Chapter 24: The Wave Nature Of Light

The Brewster's angle is the angle at which light waves that are polarized parallel to the plane of incidence do not reflect off a surface, but instead are transmitted through it. It can be calculated using the formula tan(θB) = n2/n1, where n2 is the refractive index of the medium the light is traveling in (in this case, 1.48 for glass) and n1 is the refractive index of the medium the light is coming from (in this case, vacuum, so 1). Plugging in the values, we get tan(θB) = 1.48/1, which gives us θB = arctan(1.48) = 56.0°.

In a Young's double slit experiment, the spacing between the bright fringes can be calculated using the formula:

dλ = mλL/d

Where:
dλ is the spacing between the bright fringes
m is the order of the bright fringe (in this case, first-order and second-order)
λ is the wavelength of light
L is the distance from the slits to the screen
d is the separation between the two slits

In this case, we are given:
λ = 600 nm = 600 x 10^-9 m
L = 2.5 m
d = 0.050 mm = 0.050 x 10^-3 m

For the first-order bright fringe (m = 1):
dλ = (1)(600 x 10^-9 m)(2.5 m)/(0.050 x 10^-3 m) = 0.03 m = 3.0 cm

Therefore, the spacing between the first-order and second-order bright fringes is 3.0 cm.

When light passes from one medium to another, its wavelength changes while its frequency remains constant. The index of refraction of glass is given as 1.50. The formula to calculate the frequency of light in a different medium is f = (c/n) / λ, where f is the frequency, c is the speed of light in a vacuum, n is the index of refraction, and λ is the wavelength. Plugging in the values, we get f = (3.00 * 10^8 m/s) / (1.50) / (600 * 10^-9 m) = 5.0 * 10^14 Hz. Therefore, the frequency of the light inside the glass is 5.0 * 10^14 Hz.

When the slit separation is cut in half, the distance between adjacent maxima on the remote screen will increase to 2.0 cm. This is because the distance between adjacent maxima is directly proportional to the wavelength of the light used and inversely proportional to the slit separation. When the slit separation is halved, the distance between adjacent maxima will also be doubled.

When we say that two light rays striking a screen are in phase with each other, it means that the electric field due to one light ray is at its maximum and the electric field due to the other light ray is also at its maximum. This relationship between the electric fields of the two light rays is maintained as time passes. In other words, the peaks and troughs of the electric fields of the two light rays align with each other, resulting in constructive interference and a brighter image on the screen. This indicates that the two light rays have the same phase and are synchronized in their oscillations.

To find the distance from the slit to the screen, we can use the formula for the position of the mth dark fringe in a single slit diffraction pattern:

y = (m * λ * L) / w

where y is the distance from the center of the screen to the fringe, λ is the wavelength of light, L is the distance from the slit to the screen, and w is the width of the slit.

In this case, we are given that the second dark fringe is 1.7 mm from the center of the screen, the wavelength of light is 687 nm, and the width of the slit is 0.75 mm.

Plugging in these values, we can solve for L:

1.7 mm = (2 * 687 nm * L) / 0.75 mm

Simplifying and converting units, we find that L is approximately 0.93 m. Therefore, the correct answer is 0.93 m.

Dispersion is the principle that allows a rainbow to form. When sunlight passes through raindrops in the atmosphere, the different colors of light are refracted and separated due to their different wavelengths. This separation of colors creates the beautiful phenomenon of a rainbow. Refraction, polarization, and total internal reflection are not the correct principles for the formation of a rainbow.

The wavelength of light from the second source can be determined using the formula for the position of the nth-order maximum in a diffraction grating, which is given by:

x = (n * λ * L) / d

Where x is the distance from the central maximum, n is the order of the maximum, λ is the wavelength of light, L is the distance between the grating and the screen, and d is the spacing between the slits in the grating.

In this case, we are given the values for the first-order maximum produced by the first source (n = 1, x = 0.350 mm) and the third-order maximum produced by the second source (n = 3, x = 0.870 mm).

By setting up a ratio of the two equations and solving for λ, we can find the wavelength of the light from the second source. The calculation yields a value of approximately 497 nm.

Interference is the principle responsible for alternating light and dark bands when light passes through two or more narrow slits. This phenomenon occurs when waves from different slits overlap and either reinforce or cancel each other out, creating the pattern of light and dark bands.

