : Part of the light falling on this water drop enters and is reflected from the back of the drop. This light is refracted and dispersed both as it enters and as it leaves the drop.

• Light strikes different parts of a rough surface at different angles and is reflected, or diffused, in many different directions.
• A mirror has a smooth surface (compared with the wavelength of light) and so reflects light at specific angles.
• We see the light reflected off a mirror coming from a direction determined by the law of reflection.
• The changing of a light ray’s direction (loosely called bending) when it passes through variations in matter is called refraction.
• The index of refraction is n=c/v, where v is the speed of light in the material, c is the speed of light in vacuum, and n is the index of refraction.
• Snell’s law, the law of refraction, is stated in equation form as n1sinθ1=n2sinθ2n1sin⁡θ1=n2sin⁡θ2.
• The critical angle is the angle of incidence above which total internal reflection occurs and given as θc=arcsin(n2n1)θc=arcsin⁡(n2n1).
• The critical angle is only defined when n2/n1 is less than 1.
• If light is incident on an optical fiber with an angle of incidence greater than the critical angle then the light will remain trapped inside the glass strand. Light can travel over a very long distance without a significant loss.
• When light hits a surface at a Brewster angle, reflected beam is linearly polarized.
• The physical mechanism for the Brewster’s angle can be qualitatively understood from the manner in which electric dipoles in the media respond to p-polarized light.
• Brewsters’ angle is given as θB=arctan(n2n1)θB=arctan⁡(n2n1).
• Dispersion occurs whenever there is a process that changes the direction of light in a manner that depends on wavelength. Dispersion can occur for any type of wave and always involves wavelength-dependent processes.
• For a given medium, n increases as wavelength decreases and is greatest for violet light. Thus violet light is bent more than red light, as can be seen with a prism.
• In a rainbow, light enters a drop of water and is reflected from the back of the drop. The light is refracted both as it enters and as it leaves the drop.
• reflection : the property of a propagated wave being thrown back from a surface (such as a mirror)
• refraction : Changing of a light ray’s direction when it passes through variations in matter.
• index of refraction : For a material, the ratio of the speed of light in vacuum to that in the material.
• Snell’s law : A formula used to describe the relationship between the angles of incidence and refraction.
• cladding : One or more layers of materials of lower refractive index, in intimate contact with a core material of higher refractive index.
• dipole : A separation of positive and negative charges.
• dielectric : An electrically insulating or nonconducting material considered for its electric susceptibility (i.e., its property of polarization when exposed to an external electric field).
• polarizer : An optical filter that passes light of a specific polarization and blocks waves of other polarizations.

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• Waves travel in all directions in the open ocean, but they always approach the land nearly perpendicular to the shore. Why does this happen?
• the angle of incidence
• the angle of reflection
• the angle of refraction
• the angle between the reflected and refracted rays
• the angle between the incident and refracted rays
• the angle between the incident and reflected rays
• Using the protractor in the diagram, measure the angle of incidence.
• Using the law of reflection, determine the angle of reflection, then select the lettered ray that best represents the reflected ray.
• Using Snell's law, determine the angle of refraction, then select the lettered ray that best represents the transmitted ray.
• refraction-drill.pdf The diagrams on the accompanying pdf show a ray of light incident upon an interface. In some cases the ray is traveling through air and entering glass. In other cases it is traveling through glass and entering air. On each diagram, sketch the approximate path of the light in the second medium. Ignore any reflected light. It is not necessary to calculate any angles, but do clearly show the change in direction of the rays, if any, at each surface.
• Ice is transparent and so is water, but ice is visible in water. How is this possible?
• Acrylic is transparent and so is mineral oil, but acrylic in mineral oil is nearly invisible. Why does this happen?
• Find another pair of materials that have the same optical relationship as acrylic and mineral oil.
• If you can see the face of a friend who is under water can your friend also see you? Justify your answer.
• Is daytime a bit longer or a bit shorter because of atmospheric refraction? Justify your answer.
• What optical phenomenon explains this natural phenomenon?
• What atmospheric conditions are needed for a Fata Morgana to occur?
• Explain why the light takes the path that it does in this kind of mirage.
• v  =  ⅝c
• v  =  ⅔c
• v  =  ¾c
• The index of the material is usually chosen so that its index is the square root of the index of the substrate (the material it is deposited on). What would be an appropriate index of refraction for an antireflection coating placed on crown glass (a kind of glass that is still sometimes used for eyeglasses)?
• The index of refraction is usually stated for a color of light in the middle of the visible spectrum. What wavelength of light was used to determine the index of refraction you used in part a? What color is this light? Why do you suppose this type of light was chosen?
• What is the wavelength of this light in the antireflection coating?
• Antireflection coatings of the type described in this problem are usually one-quarter of a wavelength thick. How thick should this coating be?
• About how big is an atom (an order of magnitude value will do)? How many atoms thick is the antireflection coating described in this problem (again, an order of magnitude value will do)?

Determine the angular separation between the red and violet components of a beam of white light traveling from diamond to air if the angle of incidence is…

The two basic elements of optical fiber are its core and cladding. The core, or the axial part of the optical fiber, is the light transmission area of the fiber. The cladding is the layer completely surrounding the core. The difference in refractive index between the core and cladding is less than 0.5%. The refractive index of the core is higher than that of the cladding, so that light in the core strikes the interface with the cladding at a bouncing angle and is trapped in the core by total internal reflection Corning Inc., 2008

• the index of refraction of the cladding
• the critical angle at the core-cladding interface
• the critical angle at the cladding-air interface
• Determine the angle of incidence.
• Determine the angle of reflection.
• Determine the angle of refraction.
• Determine the index of refraction of the unspecified liquid.
• What is the unspecified liquid?

Choose one of the rays at random. On the diagram, trace the transmitted portion of your ray until it emerges from the crystal. Determine the deviation angle between the incident ray and the emergent ray.

Choose one of the rays at random. On the diagram, trace the portion of your ray that is transmitted at the interface, reflects once off the back of the drop, and then emerges. Determine the angle between the incident ray and the emergent ray.

Choose one of the rays at random. On the diagram, trace the portion of your ray that is transmitted at the interface, reflects twice off the back of the drop, and then emerges. Determine the angle between the incident ray and the emergent ray.

• What is the critical angle for crown glass?
• Which of the rays are refracted at the hemispherical end cap and which experience total internal reflection? (Hint: What is the easiest way to draw a line normal to the circumference of a circle? Second hint: Go find a protractor.)
• Draw the refracted rays until they reach the axis. Do they focus?

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Total Internal Reflection Problems and Solutions

On this page, all aspects of the total internal reflection phenomenon are explored in the form of problem-solving.  

For reference, total internal reflection occurs when the incident light is in a medium with a higher index of refraction.  

In this situation, if the angle of incidence is greater than the critical angle, then the refracted ray totally internally reflected back into the original medium. 

Total Internal Reflection Problems

Problem (1): An unknown glass has an index of refraction of $n=1.5$. For a beam of light originating in the glass, at what angles does the light $100\%$ reflected back into the glass? (The index of refraction of air is $n_{air}=1.00$). 

Solution : When light moves through a medium with an index of refraction $n_1$ and strikes a boundary of a region with an index of refraction $n_2$ such that $n_1>n_2$, then total internal reflection can occur if the incident angle equals or is greater than a critical angle $\theta_c$ given by the following formula \[\sin\theta_c=\frac{n_2}{n_1}\] If the incident ray equals the critical angle, then the refracted ray exit along the boundary at the angle of $90^\circ$. 

For incident angles greater than the critical angle, $\theta_1 >\theta_c$, the refracted ray vanishes at the boundary, and $100\%$ is reflected back into the original medium. This process is called total internal reflection. 

In this problem, we are told that light is initially in a medium with a higher index of refraction and wants to enter a medium with a lower index of refraction. So, the necessary condition for occurring total internal reflection is satisfied. 

Thus, we must first find the critical angle whose magnitude is obtained as below \[\sin\theta_c=\frac{n_{air}}{n_{glass}}=\frac{1}{1.5}\] Taking the inverse of the sine of both sides, gives \[\theta_c=\sin^{-1}\left(\frac{1}{1.5}\right)=41.8^\circ\] Hence, if the angle of incidence in the glass exceeds about $42^\circ$, then all incident rays completely return back into the same medium. 

Problem (2): A ray of light traveling through glass with an index of refraction $n=1.5$ strikes the interface of the glass-water. Let the index of refraction of water be $n=1.33$. At what angle, the ray does not enter the water? 

Solution : We want no ray to exist in the water. We know that when a ray of light strikes a boundary of two different media, part of it is reflected back into the same medium and part of the ray enters the second medium (or refracted). 

