CHAPTER 15: PHYSICAL OPTICS PRACTICE TEST SOLUTION BY PHYSICS INN

 

PRACTICE TEST PAPER CHAPTER 15: PHYSICAL OPTICS

Answer Key & Detailed Solutions

VERSION 1 — DETAILED SOLUTIONS

Q.1 — MCQs (Version 1)

1. (c) Wavefront
• Explanation: By definition, a wavefront is an imaginary surface that connects all adjacent points of a wave that share the exact same phase of oscillation.
2. (b) Spherical
• Explanation: A point source in three-dimensional space emits waves uniformly in all directions. The locus of points at equal distance/phase forms concentric spheres.
3. (d)
• Explanation: The spatial period of a wave is its wavelength . Therefore, the shortest distance between two consecutive wavefronts (e.g., crest to crest) is equal to one wavelength.
4. (c) Christiaan Huygens
• Explanation: As stated in the chapter history, Christiaan Huygens first proposed the wave theory of light in 1678.
5. (b) Lamps are not coherent
• Explanation: Two independent light bulbs emit light in random, uncorrelated phase bursts. Because they lack a constant phase difference (coherence), their interference pattern changes too rapidly to be observed.
6. (c) Phase difference
• Explanation: Coherent sources are defined as sources that emit waves with a constant phase difference over time.
7. (b) Bright
• Explanation: At  (the center of the screen opposite the midpoint of the slits), the path difference between light rays from both slits is zero. The waves arrive in phase, producing constructive interference (a central bright fringe).
8. (b) Interference
• Explanation: The colors on a soap bubble arise from thin-film interference between light waves reflected from the outer and inner surfaces of the soap film.
9. (b) Circular fringes
• Explanation: Because the air film between a plano-convex lens and a flat glass plate is symmetric around the central contact point (forming regions of equal thickness in concentric circles), the resulting interference pattern forms concentric circular rings.
10. (d) Diffraction
• Explanation: Diffraction is defined as the bending of light around the sharp edges of obstacles or slits into its geometrical shadow.
11. (a) Zero order
• Explanation: The central principal maximum in a diffraction pattern occurs at  (where path difference is zero, corresponding to ), so it is designated as the zero-order maximum.
12. (b)
• Explanation: Bragg's Law relates the interplanar spacing , glancing angle , order of reflection , and wavelength  for constructive interference from crystal planes as .
13. (c) X-rays
• Explanation: X-rays have wavelengths () of the same order of magnitude as atomic spacing in crystals, making them suitable for crystal structure analysis via diffraction.

Q.2 — Short Questions (Version 1)

1. Define a wavefront and name its three types.
• Solution: A wavefront is an imaginary surface joining all points of a wave that have the same phase of oscillation. The three main types are:
1. Plane wavefronts
2. Circular wavefronts
3. Spherical wavefronts
2. State Huygens' Principle.
• Solution: Huygens' Principle consists of two main statements:

1. Every point on a primary wavefront can be considered as a source of secondary wavelets spreading out in all directions with a speed equal to the wave speed.
2. The new wavefront at a later time is the surface tangent (envelope) to these secondary wavelets.

3. What are the three basic conditions required to produce interference of light?
• Solution: The required conditions are:

1. The interfering waves must be of the same type (e.g., sound with sound, light with light).
2. The sources must be coherent (maintaining a constant phase difference).
3. The waves must propagate in the same or nearly the same direction.

4. Can Huygens' Principle be applied to sound and water waves? Explain.
• Solution: Yes, Huygens' Principle is a fundamental geometrical method describing wave propagation in general. It applies universally to all mechanical waves (such as sound and water waves) as well as electromagnetic waves.
5. Define the term "Coherent Sources."
• Solution: Coherent sources are two or more sources that emit waves having the exact same frequency (and wavelength) with a constant phase difference over time.
6. Why does the interference pattern disappear if we cover one of the two slits in Young’s experiment?
• Solution: Interference requires the superposition of waves coming from at least two coherent sources. If one slit is covered, light passes through only one slit, producing single-slit diffraction—a single broad band of light—rather than an interference pattern.
7. What is a diffraction grating?
• Solution: A diffraction grating is an optical component consisting of a large number of closely spaced, parallel, equally sized slits (or rulings) used to produce high-resolution diffraction/interference patterns.
8. Mention two applications of Newton's Rings.
• Solution:

1. Measuring the wavelength of monochromatic light.
2. Testing the surface flatness/regularity of optical lenses and plates.

9. What is the wavelength of X-rays used in diffraction experiments?
• Solution: X-rays have very short wavelengths, typically of the order of  (), which matches the atomic spacing () inside crystal lattices.
10. Distinguish between a "Ray" and a "Wavefront."
• Solution:

• Wavefront: A surface connecting points having the same phase.
• Ray: A line drawn perpendicular to a wavefront showing the direction of wave propagation and energy transfer.

