Q: Q4. Two sources of light are said to be coherent if waves produced by them have the same\n(a) amplitude\n(b) wavelength\n(c) speed\n(d) phase, or constant phase difference

The correct option is (d).\n\n**Explanation:**\nFor two sources of light to be coherent, the waves produced by them must satisfy two conditions:\n1. **Same frequency (and thus same wavelength):** This ensures that the phase relationship between the waves does not change over time due to different oscillation rates.\n2. **Constant phase difference:** The phase difference between the waves emitted by the two sources must remain constant over time. This constant phase difference can be zero.\n\nLet's analyze the given options:\n(a) **Amplitude:** While having similar amplitudes is desirable for observing distinct interference patterns, it is not a strict condition for coherence itself. Coherent sources can have different amplitudes.\n(b) **Wavelength:** Having the same wavelength is a necessary condition for coherence, as it implies the same frequency. However, it's not the *only* condition.\n(c) **Speed:** The speed of light in a given medium is a property of the medium, not the source. All light waves travel at the same speed in a uniform medium, so this is not a distinguishing factor for coherence.\n(d) **Phase, or constant phase difference:** This option encompasses both the requirement of same frequency (leading to a constant phase relationship) and the explicit condition of a constant phase difference. This is the most complete and accurate definition of coherence among the choices.

Question 1

Q1. Monochromatic light of wavelength 589 nm is incident from air on a water surface. What are the wavelength, frequency and speed of (a) reflected, and (b) refracted light? Refractive

Accepted Answer

The wavelength, frequency, and speed of the light are calculated based on whether it is reflected or refracted.

(a) For reflected light:
   - Wavelength: 589 nm
   - Frequency: 5.093 × 10¹⁴ Hz
   - Speed: 3.0 × 10⁸ m/s

(b) For refracted light:
   - Wavelength: 442.8 nm
   - Frequency: 5.093 × 10¹⁴ Hz
   - Speed: 2.255 × 10⁸ m/s

Question 2

Q2. What is the shape of the wavefront in each of the following cases:
(a) Light diverging from a point source.
(b) Light emerging out of a convex lens when a point source is placed

Accepted Answer

The shape of the wavefront depends on the geometry of the source and the optical elements involved:

(a) **Light diverging from a point source:** The wavefronts are **spherical**.
    *   Explanation: A point source emits light uniformly in all directions. At any given instant, all points equidistant from the source will be in the same phase, forming a spherical surface.

(b) **Light emerging out of a convex lens when a point source is placed at its focus:** The wavefronts are **plane**.
    *   Explanation: When a point source is placed at the principal focus of a convex lens, the rays of light, after refraction through the lens, become parallel to the principal axis. A collection of parallel rays corresponds to a plane wavefront.

(c) **The portion of the wavefront of light from a distant star intercepted by the Earth:** The wavefronts are **plane**.
    *   Explanation: For a very distant source like a star, the light rays reaching the Earth can be considered practically parallel. Therefore, the small portion of the spherical wavefront from the distant star that reaches Earth appears almost flat, forming a plane wavefront.

Question 3

Q3. The refractive index of glass is 1.5. What is the speed of light in glass? (Speed of light in vacuum is 3.0 × 10⁸ m s⁻¹)

Accepted Answer

The speed of light in glass is 2.0 × 10⁸ m/s.

Question 4

Q4. Two sources of light are said to be coherent if waves produced by them have the same
(a) amplitude
(b) wavelength
(c) speed
(d) phase, or constant phase difference

Accepted Answer

The correct option is (d).

**Explanation:**
For two sources of light to be coherent, the waves produced by them must satisfy two conditions:
1.  **Same frequency (and thus same wavelength):** This ensures that the phase relationship between the waves does not change over time due to different oscillation rates.
2.  **Constant phase difference:** The phase difference between the waves emitted by the two sources must remain constant over time. This constant phase difference can be zero.