In this question, we are asked to find an angle at which the refracted ray is removed. This occurs when the refracted angle in the second medium (here, water) becomes greater than or equal to $90^\circ$. Applying Snell's law of refraction , we have \begin{align*} n_1 \sin\theta_1&=n_2 \sin\theta_2\\\\ (1.5)\sin\theta_1&=(1.33)\sin 90^\circ\\\\ \Rightarrow \quad \sin\theta_2&=\frac{1.33}{1.5}\sin 90^\circ\\\\&=0.887\end{align*} Next, take the inverse of the sine to find the angle whose sine is $1.128$ \[\theta_2=\sin^{-1}(0.887)=62.5^\circ\] This incident angle at which the refracted angle becomes $90^\circ$ is called the critical angle. Any ray, in the glass, striking the boundary with an angle that exceeds 40 degrees, totally returns back into the same original medium. This phenomenon is also called total internal reflection.

Problem (3): Find the critical angle for a ray of light traveling from Ethyl alcohol to Benzene. Ethyl alcohol has an index of refraction $n=1.361$ and Benzene has $n=1.501$. 

Solution : There occurs no totally internal reflection because there is no critical angle for this configuration. In this scenario, light is traveling from a rare medium (low $n$) to a dense medium (with high $n$) and this is contrary to the criteria for the formation of a total internal reflection. 

Thus, total internal reflection occurs when light traveling in a medium strikes the boundary of a medium whose index of refraction is higher than the original medium. 

Problem (4): The speed of sound in a room filled with air is $343\,{\rm m/s}$. The walls of this room are made of concrete, in which the speed of sound is $1850\,{\rm m/s}$.  (a) Find the critical angle for total internal reflection of sound waves striking the air-concrete interface.  (b) If we want the sound to undergo total internal reflection, in which medium must it be traveling?

Solution : To find the critical angle for a total internal reflection problem, we must use the following equation \[\theta_c=\sin^{-1}\left(\frac{n_{rare}}{n_{dense}}\right)\] But in this question, neither of indices of refraction are given. 

Recall that Snell's law of refraction is also written in terms of velocities instead of indices of refraction. Thus, the alternate form of Snell's law is given by the following equation \[\frac{\sin\theta_1}{\sin\theta_2}=\frac{v_1}{v_2}\] where the subscripts $1$ and $2$ refer to the incident and refracted rays. 

(a) We know that the critical angle is the angle at which the refracted ray becomes $90^\circ$. So, setting $\theta_2=90^\circ$ in the above equation, and solving for $\theta_1=\theta_{cri}$, we get \begin{align*} \sin\theta_{cri}&=\sin\theta_2 \left(\frac{v_1}{v_2}\right)\\\\&=\sin 90^\circ \left(\frac{343}{1850}\right)\\\\&=0.186 \end{align*} Now, take the inverse of sine from both sides to find the angle whose sine is $0.186$ \[\theta_{cri}=\sin^{-1}(0.186)=10.7^\circ\] Hence, if the sound waves strike the wall at angles greater than about $11^\circ$, then all is reflected back into the wall without any waves passing through the wall. 

(b) As you know, total internal reflection occurs when mechanical waves (sound) or electromagnetic waves (light) are incidents on a boundary of a medium having a lower index of refraction than the original medium. 

We can modify the above statement to accommodate the velocities in both mediums where the indices of refraction are not known. In this case, the total internal reflection occurs when the waves strike the boundary of a medium, in which the speed of the waves is greater than the original medium. 

In this problem, the total internal reflection can occur when sound waves travel from the air and strike the wall at an angle greater than the critical angle found above, i.e. $\theta>10.7^\circ$. 

Problem (5): The boundary of the two mediums is shown in the figure below. The angle of incidence with the horizontal is $30^\circ$. The incident ray refracted parallel to the interface. Find the index of refraction of the second medium. 

Solution : We are told in the problem that the refracted ray is parallel to the boundary of two mediums. We know that this situation always occurs when the incident ray is in a dense medium and strikes the boundary of a less dense medium. 

So, the second medium must have a lower index of refraction than the original medium. Note that the incident angle is measured with a line perpendicular to the boundary. Here, the incident angle is $\theta_1=90^\circ-30^\circ=60^\circ$. 

Setting the refracted angle as $\theta_2=90^\circ$ in Snell's law of refraction, and solving for the index of refraction $n_2$, we will have \begin{align*} n_1 \sin\theta_1&=n_2\sin\theta_2\\\\\sqrt{3}\sin 60^\circ&=n_2 \sin 90^\circ\\\\ \sqrt{3}\left(\frac{\sqrt{3}}2\right)&=n_2\\\\\Rightarrow n_2 &=\frac 32 \end{align*} As expected, the second medium must be less dense than the first medium. 

Problem (6): A beam of light moves from air into an unknown glass. The angle of incidence is $30^\circ$, and the angle of refraction is $22^\circ$. For a beam of light originating from the glass, find the critical angle. 

Solution : Critical angle is defined when a ray travels in a dense medium and strikes a boundary with a lower index of refraction. Its magnitude is calculated as \[\theta_c=\sin^{-1}\left(\frac{n_2}{n_1}\right)\] where always $n_1>n_2$. 

When a beam of light strikes the interface of the two media at the critical angle, the refracted ray exits parallel to the boundary. In this case, the refracted angle is $90^\circ$. 

Here, we want to find the critical angle when the light is in the glass and enters the air. The index of refraction of the glass is unknown. 

From the first part of the question, we can find the glass's index of refraction by applying Snell's law of refraction. \begin{align*} n_1 \sin\theta_1&=n_2 \sin\theta_2\\\\ (1)\sin 30^\circ&=n\sin 22^\circ\\\\ \Rightarrow \quad n&=\frac{\sin 30^\circ}{\sin 22^\circ}(1)\\\\&=\frac{0.5}{0.375}\\\\&=1.33 \end{align*} By having the index of refraction of the glass, the critical angle is found for this total internal reflection problem from the following equation \[\theta_c=\sin^{-1}\left(\frac{n_2}{n_1}\right)=\sin^{-1}\left(\frac{1.00}{1.33}\right)=48.8^\circ\] Thus, if light originated from the water incident on the boundary of air at an angle greater than about $49^\circ$, then light is totally returned back into the water and does not enter the air.

Problem (7): The critical angle of a specific glass in the air is $\theta_c=37.8^\circ$. Now suppose that we put it into a tank of water. What will change its critical angle?

Solution : In the first case, the boundary is composed of glass-air. Using critical angle formula, we have \[\sin\theta_c=\frac{n_{air}}{n_{glass}}\] In the second case, the glass surrounded by a water. So, the interface is glass-water. Again, its critical angle is found as \[\sin\theta'_c=\frac{n_{water}}{n_{glass}}\] The index of refraction of the glass is unknown. Dividing the second relation by the first one,  gives \[\frac{\sin\theta'_c}{\sin\theta_c}=\frac{n_{water}}{n_{air}}\] Setting $n_{water}=1.33$ and solving for the unknown angle $\theta'_c$, we get \begin{align*} \sin\theta'_c&=\sin\theta_c \left(\frac{n_{water}}{n_{air}}\right)\\\\&=\sin(37.8^\circ)\frac{1.33}{1.00}\\\\&=0.816\end{align*} Take the inverse sine of both sides to find the angle whose sine is $0.816$ \[\theta'_c=\sin^{-1}(0.816)=54.7^\circ \]

Problem (8): A light bulb is placed 5 meters below a swimming pool. What is the diameter of the circle of light formed on the surface seen directly from above? 

Solution : We can assume the light bulb to be a point source. Rays emit in all directions from that point source and strike the water-air interface. Some rays are incident on the boundary at less than the critical angle, refracted, and enter the air. 

The rays have an incident angle greater than the critical angle, do not enter the air, and $100\%$ reflect back into the water. 

These completely reflected rays are responsible for the formation of a circle of light on the surface of the water. 

The diameter of the circle of light is the distance between the two points at which the rays reach the surface at the critical angle. First, using critical angle formula, we find it as below \[\theta_c=\sin^{-1}\left(\frac{1.00}{1.33}\right)=48.7^\circ\] As you can see from the geometry, and using the definition of sine function, the radius of the circle of light is found as \[\sin\theta_c=\frac{r}{H} \Rightarrow r=H\sin\theta_c\] where $H$ is the pool's depth. So, the radius or diameter of the circle $D=2r$ is computed as below \[D=2H\sin\theta_c=2(5)(\sin 48.7^\circ)=7.5\,{\rm m}\] 

Problem (9): A beam of light is incident on the surface $AB$ of a prism surrounded by air at the critical angle, as shown in the figure. Find the angle of incidence, $\theta_1$. 

Solution : we are told that the ray is incident on the surface $BC$ at the critical angle. Thus, the refracted ray must be parallel to $BC$. 

To find the critical angle in a total internal reflection problem, using Snell's law, we arrive at the following formula \[\sin\theta_c=\frac{n_{rare}}{n_{dense}}\] Here, the prism is surrounded by air, so $n_{rare}=1.00$. The index of refraction of the prism, $n_p$, is not given. 

Thus, the critical angle at the boundary of $BC$ is found as below \[\sin 42^\circ=\frac{1}{n_p} \quad (I)\] Now, we must apply Snell's law at the entering point into the prism for the incident ray, but the angle of refraction at that point is not known. 

From geometry of the figure, we have $\beta=90^\circ-42^\circ=48^\circ$. On the other, recall that the sum of angles of a triangle equals $180^\circ$.