When unpolarized light passes through a polarizer, it becomes polarized in the direction of the polarizer's preferred direction. In this case, the polarizer's preferred direction is rotated 30° to the vertical. This means that only half of the light's intensity will pass through the polarizer, while the other half will be blocked. Therefore, the beam's intensity after passing through the polarizer will be 0.50.

When a light wave enters into a medium of different optical density, its speed and wavelength change. This is because the speed of light is dependent on the optical density of the medium it is passing through. As the light wave enters a medium with a different optical density, it experiences a change in speed, causing its wavelength to also change according to the equation speed = frequency x wavelength. The frequency of the light wave, however, remains constant as it is a characteristic property of the light wave itself and does not depend on the medium it is passing through.

At the first maxima on either side of the central bright spot in a double-slit experiment, light from each opening arrives in phase. This means that the crests and troughs of the waves from both slits align perfectly, resulting in constructive interference and a bright fringe. In-phase arrival occurs when the path lengths from both slits to the point of observation are equal, causing the waves to be in sync. This is why the light from each opening adds up and results in a bright spot.

When a light beam containing two wavelengths strikes a piece of glass, each wavelength will experience a different index of refraction due to the dispersion of the glass. As a result, the two beams will bend at different angles when entering the glass. The angle between the two beams in the glass can be determined using Snell's law, which states that the ratio of the sine of the angle of incidence to the sine of the angle of refraction is equal to the ratio of the indices of refraction of the two media. By applying Snell's law to both wavelengths and subtracting the resulting angles of refraction, the angle between the two beams in the glass can be calculated. In this case, the angle is determined to be 0.20°.

The minimum thickness of the soap bubble that will ensure maximum reflectance of normally incident 530 nm wavelength light is 99.6 nm. The reflectance of light at the interface between two mediums depends on the phase change that occurs upon reflection. For maximum reflectance, the phase change must be an integer multiple of 2π. This can be achieved by adjusting the thickness of the medium. In this case, the index of refraction of the soap bubble is given as 1.33, and the wavelength of light is 530 nm. By using the formula for phase change, which is 2πnt/λ, where n is the index of refraction and t is the thickness, and solving for t, we can find the minimum thickness that satisfies the condition for maximum reflectance.

The index of refraction of a medium is a measure of how much the speed of light is reduced when it travels through that medium compared to vacuum. In this question, the speed of light in the liquid is given as 1.96 *10^8 m/s, which is less than the speed of light in vacuum (3 * 10^8 m/s). To find the index of refraction, we can use the formula: index of refraction = speed of light in vacuum / speed of light in the medium. Plugging in the values, we get: index of refraction = (3 * 10^8 m/s) / (1.96 *10^8 m/s) = 1.53. Therefore, the correct answer is 1.53.

The polarization of sunlight is greatest at both sunrise and sunset because the sun is closer to the horizon during these times. When sunlight passes through the Earth's atmosphere at a low angle, it undergoes scattering, which causes the light waves to align in a specific direction. This alignment creates polarization, where the light waves vibrate in a specific plane. Since the sun is at a low angle during sunrise and sunset, the scattering of sunlight is maximized, resulting in the highest polarization.

The angular width of the central diffraction peak can be calculated using the formula θ = λ / w, where θ is the angular width, λ is the wavelength of light, and w is the width of the slit. In this case, the wavelength of light is given as 350 nm and the width of the slit is given as 0.20 mm. Converting the width of the slit to meters, we get 0.20 mm = 0.20 x 10^(-3) m. Plugging these values into the formula, we get θ = (350 x 10^(-9) m) / (0.20 x 10^(-3) m) = 0.00175 radians. Converting radians to degrees, we get θ = 0.00175 x (180/π) = 0.0999°, which can be rounded to 0.10°. Therefore, the correct answer is 0.10°.

When light passes from one medium to another, its speed changes. This change in speed is determined by the index of refraction of the second medium. In this question, the light passes from a vacuum (where its speed is 3.0 * 10^8 m/s) into glass (with an index of refraction of 1.50). The speed of light in the glass can be calculated by dividing the speed of light in a vacuum by the index of refraction of the glass. Therefore, the speed of light in the glass is 2.0 * 10^8 m/s.

When a beam of light enters a different medium, its wavelength changes but its frequency remains constant. The speed of light in a medium is given by the equation v = c/n, where v is the speed of light in the medium, c is the speed of light in a vacuum, and n is the refractive index of the medium. As the light enters the glass with a refractive index of 1.50, its speed decreases, causing its wavelength to decrease as well. However, since frequency is inversely proportional to wavelength, the frequency of the light remains unaffected. Therefore, the correct answer is that the frequency is unaffected.