Let's analyze the given options:
(a) **Amplitude:** While having similar amplitudes is desirable for observing distinct interference patterns, it is not a strict condition for coherence itself. Coherent sources can have different amplitudes.
(b) **Wavelength:** Having the same wavelength is a necessary condition for coherence, as it implies the same frequency. However, it's not the *only* condition.
(c) **Speed:** The speed of light in a given medium is a property of the medium, not the source. All light waves travel at the same speed in a uniform medium, so this is not a distinguishing factor for coherence.
(d) **Phase, or constant phase difference:** This option encompasses both the requirement of same frequency (leading to a constant phase relationship) and the explicit condition of a constant phase difference. This is the most complete and accurate definition of coherence among the choices.

Question 5

Q5. The intensity of light from a point source is inversely proportional to the square of the distance from the source.
(a) How does the amplitude of the electric field from a point s

Accepted Answer

The relationship between intensity and the amplitude of the electric field, combined with how energy spreads from different source geometries, determines the dependence of the electric field amplitude on distance.

(a) **For a point source:**
    The amplitude of the electric field (E₀) from a point source depends on the distance, r, as **1/r**.

(b) **For a long cylindrical source:**
    The amplitude of the electric field (E₀) from a long cylindrical source depends on the distance, r, as **1/√r**.

Question 6

Q6. In a Young’s double-slit experiment, the slits are separated by 0.28 mm and the screen is placed 1.4 m away. The distance between the central bright fringe and the fourth bright fr

Accepted Answer

The wavelength of the light used is 600 nm. The distance of the fifth dark fringe from the central maximum is 1.35 cm.

Question 7

Q7. In Young’s double-slit experiment using monochromatic light of wavelength λ, the intensity of light at a point on the screen where path difference is λ, is K units. What is the int

Accepted Answer

The intensity of light at a point where the path difference is λ/3 is K/4 units.

Question 8

Q8. A beam of light consisting of two wavelengths, 650 nm and 520 nm, is used to obtain interference fringes in a Young’s double-slit experiment.
(a) Find the distance of the third br

Accepted Answer

(a) The distance of the third bright fringe for λ = 650 nm is 1.17 mm.
(b) The least distance from the central maximum where the bright fringes coincide is 1.56 mm.

Question 9

Q9. In a double-slit experiment, the angular width of a fringe is found to be 0.2° on a screen placed 1 m away. The wavelength of light used is 600 nm. What will be the angular width o

Accepted Answer

The new angular width of the fringe when the apparatus is immersed in water will be 0.15°.

Question 10

Q10. What is the Brewster angle for air to glass transition? (Refractive index of glass = 1.5).

Accepted Answer

The Brewster angle for the air to glass transition is approximately 56.3°.

Question 11

Q11. In Young’s double-slit experiment, the slits are separated by 0.28 mm and the screen is placed 1.4 m away. The distance between the central bright fringe and the fourth bright frin

Accepted Answer

The wavelength of light used in the experiment is 600 nm.

Question 12

Q12. In a single slit diffraction experiment, the width of the slit is made double the original width. How does this affect the size and intensity of the central diffraction band?

Accepted Answer

In a single slit diffraction experiment, when the width of the slit (a) is made double the original width:

1.  **Effect on the size (angular width) of the central maximum:**
    The angular width of the central maximum in a single-slit diffraction pattern is given by $2	heta = \frac{2\lambda}{a}$, where $\lambda$ is the wavelength of light and $a$ is the slit width.
    If the slit width 'a' is doubled, the angular width of the central maximum becomes $2	heta' = \frac{2\lambda}{2a} = \frac{\lambda}{a}$.
    Thus, the angular width of the central maximum is halved. Consequently, the linear width of the central maximum on the screen (which is proportional to the angular width) also becomes half of its original size.

2.  **Effect on the intensity of the central maximum:**
    The intensity of the central maximum is directly proportional to the square of the slit width ($I \propto a^2$).
    If the slit width 'a' is doubled, the new intensity $I'$ will be proportional to $(2a)^2 = 4a^2$.
    Therefore, the intensity of the central maximum becomes four times the original intensity. This is because doubling the slit width allows four times more light energy to pass through the slit per unit time, and this energy is now concentrated into a central maximum that is half the angular size.