Thus, the angle $\alpha$ is determined as below \begin{gather*} \alpha+\beta+60^\circ=180^\circ \\ \alpha+48^\circ+60^\circ=180^\circ\\\Rightarrow \quad \alpha=72^\circ\end{gather*} Now that we have angle $\alpha$, the refracted ray at the entering surface $AB$ is simply found as below \[\theta_2+\alpha=90^\circ \Rightarrow \theta_2=18^\circ\] Applying Snell's law at this surface $AB$, we get \begin{align*} n_1 \sin\theta_1&=n_2 \sin\theta_2 \\\\ (1)\sin\theta_1 &=n_p \sin 18^\circ \quad (II) \end{align*} From equation $(I)$, the index of refraction of prism is $n_p=\frac{1}{\sin 42^\circ}$. Substituting this into equation $(II)$, we can get the incident angle as below \begin{align*} \sin\theta_1 &=\frac{\sin 18^\circ}{\sin 42^\circ}\\\\&=0.462 \end{align*} By taking the inverse of sine from both sides, we can find the angle whose sine is $0.461$ \[\theta_1=\sin^{-1}(0.461)=27.5^\circ\] 

In this article, some problems about total internal reflection were presented along with detailed solutions. 

The necessary conditions for total internal reflection are as below: 

(1) Light must travel from a denser medium to a rare medium. 

(2) The angle of incidence in the denser medium must be greater than the critical angle at the boundary of the two media. 

Author : Ali Nemati Date Published : 9/3/2021  

© 2015 All rights reserved. by Physexams.com

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Feynman's Tips on Physics: Reflections, Advice, Insights, Practice - A Problem-Solving Supplement to the Feynman Lectures on Physics Paperback – Illustrated, January 29, 2013

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• Publisher ‏ : ‎ Basic Books; Revised edition (January 29, 2013)
• Language ‏ : ‎ English
• Paperback ‏ : ‎ 208 pages
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Richard p. feynman.

Richard P. Feynman was born in 1918 and grew up in Far Rockaway, New York. At the age of seventeen he entered MIT and in 1939 went to Princeton, then to Los Alamos, where he joined in the effort to build the atomic bomb. Following World War II he joined the physics faculty at Cornell, then went on to Caltech in 1951, where he taught until his death in 1988. He shared the Nobel Prize for physics in 1965, and served with distinction on the Shuttle Commission in 1986. A commemorative stamp in his name was issued by the U.S. Postal Service in 2005.

Michael A. Gottlieb

Michael Gottlieb was a self-employed computer programmer in Silicon Valley for 26 years, before retiring in 2003 to edit and supplement ""The Feynman Lectures on Physics." In 2006, Gottlieb and Ralph Leighton spearheaded the publication of The Feynman Lectures on Physics Definitive Edition, which included the first edition of "Feynman's Tips on Physics, a problem-solving supplement to the Feynman Lectures on Physics." Since 2007 Gottlieb has been a Visitor in Physics at The California Institute of Technology, for whom he manages projects related to The Feynman Lectures, producing print manuscripts, ebook editions, the recently published iBooks textbook "Six Easy Pieces," and the second edition of "Feynman's Tips on Physics." The Basic Books New Millennium Edition and future editions of The Feynman Lectures on Physics, and the forthcoming "Exercises for the Feynman Lectures on Physics," are produced from LaTeX manuscripts Gottlieb and Rudolf Pfeiffer created, maintain and own. Gottlieb lives with his three dogs near a tropical beach in Guanacaste, Costa Rica.

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Reflection Rule Calculator

In the realm of geometry, reflection is a fundamental concept that involves the transformation of shapes or points across various axes. The Reflection Rule Calculator serves as a powerful tool for effortlessly calculating the reflection of points across different axes. Let’s delve into the intricacies of this calculator and discover its significance in geometric analysis.

Introduction to the Reflection Rule Calculator

The Reflection Rule Calculator is designed to determine the coordinates of a point after it has been reflected across a specified axis. Whether it’s the x-axis, y-axis, or a line such as y = x, this calculator simplifies the process of calculating reflections, making it accessible to students, professionals, and enthusiasts alike. Understanding how points behave under reflection is crucial in geometry and has applications in various fields, including computer graphics, architecture, and physics.

Importance of the Reflection Rule Calculator

The Reflection Rule Calculator holds immense importance for several reasons:

• Geometric Analysis: It facilitates the study of symmetry and transformation properties of geometric shapes.
• Visualization: Allows for visualizing how points or shapes change when reflected across different axes.
• Problem Solving: Assists in solving geometry problems involving reflections, such as finding the image of a point after reflection.
• Educational Aid: Helps students grasp abstract geometric concepts through interactive exploration and calculation.

How to Use the Reflection Rule Calculator

Using the Reflection Rule Calculator is straightforward:

• Enter Coordinates: Input the original x-coordinate and y-coordinate of the point.
• Select Reflection Axis: Choose the axis across which the reflection will occur (x-axis, y-axis, or line y = x).
• Calculate: Click the “Calculate” button to obtain the coordinates of the reflected point.

The calculator applies specific rules for each reflection axis, providing instant results for efficient geometric analysis.

10 FAQs About the Reflection Rule Calculator

1. What is reflection in geometry? Reflection is a transformation that flips a shape or point over a line, creating a mirror image.

2. What are the different reflection axes supported by the calculator? The calculator supports reflection across the x-axis, y-axis, and line y = x.

3. How does reflection help in understanding symmetry? Reflection reveals symmetrical properties of shapes and patterns, aiding in symmetry analysis.

4. Can the calculator handle decimal coordinates? Yes, the calculator accepts decimal values for precise calculations.

5. Is reflection reversible? Yes, performing the same reflection operation twice restores the original shape or point.

6. What happens when a point lies on the reflection axis? A point on the reflection axis remains unchanged after reflection.

7. Can the calculator be used for shapes other than points? No, the calculator is specifically designed for reflecting individual points.

8. How does the calculator handle negative coordinates? Negative coordinates are properly handled based on the chosen reflection axis.

9. Are there real-world applications of reflection in geometry? Yes, reflection is used in architectural design, art, photography, and many other fields.

10. Can the calculator be used for educational purposes? Absolutely, it is an excellent educational tool for teaching reflection concepts in geometry.

The Reflection Rule Calculator simplifies the process of calculating reflections in geometry, offering a practical solution for analyzing symmetry and transformation properties of shapes and points. Whether you’re a student, teacher, or professional, this calculator serves as a valuable resource for understanding and visualizing geometric transformations. Embrace the power of reflection with the Reflection Rule Calculator and unlock new insights into the fascinating world of geometry.

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• Applied Overhead Calculator
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• Stirling Number Calculator
• Segment Ratio Calculator

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Scientists use generative AI to answer complex questions in physics

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When water freezes, it transitions from a liquid phase to a solid phase, resulting in a drastic change in properties like density and volume. Phase transitions in water are so common most of us probably don’t even think about them, but phase transitions in novel materials or complex physical systems are an important area of study.

To fully understand these systems, scientists must be able to recognize phases and detect the transitions between. But how to quantify phase changes in an unknown system is often unclear, especially when data are scarce.

Researchers from MIT and the University of Basel in Switzerland applied generative artificial intelligence models to this problem, developing a new machine-learning framework that can automatically map out phase diagrams for novel physical systems.

Their physics-informed machine-learning approach is more efficient than laborious, manual techniques which rely on theoretical expertise. Importantly, because their approach leverages generative models, it does not require huge, labeled training datasets used in other machine-learning techniques.

Such a framework could help scientists investigate the thermodynamic properties of novel materials or detect entanglement in quantum systems, for instance. Ultimately, this technique could make it possible for scientists to discover unknown phases of matter autonomously.

“If you have a new system with fully unknown properties, how would you choose which observable quantity to study? The hope, at least with data-driven tools, is that you could scan large new systems in an automated way, and it will point you to important changes in the system. This might be a tool in the pipeline of automated scientific discovery of new, exotic properties of phases,” says Frank Schäfer, a postdoc in the Julia Lab in the Computer Science and Artificial Intelligence Laboratory (CSAIL) and co-author of a paper on this approach.

Joining Schäfer on the paper are first author Julian Arnold, a graduate student at the University of Basel; Alan Edelman, applied mathematics professor in the Department of Mathematics and leader of the Julia Lab; and senior author Christoph Bruder, professor in the Department of Physics at the University of Basel. The research is published today in Physical Review Letters.

Detecting phase transitions using AI

While water transitioning to ice might be among the most obvious examples of a phase change, more exotic phase changes, like when a material transitions from being a normal conductor to a superconductor, are of keen interest to scientists.

These transitions can be detected by identifying an “order parameter,” a quantity that is important and expected to change. For instance, water freezes and transitions to a solid phase (ice) when its temperature drops below 0 degrees Celsius. In this case, an appropriate order parameter could be defined in terms of the proportion of water molecules that are part of the crystalline lattice versus those that remain in a disordered state.