VERSION 2 — DETAILED SOLUTIONS

Q.1 — MCQs (Version 2)

1. (b) Perpendicular to wavefront
• Explanation: A ray represents the direction of energy propagation of a wave and is always oriented perpendicular to the wavefront at any point.
2. (c) Huygens' principle
• Explanation: Huygens' principle is specifically a geometric construction used to determine the shape and position of a wavefront at a future time given its present shape.
3. (b) Destructive
• Explanation: Path differences equal to half-integer multiples of wavelength, i.e., , result in waves arriving out of phase ( phase difference), causing cancellation (destructive interference). For , path difference .
4. (a)
• Explanation: The chapter text explicitly notes the use of  wavelength microwave equipment with a double-gap metal barrier to demonstrate microwave interference patterns.
5. (b) Half
• Explanation: Fringe spacing is given by . Because  is inversely proportional to slit distance , doubling  reduces  to half its original value.
6. (b) Intense coherent light
• Explanation: Lasers produce monochromatic, highly concentrated, and coherent light beams, making them convenient for producing stable, clear interference fringes.
7. (a) A denser medium
• Explanation: According to electromagnetic theory, when light reflects off an optically denser medium (higher index of refraction), it undergoes a phase shift of  ( radians or a half-cycle).
8. (c) Air
• Explanation: In the standard Newton's rings setup, the thin film responsible for interference is the air wedge formed between the curved convex lens surface and the flat glass plate.
9. (b) Same order as
• Explanation: The diffraction formula  implies . For significant deviation ( to be large), the grating spacing  must be comparable in size to the wavelength .
10. (b) Fraunhofer
• Explanation: As noted in Section 15.8, the first diffraction grating was constructed by Joseph von Fraunhofer using fine wires.
11. (c) 4
• Explanation: The order index  in the diffraction grating equation  directly designates the order number. The fourth-order line corresponds to .
12. (b) Atomic planes
• Explanation: Bragg reflection occurs when X-rays scatter off parallel atomic planes separated by distance  within a crystal lattice.
13. (b) Wider/Sharper than  lines
• Explanation: Increasing the number of slits/lines in a grating sharpens the maxima and increases angular dispersion/resolution of spectral lines.

Q.2 — Short Questions (Version 2)

1. Why are plane wavefronts used to study interference and diffraction in this chapter?
• Solution: Plane wavefronts consist of parallel rays traveling in a single direction. This simplifies mathematical analysis (like calculating path differences ) compared to diverging spherical wavefronts.
2. Explain why two independent lasers might not produce a steady interference pattern.
• Solution: Independent lasers emit light through independent atomic transitions, causing random, abrupt phase changes over time. Lacking a constant phase relationship, their interference pattern shifts too rapidly for human detection, yielding only uniform illumination.
3. What is the "envelope" of wavelets in Huygens' Principle?
• Solution: The envelope is the tangential surface bounding all the individual secondary spherical wavelets emitted by points on the original wavefront. This envelope defines the position of the new primary wavefront.
4. Under what condition is the formula  valid in Young's experiment?
• Solution: This approximation holds when the screen-to-slit distance  is much greater than the slit separation  (), meaning the angle  to the observed fringe on the screen is very small ().
5. Why is the central spot in Newton's rings dark?
• Solution: At the point of contact between the lens and glass plate, the film thickness , so path difference . However, light reflecting at the lower air-glass boundary undergoes a  () phase shift. This phase shift causes destructive interference, creating a central dark spot.
6. How does the fringe spacing change if the wavelength of light used is increased?
• Solution: From , fringe spacing  is directly proportional to wavelength . Increasing the wavelength causes the fringes to spread further apart (fringe width increases).
7. What are "rulings" or "lines" in a diffraction grating?
• Solution: Rulings or lines are the fine, equally spaced parallel scratches made on a glass or metal surface using a diamond tip, which act as opaque boundaries separating transparent slits.
8. Describe the diffraction of radio waves around obstacles.
• Solution: Radio waves have long wavelengths (meters to kilometers). Because their wavelengths are comparable to large obstacles like hills and buildings, radio waves readily bend (diffract) around them, allowing reception in geometrical shadows.
9. What is the value of "n" for the second-order maximum in Bragg’s equation?
• Solution: In Bragg's equation ,  represents the order of reflection. For the second-order maximum, .
10. If the distance between slits  is decreased, how does it affect the interference pattern?
• Solution: Since , decreasing  causes the fringe spacing  to increase, meaning the interference pattern spreads out wider across the screen.