Question 13

Q13. A parallel beam of light of wavelength 500 nm falls on a narrow slit and the resulting diffraction pattern is observed on a screen 1 m away. It is observed that the first minimum i

Accepted Answer

The width of the slit is 0.2 mm.

Question 14

Q14. Two slits are made 1 mm apart and the screen is placed 1 m away. What is the fringe width when blue-green light of wavelength 500 nm is used?

Accepted Answer

The fringe width is 0.5 mm.

Question 15

Q15. In a Young's double-slit experiment, the slits are separated by 0.28 mm and the screen is placed 1.4 m away. The distance between the central bright fringe and the fourth bright fr

Accepted Answer

The wavelength of light used in the experiment is 600 nm.

Question 16

Q16. In a single slit diffraction experiment, the width of the slit is made double the original width. How does this affect the size and intensity of the central diffraction band?

Accepted Answer

In a single-slit diffraction experiment, the angular width of the central maximum is given by 2λ/a, where λ is the wavelength of light and 'a' is the width of the slit.

1.  **Effect on Size (Angular Width) of the Central Band:**
    When the width of the slit 'a' is doubled (let the new width be a' = 2a), the new angular width of the central maximum becomes:
    Angular width' = 2λ/a' = 2λ/(2a) = λ/a
    This means the angular width of the central maximum becomes half of its original value. Consequently, the linear size (width) of the central diffraction band on the screen also becomes half.

2.  **Effect on Intensity of the Central Band:**
    The intensity of the central maximum in a single-slit diffraction pattern is proportional to the square of the amplitude of the diffracted light. The amplitude of the diffracted light is directly proportional to the slit width 'a'. Therefore, the intensity (I) is proportional to a².
    When the slit width 'a' is doubled (a' = 2a), the new intensity (I') will be proportional to (2a)² = 4a².
    Thus, the intensity of the central maximum increases fourfold compared to the original intensity.

**In summary, doubling the slit width leads to:**
*   **Size of central band:** Halved.
*   **Intensity of central band:** Increased fourfold.

Question 17

Q17. In what way is the phenomenon of diffraction from a single slit different from interference in a double-slit experiment?

Accepted Answer

The phenomena of interference from a double-slit experiment and diffraction from a single slit, while both involving the superposition of waves and demonstrating the wave nature of light, exhibit distinct characteristics. Here's a comparison of their key differences:

### **Comparison between Interference and Diffraction**

| Feature                 | Interference (Young's Double Slit Experiment)                               | Diffraction (Single Slit Experiment)                                    |
| :---------------------- | :-------------------------------------------------------------------------- | :---------------------------------------------------------------------- |
| **Origin**              | Arises from the superposition of waves originating from two coherent sources (two slits). | Arises from the superposition of secondary wavelets originating from different parts of the same wavefront passing through a single slit. |
| **Number of Sources**   | Two coherent sources (slits) act as point sources.                          | Effectively, an infinite number of secondary wavelets originating from different points within the single slit act as coherent sources. |
| **Fringe Width**        | All bright and dark fringes (maxima and minima) are of equal width (β = λD/d). | The central bright maximum is twice as wide as the secondary maxima (angular width = 2λ/a). The width of secondary maxima decreases with increasing order. |
| **Intensity Distribution** | All bright fringes have approximately the same intensity. The intensity is uniform across all principal maxima. | The central maximum is the brightest, and its intensity is significantly higher than the secondary maxima. The intensity of secondary maxima decreases rapidly as we move away from the center. |
| **Nature of Fringes**   | Bright fringes are equally spaced and of equal intensity. Dark fringes are perfectly dark (zero intensity) if the amplitudes of interfering waves are equal. | Bright fringes (maxima) are not equally intense. Dark fringes (minima) are not perfectly dark, especially for higher orders, as some light still reaches these regions. |
| **Conditions for Maxima/Minima** | For constructive interference (bright fringes): path difference = nλ. For destructive interference (dark fringes): path difference = (n+1/2)λ. | For destructive interference (minima): a sinθ = nλ (where 'a' is slit width). For constructive interference (secondary maxima): a sinθ ≈ (n+1/2)λ. |
| **Dependence on Slit Width** | Fringe width (β) depends on the distance between the two slits (d), not the individual slit width (assuming narrow slits). | The width of the central maximum is inversely proportional to the slit width (a). A narrower slit produces a wider central maximum. |