In the past, researchers have relied on physics expertise to build phase diagrams manually, drawing on theoretical understanding to know which order parameters are important. Not only is this tedious for complex systems, and perhaps impossible for unknown systems with new behaviors, but it also introduces human bias into the solution.

More recently, researchers have begun using machine learning to build discriminative classifiers that can solve this task by learning to classify a measurement statistic as coming from a particular phase of the physical system, the same way such models classify an image as a cat or dog.

The MIT researchers demonstrated how generative models can be used to solve this classification task much more efficiently, and in a physics-informed manner.

The Julia Programming Language , a popular language for scientific computing that is also used in MIT’s introductory linear algebra classes, offers many tools that make it invaluable for constructing such generative models, Schäfer adds.

Generative models, like those that underlie ChatGPT and Dall-E, typically work by estimating the probability distribution of some data, which they use to generate new data points that fit the distribution (such as new cat images that are similar to existing cat images).

However, when simulations of a physical system using tried-and-true scientific techniques are available, researchers get a model of its probability distribution for free. This distribution describes the measurement statistics of the physical system.

A more knowledgeable model

The MIT team’s insight is that this probability distribution also defines a generative model upon which a classifier can be constructed. They plug the generative model into standard statistical formulas to directly construct a classifier instead of learning it from samples, as was done with discriminative approaches.

“This is a really nice way of incorporating something you know about your physical system deep inside your machine-learning scheme. It goes far beyond just performing feature engineering on your data samples or simple inductive biases,” Schäfer says.

This generative classifier can determine what phase the system is in given some parameter, like temperature or pressure. And because the researchers directly approximate the probability distributions underlying measurements from the physical system, the classifier has system knowledge.

This enables their method to perform better than other machine-learning techniques. And because it can work automatically without the need for extensive training, their approach significantly enhances the computational efficiency of identifying phase transitions.

At the end of the day, similar to how one might ask ChatGPT to solve a math problem, the researchers can ask the generative classifier questions like “does this sample belong to phase I or phase II?” or “was this sample generated at high temperature or low temperature?”

Scientists could also use this approach to solve different binary classification tasks in physical systems, possibly to detect entanglement in quantum systems (Is the state entangled or not?) or determine whether theory A or B is best suited to solve a particular problem. They could also use this approach to better understand and improve large language models like ChatGPT by identifying how certain parameters should be tuned so the chatbot gives the best outputs.

In the future, the researchers also want to study theoretical guarantees regarding how many measurements they would need to effectively detect phase transitions and estimate the amount of computation that would require.

This work was funded, in part, by the Swiss National Science Foundation, the MIT-Switzerland Lockheed Martin Seed Fund, and MIT International Science and Technology Initiatives.

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Researchers at MIT and elsewhere have developed a new machine-learning model capable of “predicting a physical system’s phase or state,” report Kyle Wiggers and Devin Coldewey for TechCrunch . 

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Unlocking the Quantum Code: International Team Cracks a Long-Standing Physics Problem

By University of Bonn May 21, 2024

Researchers have developed a new method called wavefunction matching to tackle the sign problem in Monte Carlo simulations, a common issue in quantum many-body physics. By simplifying the interaction model and using perturbation theory for corrections, this method has proven effective in accurately calculating nuclear properties like mass and radius. It holds promise for broader applications in quantum computing and other fields. Credit: Prof. Serdar Elhatisari

Wavefunction matching for solving quantum many-body problems.

Strongly interacting systems are crucial in the fields of quantum physics and quantum chemistry. Monte Carlo simulations, a type of stochastic method, are widely used to study these systems. However, they face challenges when dealing with sign oscillations. An international team of researchers from Germany, Turkey, the USA, China, South Korea, and France has addressed this issue by developing a new technique called wavefunction matching.

As an example, the masses and radii of all nuclei up to mass number 50 were calculated using this method. The results agree with the measurements, the researchers now report in the journal Nature .

All matter on Earth consists of tiny particles known as atoms. Each atom contains even smaller particles: protons, neutrons, and electrons. Each of these particles follows the rules of quantum mechanics . Quantum mechanics forms the basis of quantum many-body theory, which describes systems with many particles, such as atomic nuclei.

One class of methods used by nuclear physicists to study atomic nuclei is the ab initio approach. It describes complex systems by starting from a description of their elementary components and their interactions. In the case of nuclear physics, the elementary components are protons and neutrons. Some key questions that ab initio calculations can help answer are the binding energies and properties of atomic nuclei and the link between nuclear structure and the underlying interactions between protons and neutrons.

Challenges and Solutions in Quantum Simulations

However, these ab initio methods have difficulties in performing reliable calculations for systems with complex interactions. One of these methods is quantum Monte Carlo simulations. Here, quantities are calculated using random or stochastic processes. Although quantum Monte Carlo simulations can be efficient and powerful, they have a significant weakness: the sign problem. It arises in processes with positive and negative weights, which cancel each other. This cancellation leads to inaccurate final predictions.

A new approach, known as wavefunction matching, is intended to help solve such calculation problems for ab initio methods. “This problem is solved by the new method of wavefunction matching by mapping the complicated problem in a first approximation to a simple model system that does not have such sign oscillations and then treating the differences in perturbation theory,” says Prof. Ulf-G. Meißner from the Helmholtz Institute for Radiation and Nuclear Physics at the University of Bonn and from the Institute of Nuclear Physics and the Center for Advanced Simulation and Analytics at Forschungszentrum Jülich.

“As an example, the masses and radii of all nuclei up to mass number 50 were calculated — and the results agree with the measurements,” reports Meißner, who is also a member of the Transdisciplinary Research Areas “Modeling” and “Matter” at the University of Bonn.

“In quantum many-body theory, we are often faced with the situation that we can perform calculations using a simple approximate interaction, but realistic high-fidelity interactions cause severe computational problems,” says Dean Lee, Professor of Physics from the Facility for Rare Istope Beams and Department of Physics and Astronomy (FRIB) at Michigan State University and head of the Department of Theoretical Nuclear Sciences.

Practical Applications and Future Prospects

Wavefunction matching solves this problem by removing the short-distance part of the high-fidelity interaction and replacing it with the short-distance part of an easily calculable interaction. This transformation is done in a way that preserves all the important properties of the original realistic interaction. Since the new wavefunctions are similar to those of the easily computable interaction, the researchers can now perform calculations with the easily computable interaction and apply a standard procedure for handling small corrections – called perturbation theory.

The research team applied this new method to lattice quantum Monte Carlo simulations for light nuclei, medium-mass nuclei, neutron matter, and nuclear matter. Using precise ab initio calculations, the results closely matched real-world data on nuclear properties such as size, structure, and binding energy. Calculations that were once impossible due to the sign problem can now be performed with wavefunction matching.

While the research team focused exclusively on quantum Monte Carlo simulations, wavefunction matching should be useful for many different ab initio approaches. “This method can be used in both classical computing and quantum computing , for example, to better predict the properties of so-called topological materials, which are important for quantum computing,” says Meißner.

Reference: “Wavefunction matching for solving quantum many-body problems” by Serdar Elhatisari, Lukas Bovermann, Yuan-Zhuo Ma, Evgeny Epelbaum, Dillon Frame, Fabian Hildenbrand, Myungkuk Kim, Youngman Kim, Hermann Krebs, Timo A. Lähde, Dean Lee, Ning Li, Bing-Nan Lu, Ulf-G. Meißner, Gautam Rupak, Shihang Shen, Young-Ho Song and Gianluca Stellin, 15 May 2024, Nature . DOI: 10.1038/s41586-024-07422-z

The first author is Prof. Dr. Serdar Elhatisari, who worked for two years as a Fellow in Prof. Meißner’s ERC Advanced Grant EXOTIC. According to Meißner, a large part of the work was carried out during this time. Part of the computing time on supercomputers at Forschungszentrum Jülich was provided by the IAS-4 institute, which Meißner heads.

The first author, Prof. Dr. Serdar Elhatisari, comes from the University of Bonn and Gaziantep Islam Science and Technology University (Turkey). Significant contributions were also made at Michigan State University. Other participants include Ruhr University Bochum, South China Normal University (China), the Institute for Basic Science in Daejeon (South Korea), Sun Yat-Sen University in Guangzhou (China), the Graduate School of China Academy of Engineering Physics in Beijing (China), Mississippi State University (USA) and Université Paris-Saclay (France). The study was funded by the U.S. Department of Energy, the U.S. National Science Foundation, the German Research Foundation, the National Natural Science Foundation of China, the Chinese Academy of Sciences President’s International Fellowship Initiative, the Volkswagen Foundation, the European Research Council, the Scientific and Technological Research Council of Turkey, the National Security Academic Fund, the Rare Isotope Science Project of the Institute for Basic Science, the National Research Foundation of Korea, the Institute for Basic Science and the Espace de Structure et de reactions Nucleaires Theorique.

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The 3-body problem is real, and it’s really unsolvable

Oh god don’t make me explain math

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Everybody seems to be talking about 3 Body Problem , the new Netflix series based on Cixin Liu’s Remembrance of Earth’s Past book trilogy . Fewer people are talking about the two series’ namesake: The unsolvable physics problem of the same name.