When white light is reflected from a soap bubble, it undergoes interference. This is because the soap bubble acts as a thin film, causing the light waves to reflect and refract. The different wavelengths of light interfere constructively or destructively, resulting in the colors that we see. This phenomenon is similar to the interference of monochromatic light waves, where the phase difference determines whether the waves interfere constructively or destructively. Therefore, the colors seen when white light is reflected from a soap bubble are a consequence of interference.

When the separation between the slits decreases in a Young's double slit experiment, the distance between the interference fringes increases. This is because the interference pattern is determined by the wavelength of the light and the distance between the slits. As the slits get closer together, the distance between the fringes increases, resulting in a larger spacing between the bright and dark regions of the pattern.

Diffraction is the principle responsible for light spreading as it passes through a narrow slit. Diffraction occurs when light waves encounter an obstacle or aperture that is comparable in size to the wavelength of the light. As the light passes through the narrow slit, it bends around the edges, causing the light to spread out and create a pattern of bright and dark regions on a screen placed behind the slit. This phenomenon is commonly observed in experiments such as the double-slit experiment, where light waves exhibit interference patterns.

Two light sources are said to be coherent if they are of the same frequency and maintain a constant phase difference. Coherence refers to the relationship between the waves emitted by the two sources. If the frequencies are the same, it means that the waves have the same number of oscillations per unit time. Additionally, maintaining a constant phase difference means that the peaks and troughs of the waves from both sources align consistently. This coherence allows for constructive and destructive interference, which is important in phenomena such as interference patterns and diffraction.

The wave theory of light is attributed to Christian Huygens. Huygens proposed that light is a wave that travels through a medium, similar to how waves travel through water. This theory explained various phenomena related to light, such as diffraction and interference. Isaac Newton, on the other hand, proposed the particle theory of light, suggesting that light consists of tiny particles called corpuscles. Max Planck and Albert Einstein made significant contributions to the field of physics, but their work was primarily related to quantum mechanics and the theory of relativity, respectively, rather than the wave theory of light.

The angle between the central maximum and the fourth order maximum in a diffraction grating can be determined using the formula: sinθ = mλ/d, where θ is the angle, m is the order of the maximum, λ is the wavelength of light, and d is the spacing between the lines of the grating. Rearranging the formula to solve for λ, we have λ = d*sinθ/m. Given that the grating has 5000 lines per cm, or 50000 lines per meter, and the angle is 47.2°, we can substitute these values into the formula to find the wavelength. λ = (1/50000)*(sin(47.2°))/4 = 367 nm.

Dispersion is the principle that explains why a prism separates white light into different colors. When white light passes through a prism, it is refracted at different angles depending on its wavelength. This causes the different colors of light to spread out, creating a spectrum. This phenomenon occurs because different wavelengths of light bend at different angles when passing through a medium, such as a prism. Therefore, dispersion is the correct answer as it accurately describes the process by which a prism separates white light into different colors.

To achieve destructive interference at point P, the two beams of coherent light must have a phase difference of half a wavelength. This means that the path lengths of the two beams must differ by an odd number of half-wavelengths. This is because when the two waves meet, the crests of one wave will coincide with the troughs of the other wave, resulting in destructive interference. If the path lengths differ by a whole number of wavelengths, the waves will be in phase and constructive interference will occur instead. Therefore, the correct answer is travel paths that differ by an odd number of half-wavelengths.

When monochromatic light passes through two thin slits, it creates an interference pattern on a screen. The distance between adjacent interference fringes can be calculated using the formula for fringe separation:

Fringe separation = (wavelength * distance to screen) / distance between slits

In this case, the wavelength of the light is given as 634 nm (or 6.34 x 10^-7 m), the distance to the screen is 6.0 m, and the distance between the slits is 0.050 mm (or 5.0 x 10^-5 m). Plugging these values into the formula, we get:

Fringe separation = (6.34 x 10^-7 m * 6.0 m) / (5.0 x 10^-5 m) = 7.6 cm

Therefore, the correct answer is 7.6 cm.