**Conclusion:**
While both phenomena are manifestations of the superposition principle and the wave nature of light, interference involves the superposition of waves from distinct coherent sources, leading to a pattern of equally spaced and equally intense fringes. Diffraction, on the other hand, involves the superposition of wavelets from different parts of a single wavefront, resulting in a central maximum that is significantly wider and brighter than the rapidly diminishing secondary maxima.

Question 18

Q18. A parallel beam of light of wavelength 500 nm falls on a narrow slit and the resulting diffraction pattern is observed on a screen 1 m away. It is observed that the first minimum i

Accepted Answer

The width of the slit is calculated using the formula for the position of the first minimum in a single-slit diffraction pattern. Given the wavelength of light, the distance to the screen, and the distance of the first minimum from the center, we can find the slit width. The calculated width of the slit is 0.2 mm.

Question 19

Q19. Answer the following questions:
(a) When a tiny circular obstacle is placed in the path of light from a distant source, a bright spot is seen at the centre of the shadow of the ob

Accepted Answer

Here are the explanations for the given questions:

### (a) Bright Spot at the Centre of the Shadow of a Tiny Circular Obstacle

This phenomenon is known as **Poisson's spot** or **Arago's spot**, and it is a classic demonstration of the wave nature of light, specifically **Fresnel diffraction**. According to Huygens' principle, every point on a wavefront acts as a source of secondary wavelets. When a tiny circular obstacle blocks a parallel beam of light, the light waves diffract around the edges of the obstacle. The wavelets originating from different points on the circumference of the obstacle's shadow boundary travel different path lengths to reach the exact center of the shadow. Due to the circular symmetry, all the wavelets originating from the edge of the obstacle's shadow travel equal path lengths to reach the center of the shadow. Consequently, these wavelets arrive at the center of the shadow in phase and interfere constructively. This constructive interference results in a bright spot being observed at the center of the geometric shadow, surrounded by concentric diffraction rings.

### (b) Inability to See Each Other but Ability to Converse Easily

Both light and sound are waves and exhibit diffraction (bending around obstacles). However, the extent of diffraction depends critically on the ratio of the wavelength (λ) of the wave to the size of the obstacle or aperture (a). Significant diffraction occurs when the wavelength is comparable to or larger than the obstacle/aperture size (λ ≈ a or λ > a).

*   **Sound Waves:** The wavelength of audible sound waves typically ranges from a few centimeters to several meters (e.g., for a frequency of 340 Hz, λ = v/f = 340 m/s / 340 Hz = 1 m). The 7 m partition wall is comparable to or smaller than the wavelength of many sound waves. Therefore, sound waves diffract significantly around the wall, allowing them to bend and reach the other side, enabling the students to converse easily.
*   **Light Waves:** The wavelength of visible light is very small, ranging from approximately 400 nm to 700 nm (4 × 10⁻⁷ m to 7 × 10⁻⁷ m). The 7 m partition wall is vastly larger than the wavelength of light (a >> λ). Consequently, light waves undergo negligible diffraction around the wall. They essentially travel in straight lines, and the wall casts a sharp shadow, preventing the students from seeing each other.

In summary, the difference in observable diffraction effects is due to the enormous difference in the wavelengths of sound and light relative to the size of common obstacles like a partition wall.

### (c) Use of Ray Optics Approximation Even When Diffraction is Significant

Ray optics (or geometrical optics) is an approximation of wave optics. It assumes that light travels in straight lines (rays) and that diffraction effects are negligible. This approximation is valid when the size of the obstacles or apertures (a) in the path of light is much larger than the wavelength (λ) of light (a >> λ). In such cases, the bending of light around edges is so small that it can be ignored, and light can be considered to propagate along straight lines.