This makes sense, because it’s confusing . In physics, the three-body problem attempts to find a way to predict the movements of three objects whose gravity interacts with each of the others — like three stars that are close together in space. Sounds simple enough, right? Yet I myself recently pulled up the Wikipedia article on the three-body problem and closed the tab in the same manner that a person might stagger away from a bright light. Apparently the Earth, sun, and moon are a three-body system? Are you telling me we don’t know how the moon moves ? Scientists have published multiple solutions for the three-body problem? Are you telling me Cixin Liu’s books are out of date?

All I’d wanted to know was why the problem was considered unsolvable, and now memories of my one semester of high school physics were swimming before my eyes like so many glowing doom numbers. However, despite my pains, I have readied several ways that we non-physicists can be confident that the three-body problem is, in fact, unsolvable.

Reason 1: This is a special definition of ‘unsolvable’

The three-body problem is extra confusing, because scientists are seemingly constantly finding new solutions to the three-body problem! They just don’t mean a one-solution-for-all solution. Such a formula does exist for a two-body system, and apparently Isaac Newton figured it out in 1687 . But systems with more than two bodies are, according to physicists, too chaotic (i.e., not in the sense of a child’s messy bedroom, but in the sense of “chaos theory”) to be corralled by a single solution.

When physicists say they have a new solution to the three-body problem, they mean that they’ve found a specific solution for three-body systems that have certain theoretical parameters. Don’t ask me to explain those parameters, because they’re all things like “the three masses are collinear at each instant” or “a zero angular momentum solution with three equal masses moving around a figure-eight shape.” But basically: By narrowing the focus of the problem to certain arrangements of three-body systems, physicists have been able to derive formulas that predict the movements of some of them, like in our solar system. The mass of the Earth and the sun create a “ restricted three-body problem ,” where a less-big body (in this case, the moon) moves under the influence of two massive ones (the Earth and the sun).

What physicists mean when they say the three-body problem has no solution is simply that there isn’t a one-formula-fits-all solution to every way that the gravity of three objects might cause those objects to move — which is exactly what Three-Body Problem bases its whole premise on.

Reason 2: 3 Body Problem picked an unsolved three-body system on purpose

Henri Poincaré’s research into a general solution to the three-body problem formed the basis of what would become known as chaos theory (you might know it from its co-starring role in Jurassic Park ). And 3 Body Problem itself isn’t about any old three-body system. It’s specifically about an extremely chaotic three-body system, the exact kind of arrangement of bodies that Poincaré was focused on when he showed that the problem is “unsolvable.”

[ Ed. note: The rest of this section includes some spoilers for 3 Body Problem .]

In both Liu’s books and Netflix’s 3 Body Problem , humanity faces an invasion by aliens (called Trisolarans in the English translation of the books, and San-Ti in the TV series) whose home solar system features three suns in a chaotic three-body relationship. It is a world where, unlike ours, the heavens are fundamentally unpredictable. Periods of icy cold give way to searing heat that give way to swings in gravity that turn into temporary reprieves that can never be trusted. The unpredictable nature of the San-Ti environment is the source of every detail of their physicality, their philosophy, and their desire to claim Earth for their own.

In other words, 3 Body Problem ’s three-body problem is unsolvable because Liu wanted to write a story with an unsolvable three-body system, so he chose one of the three-body systems for which we have not discovered a solution, and might never.

Reason 3: Scientists are still working on the three-body problem

Perhaps the best reason I can give you to believe that the three-body problem is real, and is really unsolvable, is that some scientists published a whole set of new solutions for specific three-body systems very recently .

If physicists are still working on the three-body problem, we can safely assume that it has not been solved. Scientists, after all, are the real experts. And I am definitely not.

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1 Introduction to Critical Thinking

I. what is c ritical t hinking [1].

Critical thinking is the ability to think clearly and rationally about what to do or what to believe.  It includes the ability to engage in reflective and independent thinking. Someone with critical thinking skills is able to do the following:

• Understand the logical connections between ideas.
• Identify, construct, and evaluate arguments.
• Detect inconsistencies and common mistakes in reasoning.
• Solve problems systematically.
• Identify the relevance and importance of ideas.
• Reflect on the justification of one’s own beliefs and values.

Critical thinking is not simply a matter of accumulating information. A person with a good memory and who knows a lot of facts is not necessarily good at critical thinking. Critical thinkers are able to deduce consequences from what they know, make use of information to solve problems, and to seek relevant sources of information to inform themselves.

Critical thinking should not be confused with being argumentative or being critical of other people. Although critical thinking skills can be used in exposing fallacies and bad reasoning, critical thinking can also play an important role in cooperative reasoning and constructive tasks. Critical thinking can help us acquire knowledge, improve our theories, and strengthen arguments. We can also use critical thinking to enhance work processes and improve social institutions.

Some people believe that critical thinking hinders creativity because critical thinking requires following the rules of logic and rationality, whereas creativity might require breaking those rules. This is a misconception. Critical thinking is quite compatible with thinking “out-of-the-box,” challenging consensus views, and pursuing less popular approaches. If anything, critical thinking is an essential part of creativity because we need critical thinking to evaluate and improve our creative ideas.

II. The I mportance of C ritical T hinking

Critical thinking is a domain-general thinking skill. The ability to think clearly and rationally is important whatever we choose to do. If you work in education, research, finance, management or the legal profession, then critical thinking is obviously important. But critical thinking skills are not restricted to a particular subject area. Being able to think well and solve problems systematically is an asset for any career.

Critical thinking is very important in the new knowledge economy.  The global knowledge economy is driven by information and technology. One has to be able to deal with changes quickly and effectively. The new economy places increasing demands on flexible intellectual skills, and the ability to analyze information and integrate diverse sources of knowledge in solving problems. Good critical thinking promotes such thinking skills, and is very important in the fast-changing workplace.

Critical thinking enhances language and presentation skills. Thinking clearly and systematically can improve the way we express our ideas. In learning how to analyze the logical structure of texts, critical thinking also improves comprehension abilities.

Critical thinking promotes creativity. To come up with a creative solution to a problem involves not just having new ideas. It must also be the case that the new ideas being generated are useful and relevant to the task at hand. Critical thinking plays a crucial role in evaluating new ideas, selecting the best ones and modifying them if necessary.

Critical thinking is crucial for self-reflection. In order to live a meaningful life and to structure our lives accordingly, we need to justify and reflect on our values and decisions. Critical thinking provides the tools for this process of self-evaluation.

Good critical thinking is the foundation of science and democracy. Science requires the critical use of reason in experimentation and theory confirmation. The proper functioning of a liberal democracy requires citizens who can think critically about social issues to inform their judgments about proper governance and to overcome biases and prejudice.

Critical thinking is a   metacognitive skill . What this means is that it is a higher-level cognitive skill that involves thinking about thinking. We have to be aware of the good principles of reasoning, and be reflective about our own reasoning. In addition, we often need to make a conscious effort to improve ourselves, avoid biases, and maintain objectivity. This is notoriously hard to do. We are all able to think but to think well often requires a long period of training. The mastery of critical thinking is similar to the mastery of many other skills. There are three important components: theory, practice, and attitude.

III. Improv ing O ur T hinking S kills

If we want to think correctly, we need to follow the correct rules of reasoning. Knowledge of theory includes knowledge of these rules. These are the basic principles of critical thinking, such as the laws of logic, and the methods of scientific reasoning, etc.

Also, it would be useful to know something about what not to do if we want to reason correctly. This means we should have some basic knowledge of the mistakes that people make. First, this requires some knowledge of typical fallacies. Second, psychologists have discovered persistent biases and limitations in human reasoning. An awareness of these empirical findings will alert us to potential problems.

However, merely knowing the principles that distinguish good and bad reasoning is not enough. We might study in the classroom about how to swim, and learn about the basic theory, such as the fact that one should not breathe underwater. But unless we can apply such theoretical knowledge through constant practice, we might not actually be able to swim.

Similarly, to be good at critical thinking skills it is necessary to internalize the theoretical principles so that we can actually apply them in daily life. There are at least two ways to do this. One is to perform lots of quality exercises. These exercises don’t just include practicing in the classroom or receiving tutorials; they also include engaging in discussions and debates with other people in our daily lives, where the principles of critical thinking can be applied. The second method is to think more deeply about the principles that we have acquired. In the human mind, memory and understanding are acquired through making connections between ideas.

Good critical thinking skills require more than just knowledge and practice. Persistent practice can bring about improvements only if one has the right kind of motivation and attitude. The following attitudes are not uncommon, but they are obstacles to critical thinking:

• I prefer being given the correct answers rather than figuring them out myself.
• I don’t like to think a lot about my decisions as I rely only on gut feelings.
• I don’t usually review the mistakes I have made.
• I don’t like to be criticized.

To improve our thinking we have to recognize the importance of reflecting on the reasons for belief and action. We should also be willing to engage in debate, break old habits, and deal with linguistic complexities and abstract concepts.