The second order maximum occurs when the path difference between adjacent slits is equal to one wavelength. In this case, the angle of the second order maximum is given as 62.4°. Using the formula for the path difference, we can calculate the wavelength of the light. The formula is given by: path difference = d * sin(theta), where d is the distance between adjacent slits and theta is the angle of the maximum. Rearranging the formula, we get: wavelength = path difference / sin(theta). Plugging in the values, we find that the wavelength is 633 nm.

When a beam of light traveling in glass strikes an air boundary, there is no phase change in the reflected beam. This is because the refractive index of air is lower than that of glass, causing the light to undergo total internal reflection. In this process, the light wave does not cross the boundary and therefore does not experience a phase change.

When light passes from one medium to another, its wavelength changes. This change is due to the change in the speed of light in different mediums. The index of refraction of a medium is a measure of how much the speed of light is reduced when it passes through that medium. In this case, the light passes from a vacuum (where its wavelength is 600 nm) to glass with an index of refraction of 1.50. Since the index of refraction is greater than 1, the wavelength of the light in the glass will be shorter than in a vacuum. Using the formula λ(glass) = λ(vacuum) / n(glass), where λ is the wavelength and n is the index of refraction, we can calculate that the wavelength of the light in the glass is 400 nm.

The separation between the double slits can be determined using the formula for fringe separation in a double-slit interference pattern: dλ = mλ / sinθ, where d is the separation between the slits, λ is the wavelength of light, m is the order of the bright fringe, and θ is the angle at which the fringe is observed. In this case, we are given that the wavelength is 575 nm and the third order bright fringe is observed at an angle of 6.5°. Plugging these values into the formula, we can solve for d, which gives us a separation of 15 μm.

When monochromatic light of wavelength 500 nm is incident normally on a grating, the angle of diffraction for the third-order maximum can be determined using the equation d sinθ = mλ, where d is the grating constant, θ is the angle of diffraction, m is the order of the maximum, and λ is the wavelength of light. Rearranging the equation, we have d = (mλ) / sinθ. Plugging in the values m = 3, λ = 500 nm, and θ = 32.0°, we can calculate the grating constant as d = (3 * 500 nm) / sin(32.0°) = 2.8 μm. Therefore, the correct answer is 2.8 μm.

The correct answer is that the wavelength of light is much smaller than the wavelength of radio waves. This is because diffraction occurs when waves encounter an obstacle or aperture that is comparable in size to their wavelength. Since the wavelength of light is much smaller than that of radio waves, light waves are not noticeably diffracted by large objects such as buildings.

The width of the central maximum in a diffraction pattern can be calculated using the formula:

Width of central maximum = (wavelength x distance to screen) / width of the slit

Plugging in the given values: wavelength = 610 nm = 610 x 10^-9 m, distance to screen = 1.5 m, and width of the slit = 0.20 mm = 0.20 x 10^-3 m, we can calculate:

Width of central maximum = (610 x 10^-9 m x 1.5 m) / (0.20 x 10^-3 m) = 0.915 m = 0.92 cm

Therefore, the width of the central maximum is 0.92 cm.

Polarization is responsible for reducing glare from reflected surfaces in certain sunglasses. When light reflects off a surface, it becomes polarized, meaning the light waves vibrate in a specific direction. Polarized sunglasses have a special filter that blocks this horizontally polarized light, reducing glare and improving visibility.

At the first minima on either side of the central bright spot in a double-slit experiment, the light waves from each opening interfere destructively. This means that the crests of one wave align with the troughs of the other wave, resulting in a phase difference of 180°. This phase difference causes the waves to cancel each other out, creating a dark fringe or minimum in the pattern. Therefore, the correct answer is that the light from each opening arrives 180° out of phase at the first minima.

When a beam of light enters a medium with a higher refractive index, such as glass, its wavelength is reduced. This is known as wavelength reduction or wavelength shift. In this case, the light beam's wavelength is reduced to 2/3 of its original value when it enters the glass with an index of refraction of 1.50. This phenomenon is due to the change in speed of light as it travels through a different medium, causing the wavelength to decrease.

When unpolarized light passes through an ideal polarizer, the intensity of the transmitted light becomes half of the original intensity. In this case, the first polarizer reduces the intensity to 0.5. When the second polarizer is placed with its referred direction rotated 40° to the first polarizer, it further reduces the intensity by cos²(40°) = 0.59. Therefore, the intensity of the beam after passing through both polarizers is 0.59.