However, the question states, 'even in situations where diffraction is significant'. This phrasing is a bit contradictory to the premise of ray optics. If diffraction is truly significant (i.e., a is comparable to λ), then ray optics is *not* a valid approximation, and wave optics must be used. The statement 'How then is the ray optics approximation often used to analyse the propagation of light even in situations where diffraction is significant?' might be interpreted in two ways:

1.  **Misconception in the premise:** Ray optics is *not* used when diffraction is significant. It is precisely when diffraction is *not* significant that ray optics is applicable.
2.  **Contextual interpretation:** Perhaps it refers to situations where diffraction *occurs* but its *effects* on the overall light path are still small enough for ray optics to provide a useful, albeit approximate, description for certain macroscopic phenomena. For example, when designing lenses or mirrors, we primarily use ray optics, even though diffraction ultimately limits the resolution. The 'significance' of diffraction here might refer to its fundamental presence rather than its dominant observable effect. However, for a precise analysis where diffraction patterns are the primary concern (e.g., resolving power of telescopes, grating spectra), ray optics fails completely.

Therefore, the correct understanding is that **ray optics is an approximation that breaks down when diffraction effects become significant.** It is used *only* when diffraction is negligible (a >> λ). When diffraction is significant (a ≈ λ), wave optics is indispensable. The question's premise might be slightly misleading in implying ray optics is used when diffraction is significant; rather, it's used when diffraction is *not* significant enough to alter the macroscopic path of light.

### (d) Effect of the Width of the Source Slit on the Width of the Central Maximum in a Single Slit Diffraction Pattern

In the ideal single-slit diffraction pattern, we assume a point source of light, which produces a clear and distinct diffraction pattern with a well-defined central maximum. However, if the source slit has a finite width (i.e., it's an extended source), it can be considered as a collection of many independent point sources, each producing its own diffraction pattern.

*   **Superposition of Patterns:** Each point on the extended source produces its own diffraction pattern, slightly shifted relative to the others. These individual patterns superimpose on the screen.
*   **Blurring and Loss of Distinctness:** If the source slit is too wide, the individual diffraction patterns from different parts of the source will overlap significantly. This superposition leads to a blurring of the overall diffraction pattern. The minima and maxima become less distinct, and the intensity distribution becomes flatter.
*   **Effect on Central Maximum:** As the source slit width increases, the central maximum from one part of the source starts to overlap with the first minimum or even other parts of the central maximum from another part of the source. This effectively causes the central maximum to appear broader and less intense at its peak, and its edges become less sharp. In essence, the distinctness of the central maximum (and the entire diffraction pattern) decreases, and it appears to 'widen' due to the blurring and overlapping of multiple patterns. If the source is extremely wide, the diffraction pattern may be completely washed out, resulting in uniform illumination.

Therefore, an increase in the width of the source slit leads to a blurring and effective broadening of the central maximum, reducing its peak intensity and overall distinctness.

Question 20

Q20. An unpolarised light is incident on a plane glass surface. What is the angle of incidence at which the reflected and refracted rays are mutually perpendicular?

Accepted Answer

When unpolarised light is incident on a plane glass surface, the reflected and refracted rays are mutually perpendicular at a specific angle of incidence known as Brewster's angle (or polarizing angle). At this angle, the reflected light is completely plane-polarised. The angle of incidence at which this occurs is given by the condition that the sum of the angle of incidence and the angle of refraction is 90 degrees. Using Snell's law, this condition leads to Brewster's law, tan(i_p) = n, where n is the refractive index of the medium. Without a specific value for the refractive index of glass, the angle of incidence is given by arctan(n). Assuming a typical refractive index for glass (n ≈ 1.5), the angle of incidence would be approximately 56.3 degrees.

Question 21

Q21. What is the Brewster's angle for a transparent medium of refractive index 1.5? (Note: The angle is measured with respect to the normal to the interface.)

Accepted Answer

Brewster's angle (or polarizing angle) is the angle of incidence at which light reflected from a transparent medium is completely plane-polarised. It is given by Brewster's Law, which states that the tangent of Brewster's angle is equal to the refractive index of the medium. For a medium with a refractive index of 1.5, Brewster's angle is approximately 56.3 degrees.

Chapter 10: Wave Optics