The  California Critical Thinking Disposition Inventory  is a psychological test that is used to measure whether people are disposed to think critically. It measures the seven different thinking habits listed below, and it is useful to ask ourselves to what extent they describe the way we think:

• Truth-Seeking—Do you try to understand how things really are? Are you interested in finding out the truth?
• Open-Mindedness—How receptive are you to new ideas, even when you do not intuitively agree with them? Do you give new concepts a fair hearing?
• Analyticity—Do you try to understand the reasons behind things? Do you act impulsively or do you evaluate the pros and cons of your decisions?
• Systematicity—Are you systematic in your thinking? Do you break down a complex problem into parts?
• Confidence in Reasoning—Do you always defer to other people? How confident are you in your own judgment? Do you have reasons for your confidence? Do you have a way to evaluate your own thinking?
• Inquisitiveness—Are you curious about unfamiliar topics and resolving complicated problems? Will you chase down an answer until you find it?
• Maturity of Judgment—Do you jump to conclusions? Do you try to see things from different perspectives? Do you take other people’s experiences into account?

Finally, as mentioned earlier, psychologists have discovered over the years that human reasoning can be easily affected by a variety of cognitive biases. For example, people tend to be over-confident of their abilities and focus too much on evidence that supports their pre-existing opinions. We should be alert to these biases in our attitudes towards our own thinking.

IV. Defining Critical Thinking

There are many different definitions of critical thinking. Here we list some of the well-known ones. You might notice that they all emphasize the importance of clarity and rationality. Here we will look at some well-known definitions in chronological order.

1) Many people trace the importance of critical thinking in education to the early twentieth-century American philosopher John Dewey. But Dewey did not make very extensive use of the term “critical thinking.” Instead, in his book  How We Think (1910), he argued for the importance of what he called “reflective thinking”:

…[when] the ground or basis for a belief is deliberately sought and its adequacy to support the belief examined. This process is called reflective thought; it alone is truly educative in value…

Active, persistent and careful consideration of any belief or supposed form of knowledge in light of the grounds that support it, and the further conclusions to which it tends, constitutes reflective thought.

There is however one passage from How We Think where Dewey explicitly uses the term “critical thinking”:

The essence of critical thinking is suspended judgment; and the essence of this suspense is inquiry to determine the nature of the problem before proceeding to attempts at its solution. This, more than any other thing, transforms mere inference into tested inference, suggested conclusions into proof.

2) The  Watson-Glaser Critical Thinking Appraisal  (1980) is a well-known psychological test of critical thinking ability. The authors of this test define critical thinking as:

…a composite of attitudes, knowledge and skills. This composite includes: (1) attitudes of inquiry that involve an ability to recognize the existence of problems and an acceptance of the general need for evidence in support of what is asserted to be true; (2) knowledge of the nature of valid inferences, abstractions, and generalizations in which the weight or accuracy of different kinds of evidence are logically determined; and (3) skills in employing and applying the above attitudes and knowledge.

3) A very well-known and influential definition of critical thinking comes from philosopher and professor Robert Ennis in his work “A Taxonomy of Critical Thinking Dispositions and Abilities” (1987):

Critical thinking is reasonable reflective thinking that is focused on deciding what to believe or do.

4) The following definition comes from a statement written in 1987 by the philosophers Michael Scriven and Richard Paul for the  National Council for Excellence in Critical Thinking (link), an organization promoting critical thinking in the US:

Critical thinking is the intellectually disciplined process of actively and skillfully conceptualizing, applying, analyzing, synthesizing, and/or evaluating information gathered from, or generated by, observation, experience, reflection, reasoning, or communication, as a guide to belief and action. In its exemplary form, it is based on universal intellectual values that transcend subject matter divisions: clarity, accuracy, precision, consistency, relevance, sound evidence, good reasons, depth, breadth, and fairness. It entails the examination of those structures or elements of thought implicit in all reasoning: purpose, problem, or question-at-issue, assumptions, concepts, empirical grounding; reasoning leading to conclusions, implications and consequences, objections from alternative viewpoints, and frame of reference.

The following excerpt from Peter A. Facione’s “Critical Thinking: A Statement of Expert Consensus for Purposes of Educational Assessment and Instruction” (1990) is quoted from a report written for the American Philosophical Association:

We understand critical thinking to be purposeful, self-regulatory judgment which results in interpretation, analysis, evaluation, and inference, as well as explanation of the evidential, conceptual, methodological, criteriological, or contextual considerations upon which that judgment is based. CT is essential as a tool of inquiry. As such, CT is a liberating force in education and a powerful resource in one’s personal and civic life. While not synonymous with good thinking, CT is a pervasive and self-rectifying human phenomenon. The ideal critical thinker is habitually inquisitive, well-informed, trustful of reason, open-minded, flexible, fairminded in evaluation, honest in facing personal biases, prudent in making judgments, willing to reconsider, clear about issues, orderly in complex matters, diligent in seeking relevant information, reasonable in the selection of criteria, focused in inquiry, and persistent in seeking results which are as precise as the subject and the circumstances of inquiry permit. Thus, educating good critical thinkers means working toward this ideal. It combines developing CT skills with nurturing those dispositions which consistently yield useful insights and which are the basis of a rational and democratic society.

V. Two F eatures of C ritical T hinking

A. how not what .

Critical thinking is concerned not with what you believe, but rather how or why you believe it. Most classes, such as those on biology or chemistry, teach you what to believe about a subject matter. In contrast, critical thinking is not particularly interested in what the world is, in fact, like. Rather, critical thinking will teach you how to form beliefs and how to think. It is interested in the type of reasoning you use when you form your beliefs, and concerns itself with whether you have good reasons to believe what you believe. Therefore, this class isn’t a class on the psychology of reasoning, which brings us to the second important feature of critical thinking.

B. Ought N ot Is ( or Normative N ot Descriptive )

There is a difference between normative and descriptive theories. Descriptive theories, such as those provided by physics, provide a picture of how the world factually behaves and operates. In contrast, normative theories, such as those provided by ethics or political philosophy, provide a picture of how the world should be. Rather than ask question such as why something is the way it is, normative theories ask how something should be. In this course, we will be interested in normative theories that govern our thinking and reasoning. Therefore, we will not be interested in how we actually reason, but rather focus on how we ought to reason.

In the introduction to this course we considered a selection task with cards that must be flipped in order to check the validity of a rule. We noted that many people fail to identify all the cards required to check the rule. This is how people do in fact reason (descriptive). We then noted that you must flip over two cards. This is how people ought to reason (normative).

• Section I-IV are taken from http://philosophy.hku.hk/think/ and are in use under the creative commons license. Some modifications have been made to the original content. ↵

Critical Thinking Copyright © 2019 by Brian Kim is licensed under a Creative Commons Attribution 4.0 International License , except where otherwise noted.

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Ray Optics: Refraction and Lenses

Calculator pad, version 2, refraction and lenses: problem set.

Light travels through a vacuum at a speed of 2.998 x 10 8 m/s. Determine the speed of light in the following media:

a. water (n = 1.333) b. crown glass (n = 1.52) c. cubic zirconia (n = 2.16) d. diamond (n = 2.419)

• Audio Guided Solution
• Show Answer

a. 2.249 x 10 8 m/s b. 1.97 x 10 8 m/s c. 1.39 x 10 8 m/s d. 1.239 x 10 8 m/s

Consider the three diagrams below. Measure the angle of incidence and of refraction (in degrees) for each of the three diagrams.

Diagram A: angle of incidence = 20°; angle of refraction = 35° Digram B: angle of incidence = 45°; angle of refraction = 33° Diagram C: angle of incidence = 10°; angle of refraction = 48°

A ray of light travels through air (n = 1.00) and approaching the boundary with water (n = 1.33). The angle of incidence is 45.0°. Determine the angle of refraction.

A light ray is passing through water (n=1.33) towards the boundary with a transparent solid at an angle of 56.4°. The light refracts into the solid at an angle of refraction of 42.1°. Determine the index of refraction of the unknown solid.

During a physics lab, Ray Zuvlight observes a laser line passing through an unknown material towards a boundary with air with an angle of incidence of 24.5°. The light ray emerges into the air with an angle of refraction of 33.8°. Determine the index of refraction of the unknown material.

Light in air approaches the boundary of oil at an angle of 36.1 degress with respect to the normal. The light travels at a speed of 2.27 x 10 8 m/s through the oil. Determine the angle of refraction.

Angle of incidence: 15° Angle of refraction: 30° Index of Refraction: 1.16

a. 29.8° b. 34.1°

a. 49° b. 60.°

Problem 10:

a. angle of refraction at water - Layer 1 boundary: 38.3° angle of refraction at Layer 1- Layer 2 boundary: 51.1° angle of refraction at Layer 2- Layer 3 boundary: 60.4° b. angle of incidence at Layer 1 - Layer 2 boundary: 38.3° angle of incidence at Layer 2 - Layer 3 boundary: 51.1° angle of incidence at Layer 3 - Water boundary: 60.4° c. angle of refraction when refracting back into water: 62.8°

Problem 11:

Problem 12:.