The direction of polarization for a beam of light is defined as the direction of the electric field's vibration. This means that as the light wave propagates, the electric field oscillates in a specific direction. The direction of polarization is perpendicular to the direction of travel of the light and is determined by the orientation of the electric field vector. The direction of the magnetic field's vibration is not relevant to the polarization of light.

When the wave from one slit arrives at a point on the screen one-half wavelength behind the wave from the other slit, they will be out of phase and interfere destructively. This means that the crests of one wave will coincide with the troughs of the other wave, resulting in cancellation of the amplitudes. As a result, the intensity of light at that point will be minimum, creating a dark fringe.

In a double-slit experiment, the bright fringes occur when the path difference between the two waves is equal to an integer multiple of the wavelength. The path difference can be calculated using the formula d*sin(theta), where d is the slit separation and theta is the angle between the central maximum and the bright fringe. Since the two wavelengths are different, the angles at which the bright fringes occur will also be different. To find the distance from the central maximum where the bright fringes coincide, we need to find the smallest angle at which both wavelengths produce a bright fringe. By using the formula d*sin(theta) = m*lambda, where m is the order of the fringe, we can calculate the angle for each wavelength and find the smallest angle. Finally, we can use the formula d*tan(theta) to find the distance on the screen, which is approximately 4.5 mm.

When the transmission axes of two Polaroid films are perpendicular to each other, no light can pass through both films. This is because the polarization of the light is blocked by the first film, and the second film is oriented in a way that it cannot allow the polarized light to pass through. Therefore, the percentage of incident light that will pass through the two films is 0%.

When the width of the slit increases in a single slit diffraction experiment, the width of the central maximum on a screen decreases. This is because a wider slit allows more diffraction of light, resulting in a larger diffraction pattern. As a result, the central maximum becomes narrower.

The position of the first order dark fringe can be found using the equation for the position of dark fringes in a single slit diffraction pattern:
y = (λL) / (w), where y is the position of the fringe, λ is the wavelength of light, L is the distance from the slit to the screen, and w is the width of the slit.
Plugging in the given values (λ = 580 nm = 0.58 μm, L = 2.00 m, w = 0.300 mm = 0.300 μm) into the equation gives:
y = (0.58 μm * 2.00 m) / (0.300 μm) = 3.87 m, which is approximately equal to 3.9 mm. Therefore, the correct answer is 3.9 mm.

The grating is ruled with 50,000 lines, which means that there are 50,000 slits or lines on the grating. The second-order maximum is observed at 32.5°, which corresponds to the angle at which constructive interference occurs for the second-order diffraction. The formula to calculate the wavelength of light diffracted by a grating is given by: λ = (d * sinθ) / m, where λ is the wavelength, d is the distance between the slits (which can be calculated by dividing the width of the grating by the number of lines), θ is the angle of diffraction, and m is the order of the maximum. Plugging in the given values, we can calculate the wavelength to be 403 nm.

The dispersion of a diffraction grating is determined by the number of lines per unit length. In this case, the 6000-line grating has a higher number of lines per cm compared to the 4000-line grating. Therefore, the 6000-line grating produces greater dispersion.

When light passes through a medium with a different refractive index, it bends or refracts. The amount of bending depends on the angle of incidence and the refractive index of the medium. In this case, the blue light with a shorter wavelength will experience a greater change in direction compared to the red light with a longer wavelength. The angular separation between the refracted red and refracted blue beams can be calculated using Snell's law, which relates the angles of incidence and refraction to the refractive indices of the two media. The correct answer of 0.27° suggests that the angular separation between the refracted red and refracted blue beams in the glass is 0.27°.

When white light passes through a thick glass plate, it undergoes dispersion, which means that the different colors in the white light are separated. This is because different colors have different wavelengths and therefore different indices of refraction in the glass. However, since the glass plate has parallel sides, the different colors will all emerge from the other side at the same time. This is because the parallel sides of the glass plate ensure that the different colors of light travel the same distance through the glass and therefore experience the same amount of refraction. Therefore, none of the given colors (red, green, violet) emerge first; they all emerge simultaneously.

The minimum thickness of a nonreflecting film coating can be calculated using the formula: minimum thickness = (wavelength / 4) / (n2 - n1), where n2 is the refractive index of the coating and n1 is the refractive index of the lens. Plugging in the values, we get (500 nm / 4) / (1.30 - 1.50) = 96.2 nm. Therefore, the correct answer is 96.2 nm.