VERSION 3 — DETAILED SOLUTIONS

Q.1 — MCQs (Version 3)

1. (b) Plane
• Explanation: Spherical wavefronts expanding from distant sources (like stars) have extremely large radii of curvature by the time they reach Earth, making small sections of them effectively flat (plane wavefronts).
2. (b) Secondary wavelets
• Explanation: Huygens' principle posits that every point on a primary wavefront acts as a fresh point source of secondary spherical wavelets.
3. (b) Coherent
• Explanation: Coherent sources are explicitly defined as sources maintaining a constant phase relationship over time.
4. (b)
• Explanation: Destructive interference occurs when the path difference between two overlapping waves is an odd half-integer multiple of the wavelength,  where
5. (b) Decrease
• Explanation: When submerged in water, the refractive index increases (), so the effective wavelength decreases (). Since , the fringe width  decreases.
6. (b)
• Explanation: Light reflected from the back surface travels down through the film of thickness  and back up, adding an additional distance of approximately  at near-normal incidence.
7. (b) Red end
• Explanation: For a grating, . Longer wavelengths deviate at larger angles . Because red light has a longer wavelength than blue/violet light, red light is deviated more.
8. (b) Reciprocal of number of lines per unit length
• Explanation: If a grating has  lines per unit length (e.g., lines/mm), the grating spacing  is given by .
9. (b) The crystal plane
• Explanation: In the Bragg diffraction geometry,  is the glancing angle measured relative to the atomic reflecting planes (not the surface normal).
10. (b) Sound has longer wavelength
• Explanation: Diffraction is pronounced when the obstacle/gap size is comparable to the wave's wavelength. Sound waves have wavelengths of centimeters to meters (matching everyday obstacles), whereas visible light has wavelengths of nanometers.
11. (b) Radius of rings and radius of curvature
• Explanation: Newton's rings geometry links the ring radius , lens radius of curvature , and order  to determine wavelength via relations like .
12. (b)
• Explanation: . Thus,  ().
13. (c) X-ray diffraction
• Explanation: As noted in Section 15.9, X-ray crystallography was pivotal in resolving biological macromolecular structures, including hemoglobin and the DNA double helix.

Q.2 — Short Questions (Version 3)

1. How did Young's experiment establish the Huygens wave theory over Newton's corpuscular theory?
• Solution: Newton's corpuscular theory treated light as particles, which predicted sharp shadows and could not account for constructive/destructive interference fringes. Young's double-slit experiment demonstrated interference fringes (bright and dark bands), proving light undergoes wave superposition as predicted by Huygens.
2. If the amplitudes of two interfering waves are different, how does it affect the pattern?
• Solution: Interference will still occur, but total cancellation at dark fringes will not happen. The minimum intensity will not be zero, resulting in reduced contrast between bright and dark fringes.
3. Explain why we see Haloes around street lamps on a foggy night.
• Solution: Water droplets in fog act as tiny obstacles/apertures. When light from a street lamp passes around these tiny droplets, it undergoes diffraction and interference, producing circular rings or haloes of light around the lamp.
4. Compare the interference of sound waves and light waves in terms of experimental arrangement.
• Solution:
• Sound Waves: Two separate loudspeakers can be connected to the same signal generator to act as coherent sources because electronic frequencies are easily locked. Slit spacings are large (meters).
• Light Waves: Two independent lamps cannot be electronically locked. Coherence must be achieved by splitting a single light beam using a double slit. Slit spacings must be very small (sub-millimeter).
5. Why is a single laser beam divided by two slits in YDSE instead of using two separate lasers?
• Solution: Independent lasers undergo random phase jumps, breaking coherence. Splitting a single wavefront with a double slit guarantees that any phase variation in the incident beam affects both slits simultaneously, keeping them coherent.
6. What happens to the fringes in YDSE if the screen is moved further away from the slits?
• Solution: The fringe spacing is given by . Because  is directly proportional to screen distance , moving the screen further away increases , making the fringes wider and more spread out.
7. How can the number of orders observed in a diffraction grating be increased?
• Solution: The condition for grating principal maxima is . Since , the maximum order is . To increase , one can:

1. Increase the grating spacing  (fewer lines per mm).
2. Decrease the wavelength  of light used.

8. Explain the significance of path difference in deriving the Bragg equation.
• Solution: When X-rays reflect off adjacent parallel crystal planes, the ray penetrating to the lower plane travels an extra distance equal to . Setting this extra path difference equal to an integer number of wavelengths () determines the condition for constructive interference (reinforcement).
9. Why is it necessary for the grating spacing  to be of the same order as  for diffraction?
• Solution: From , if , then , meaning diffraction angles are so tiny that all orders blend together around the straight-through direction. When ,  is large enough to spread the spectral orders out measurably.
10. What would be observed if white light falls on a double slit where one slit has a red filter and the other a blue filter?
• Solution: No interference pattern will be observed on the screen. The red filter passes only red wavelengths () while the blue filter passes blue wavelengths (). Because waves of different frequencies/wavelengths cannot maintain a constant phase difference, they cannot produce a stable interference pattern. Instead, a uniform colored illumination will appear.