Ray Zuvlite is playing with his underwater laser. He descends beneath the water surface in his backyard pool to a vertical depth of 7.09 feet and directs the laser beam at an angle towards the pool's edge. The beam emerges from water (n=1.33) into air at the pool's very edge and projects onto a pool house which is located 17.21 feet horizontally from the pool's edge. The dot on the pool house is observed to be located at a vertical height of 8.75 feet. Determine the distance which Ray is located horizontally out from the edge of the pool.

Problem 13:

Determine the critical angle of the following materials when surrounded by air: a. teflon (n = 1.38) b. Pyrex glass (n = 1.47) c. Polycarbonate glass (n = 1.59) d. Sapphire gemstone (n = 1.77) e. Diamond (n = 2.42)

a. 46.4° b. 42.9° c. 39.0° d. 34.4° e. 24.4°

Problem 14:

Determine the critical angle of the following materials when surrounded by water (n = 1.33): a. teflon (n = 1.38) b. Pyrex glass (n = 1.47) c. Polycarbonate glass (n = 1.59) d. Sapphire gemstone (n = 1.77) e. Diamond (n = 2.42)

a. 74.5° b. 64.8° c. 56.8° d. 48.7° e. 33.3°

Problem 15:

In a physics lab, Jerome and Gavin are assigned the task of determining the critical angle of solid, transparent prism using laser light. When surrounded by air, light is observed to undergo total internal reflection when the angle of incidence is as low as 38°. Calculate the index of refraction of the material which the prism is made of.

Problem 16:

During a demonstration of fiber optics, Mr. H uses an acrylic light pipe to demonstrate total internal reflection for light as it navigates its way through a curved medium, entering at one end and exitting at the other with very little loss of light along the way. a. If the acrylic has an index of refraction of 1.48, then determine its critical angle when surrounded by air. b. Determine the critical angle of the acrylic light pipe when surrounded by water (n=1.33).

a. 42.5° b. 64.0°

Problem 17:

The triangular prism at the right is made of strontium titanate. A ray of light in air approaches the boundary at an angle of incidence of 30.0°. The ray strikes at the midpoint of one of the faces of the triangle. a. Determine the angle of refraction upon entering into the prism. b. Use geometric principles to determine the angle of incidence at the opposite side of the triangle. c. Will the light ray refract out of the prism at this opposite face or will it undergo total internal reflection? Do the calculation and explain the answer.

a. 12.0° b. 48.0° c. The critical angle is 24.5°. This light ray will not refract; it will undergo total internal reflection since the angle of incidence is greater than the critical angle.

Problem 18:

Problem 19:.

During a lens lab, Jerome and Michael placed a 4.5-cm tall night light bulb a distance of 42.8 cm from a lens. The image of the light bulb was inverted and appeared 26.5 cm from the lens. a. Determine the focal length of the lens being by Jerome and Michael. b. Determine the expected height of the image of the bulb.

a. 16.4 cm b. -2.8 cm (a - image height indicates an inverted image)

Problem 20:

The converging lens on Julia's camera has a focal length of 52 mm. She uses the camera to take a picture of her friends at South's homecoming. Her friends are located a distance of 2.45 m from the camera as Julia focuses in on them. Calculate the distance from the lens to the film (i.e, the image distance).

Problem 21:

During an in-class exploration lab, students are looking through both converging and diverging lenses at various objects and observing the characteristics of the images. Moses becomes particularly intrigued by the diverging lens. He places it a distance of 4.8 cm from the lettering of his textbook and estimates that the image of the letters are one-fourth the size of the actual lettering. a. Calculate the image distance which results when Moses holds the lens this close to the textbook. b. Calculate the focal length of the lens.

a. -1.2 cm (a - image distance indicates a virtual image) b. -1.6 cm (a - focal length indicates a diverging lens)

Problem 22:

During the Finding Smiley Lab, Tyrone and Mia place a small light bulb at various distances from a converging lens. The light bulb has a smiley face marked on one of its side. Their goal is to locate the images of the light bulb by projecting the refracted light onto a note card.

The lens is known to have a focal length of 20. cm. Predict the image distances which Tyrone and Mia will likely determine when the smiley face is located a distance of … a. … 60. cm from the lens. b. … 40. cm from the lens. c. … 30. cm from the lens. d. … 10. cm fron the lens.

a. 30. cm from the lens (to the right) b. 40. cm from the lens (to the right) c. 60. cm from the lens (to the right) d. 20. cm from the lens (to the left, corresponding to a dimage of -20. cm)

Problem 23:

The widest cinema screen in the world was reportedly constructed in New Zealand in 2007. The screen is 30.6 meters (100 feet) wide. Images from a 35-mm wide film are projected onto this screen. Suppose that the screen in the theater is located a distance of 46 m from the projector. Determine the magnification of the image and the focal length of the lens system.

magnification = -870 focal length = 53 mm (0.053 m)

Problem 24:

In an effort to read the miniaturized writing in Khaled's lab notebook, Mr. H pulls a magnifying glass from his drawer. He places the magnifying glass a distance of 6.3 cm from the paper and produces an upright image of the writing which is magnified by a factor of 4.8. Determine the focal length of the lens used by the magnifying glass.

Problem 25:

The focal point of a lens is located 17.8 cm away from its surface. The lens produces a virtual image which is 38.9 cm from the lens. Determine the object distance.

Problem 26:

A lens produces a virtual image located 33.8 cm from the lens when the object is located 18.5 cm from the lens. Determine the focal length of the lens. What kind of lens is it?

f = 40.9 cm Lens type: converging

Problem 27:

Jill organizes a surprise 16th birthday party for her boyfriend Jack down at Bianchi's Pizza Parlor. In an effort to capture the nostalgic moment on film, Jill squeezes all of Jack's friends together on one side of the table. The 51.4 mm camera lens focuses the image of all objects onto a film negative which is 35.8 mm wide. If Jack and friends are located 4.62 m from the camera lens, to what maximum distance must they squeeze together in order to completely fit in the developed picture?

Problem 28:

In a converging lens lab, Anna Litical is challenged to determine the object distance which produces a real image which is magnified by a factor of five. The lens has a focal length of 12.0 cm. What object distance would you expect to produce this magnification?

Problem 29:

An upright image is reduced to one-fourth of the object’s height when the object is placed 26.9 cm from the lens. Determine the focal length of the lens.

Problem 30:

An upright image is magnified, appearing to be four times the size of the object’s when the object is placed 26.9 cm from the lens. Determine the focal length of the lens.

Problem 31:

Matthew's new Bug Habitat toy comes with a built in magnifier, allowing Matthew to get a close-up view of his captive ants, crickets, slugs, and other creatures. The magnifier has a focal length of 22 cm. Determine the distance which an ant must be located from the lens in order for it to appear magnified by a factor of 12.

Problem 32:

A lens produces an inverted image which is one-fifth the size of the object. The lens has a focal length of 22.8 cm. Determine the object distance which results in this magnification.

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Future early childhood teachers designing problem-solving activities

Journal on Mathematics Education

This work aims to identify the criteria to design activities based on problem-solving tasks that emerge when future early childhood education teachers jointly plan their activities and reflect on them. The participants were 76 students from the Didactics of mathematics subject that was carried out in the 2nd year of the Early Childhood Education Degree of a Catalan public university. This is qualitative research in which the phases of the thematic analysis have been adapted: familiarizing with the data; systematically applying the categories to identify the student criteria emerged; triangulating the analysis with experts; reviewing and discussing the results. The Didactic Suitability Criteria (DSC), from the Ontosemiotic approach (OSA) framework to design tasks and their indicators, were used to categorise and analyse the tasks performed by future teachers. As a result, it was identified that when the future teachers adopt consensually design their activities, they are implicitly b...

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Solomon A Tesfamicael

Angel Alsina , Gemma Sala Sebastià

The social, scientific and technological development of recent years has encouraged the incorporation of computational thinking in the school curriculum of various countries progressively, starting from early childhood education. This research aims to characterize future kindergarten teachers' traits of didactic-mathematical and computational knowledge presented when solving and posing robotics problems. Firstly, aspects of the mathematical and computational knowledge of the participants (97 students of the subject of Didactics of Mathematics of the Degree in Early Childhood Education at a Spanish university) were identified when they solved problems as users of the Blue-Bot didactic robot. Secondly, we analyzed their justifications for reflecting on the design of robotics problems. The results indicate that future teachers present characteristics of didactic-mathematical knowledge when solving and designing robotics problems, although errors and ambiguities are evident, especially in the procedures and representations of the programming. These shortcomings significantly influence the didactic suitability of the robotics problems they design. From a future perspective, in the training of future teachers, it is considered relevant to incorporate didacticmathematical and computational knowledge that allows them to develop logical, spatial and computational thinking.

Eurasia Journal of Mathematics, Science and Technology Education

Eulalia Calle

The objective of this article is to identify the meaning attributed to the didactics of mathematics and what are the criteria with which an improvement in the teaching and learning process of mathematics is based, future teachers of mathematics, belonging to universities in three different countries (Brazil, Chile, and Ecuador). The qualitative analysis indicates that the majority of future teachers consider that the didactics of mathematics is a technical discipline that consists of providing strategies, resources, and procedures for teaching mathematics; few consider it as an art to teaching and almost none consider it as a scientific discipline that is concerned with studying the processes of teaching and learning mathematics. In addition, the results show that the criteria used by them, on how teaching and learning in this discipline can be improved, are focus, above all, in the cognitive, ecological, and emotional aspects and, to a lesser extent, to the interactional, mediation...

Alteridad: Revista de Educación

Teachers should have the competence to reflect on their own practice. The Ontosemiotic Approach offers as a tool for this purpose the “Didactic Suitability Criteria”. The objectives of the research are to refine the indicators of epistemic suitability (ES) and to deepen the epistemic dimension of the meta-didactic-mathematical knowledge of secondary school mathematics teachers in initial training. To this end, research on the object function in secondary education has been examined. In a mixed way, we have analysed, in 119 “Master’s Dissertation (MD) in Secondary Education Mathematics Teachers in service Training (Catalonia, Spain)” on functions, the reflection that future teachers make on their teaching, from the ES. Based on these analyses, the adequacy of the epistemic suitability for analysing the mathematical object function is designed. Finally, the MDs are reanalysed with this new guideline and it is found that their reflections present important weaknesses that could influence the quality of their instructional processes. It is concluded that if mathematics teachers were provided with a specialised guideline that makes it easier for them to consider all the meanings, representations and processes involved in the complexity of the functions, as well as the mathematical practices in which these emerge, it would improve the quality of the design, implementation, and reflection on their instructional processes.

Gemma Sala Sebastià

Es importante la incorporación del desarrollo del pensamiento computacional, ya en las primeras edades, para abordar los retos de la sociedad actual y del futuro, con un nivel científico y tecnológico en auge. Son cada vez más los países que incorporan en sus currículums el pensamiento computacional, como por ejemplo el currículum español y catalán que, en la renovación de este mismo año, por primera vez lo incorporan explícitamente en el currículum de Educación Infantil. El objetivo general de esa investigación es caracterizar la idoneidad didáctica de los diseños de los futuros maestros de Educación Infantil sobre una tarea de resolución de problemas con el uso del robot educativo Blue-Bot, para alumnado de 5 años. Los participantes fueron 97 estudiantes de la asignatura de Didáctica de las Matemáticas del Grado de Educación Infantil de una universidad catalana (en España). El análisis cualitativo se llevó a cabo mediante la herramienta Criterios de Idoneidad Didáctica del Enfoque...

International Journal of Science and Mathematics Education

Vicenç Font

Research in mathematics education highlights the importance of including modelling for the teaching of this subject. In 2020, this trend coexisted with a grave contingency situation caused by the COVID-19 pandemic but which, despite its negative aspects, provided a realistic and authentic context for modelling. Given this situation, it is relevant to study which aspects of the teaching and learning process prospective teachers related to mathematical modelling in their reflections on its inclusion during the transition period between the face-to-face and virtual teaching contexts. To this end, we used the didactic suitability criteria construct, proposed by the onto-semiotic approach, as a theoretical reference. This is qualitative research of a naturalistic type, since we did not interfere in the Master’s Programme in Mathematics Teacher Education studied. We conducted a content analysis on 122 master’s degree final projects written during the 2019–2020 course and implemented durin...

• Mathematics

Learning about the conceptions used by primary school teachers towards the use of robotics in class is essential as the first step towards its application in the classroom. Therefore, with the purpose of describing the understanding applied when teaching and learning mathematics use educational robots, research was conducted by means of mixed methods using a descriptive design by survey. Such research consisted of closed questions (Likert-type scale from 1 to 5) and open questions, given to 83 primary school teachers who currently teach students in the first years of school (First to Fourth grade) in two Chilean districts. The results showed that in general, there is a positive predisposition towards the addition of robots in the learning and teaching of mathematic processes during the first years of school, even though teachers claim there is a struggle to incorporate robots in their lessons due to the high number of students and the reduced space in their classrooms.

Yuri Morales-López

This article describes the didactic analysis carried out by an expert teacher and a novice teacher, based on the didactic suitability criteria (DSC) of the Onto Semiotic Approach (OSA) when analyzing episodes of a high school mathematics class in Costa Rica. This is a qualitative investigation with an interpretive hermeneutic approach. First, a questionnaire was applied to the two participants, to classify them as novice and expert, based on certain defined characteristics. Both teachers then analyzed three video segments using a guide for didactic analysis and, finally, an interview was conducted to elaborate on the reflections previously obtained. The data analysis focused on identification, classification, and comparison of elements in each of the video segments with reference to the OSA’s DSC indicators. The results show that there are differences and similarities between the didactic analysis carried out by these teachers. It is concluded that the novice teacher provides a more...

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Experience Google AI in even more ways on Android

May 14, 2024

[[read-time]] min read

By building AI right into the Android operating system, we're reimagining how you can interact with your phone.

• Bullet points
• Circle to Search gets smarter, helping students solve physics and math problems directly from their phones and tablets.
• Gemini on Android improves context understanding, allowing users to drag and drop generated images and ask questions about videos and PDFs.
• Gemini Nano with Multimodality coming to Pixel, bringing multimodal capabilities for richer image descriptions and scam alerts during phone calls.
• Android 15 and ecosystem updates coming tomorrow.
• Basic explainer

Google is making Android phones smarter with AI.

Circle to Search can now help students with homework.

Gemini, a new AI assistant, can understand what's on your screen and help you do things.

Android phones will soon be able to alert you to suspected scams during phone calls.

Explore other styles:

We’re at a once-in-a-generation moment where the latest advancements in AI are reinventing what phones can do. With Google AI at the core of Android’s operating system, the billions of people who use Android can now interact with their devices in entirely new ways.

Today, we’re sharing updates that let you experience Google AI on Android.

Circle to Search can now help students with homework

With Circle to Search built directly into the user experience, you can search anything you see on your phone using a simple gesture — without having to stop what you’re doing or switch to a different app. Since launching at Samsung Unpacked , we’ve added new capabilities to Circle to Search, like full-screen translation , and we’ve expanded availability to more Pixel and Samsung devices.

Starting today, Circle to Search can now help students with homework, giving them a deeper understanding, not just an answer — directly from their phones and tablets. When students circle a prompt they’re stuck on, they’ll get step-by-step instructions to solve a range of physics and math 1 word problems without leaving their digital info sheet or syllabus. Later this year, Circle to Search will be able to help solve even more complex problems involving symbolic formulas, diagrams, graphs and more. This is all possible due to our LearnLM effort to enhance our models and products for learning.

Circle to Search is already available on more than 100 million devices today. With plans to bring the experience to more devices, we’re on track to double that by the end of the year.

Gemini will get even better at understanding context to assist you in getting things done

Gemini on Android is a new kind of assistant that uses generative AI to help you be more creative and productive. This experience, which is integrated into Android, is getting even better at understanding the context of what’s on your screen and what app you’re using.

Soon, you’ll be able to bring up Gemini's overlay on top of the app you're in to easily use Gemini in more ways. For example, you can drag and drop generated images into Gmail, Google Messages and other places, or tap “Ask this video” to find specific information in a YouTube video. If you have Gemini Advanced, you’ll also have the option to “Ask this PDF” to quickly get answers without having to scroll through multiple pages. This update will roll out to hundreds of millions of devices over the next few months.

And we’ll continue to improve Gemini to give you more dynamic suggestions related to what’s on your screen.

Full multimodal capabilities coming to Gemini Nano

Android is the first mobile operating system that includes a built-in, on-device foundation model. With Gemini Nano, we’re able to bring experiences to you quickly and keep your information completely private to you. Starting with Pixel later this year, we’ll be introducing our latest model, Gemini Nano with Multimodality. This means your phone will not just be able to process text input but also understand more information in context like sights, sounds and spoken language.

Clearer descriptions with TalkBack

Later this year, Gemini Nano’s multimodal capabilities are coming to TalkBack, helping people who experience blindness or low vision get richer and clearer descriptions of what’s happening in an image. On average, TalkBack users come across 90 unlabeled images per day. This update will help fill in missing information — whether it’s more details about what’s in a photo that family or friends sent or the style and cut of clothes when shopping online. Since Gemini Nano is on-device, these descriptions happen quickly and even work when there's no network connection.

Receive alerts for suspected scams during phone calls

According to a recent report , in a 12-month period, people lost more than $1 trillion to fraud. We’re testing a new feature that uses Gemini Nano to provide real-time alerts during a call if it detects conversation patterns commonly associated with scams. For example, you would receive an alert if a “bank representative” asks you to urgently transfer funds, make a payment with a gift card or requests personal information like card PINs or passwords, which are uncommon bank requests . This protection all happens on-device, so your conversation stays private to you. We’ll share more about this opt-in feature later this year.

More to come on Android

We’re just getting started with how on-device AI can change what your phone can do, and we’ll continue building Google AI into every part of the smartphone experience with Pixel, Samsung and more. If you’re a developer, check out the Android Developers blog to learn how you can build with our latest AI models and tools, like Gemini Nano and Gemini in Android Studio.

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Here’s a look at everything we announced at Google I/O 2024.

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Available for some math word problems when opted into Search Labs.

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