Vardaan Institute - Wave Optics Worksheet (Level 3)

Part I: Huygens Principle & Wave Theory Challenges

Using Huygens’ construction, derive the expression for the intensity of light reflected from a moving mirror. How does the Doppler effect manifest in the reflected wavefronts?

A plane wavefront is incident at an angle $\alpha$ on a glass prism of refracting angle $A$. Use Huygens' principle to derive the expression for the total deviation suffered by the wavefront.

Show that a spherical wavefront incident on a plane interface separating two media transforms into a non-spherical wavefront (hyperboloid/ellipsoid) upon refraction.

AI Prompt: Create a mathematically accurate physics diagram showing a spherical wavefront originating from a point source in a denser medium, being refracted at a plane interface into a rarer medium. Show that the refracted wavefront is no longer perfectly spherical but takes a flattened shape (hyperboloid of revolution). Background fully white. Landscape mode. High quality lines.

File Name: Level3_Q3_RefractedSphericalWavefront.png

Assume the refractive index of a medium varies as $n(y) = n_0 e^{-ky}$. Trace the path of a plane wavefront entering this medium obliquely. What happens to the wavefront curvature?

The speed of light in a certain liquid is $v$. A point source is placed at depth $h$. Determine the radius of the spherical wavefront at the instant it just begins to emerge into air. Use this to calculate the angle of the "cone of light".

Part II: Interference & YDSE Challenger Problems

In a YDSE, the slits are illuminated by two monochromatic wavelengths $\lambda_1 = 400 \text{ nm}$ and $\lambda_2 = 560 \text{ nm}$. Find the distance from the central maximum where the $n^{th}$ maximum of $\lambda_1$ coincides with the $m^{th}$ maximum of $\lambda_2$ for the first time.

A thin glass slab of thickness $t = 2 \mu\text{m}$ and refractive index $1.5$ is placed in front of one of the slits in YDSE. If the wavelength of light used is $500 \text{ nm}$, by how many fringe widths does the central maximum shift?

In YDSE, if the intensity of one slit is twice that of the other, calculate the ratio of $I_{max}$ to $I_{min}$ in the interference pattern. Also, find the percentage contrast of the fringes.

Consider a YDSE setup where the screen is moved away from the slits with a constant velocity $v$. Derive the rate of change of fringe width as a function of time.

10.

What happens to the interference pattern if the monochromatic source is replaced by a source of white light? Discuss the color of the 1st and 10th order fringes.

11.

Find the minimum thickness of a soap film ($n=1.33$) that will result in constructive interference for reflected light of wavelength $600 \text{ nm}$ incident normally.

12.

Draw a graph of the resultant intensity $I$ versus the phase difference $\phi$ for three coherent sources of equal amplitude $a$ and constant phase difference between successive sources.

AI Prompt: Create a mathematically accurate physics graph showing the intensity distribution for the interference of three coherent sources. The graph should show a primary peak at multiples of 2*pi, with one smaller secondary peak between each pair of primary peaks. Label axes Intensity I and Phase difference phi. Background white. Landscape mode.

File Name: Level3_Q12_ThreeSourceInterference.png

13.

In YDSE, the fringe width is $0.1 \text{ mm}$. If the distance between the slits is reduced to half and the distance of the screen is doubled, what is the new fringe width?

14.

Calculate the distance between the two slits in YDSE if the 5th dark fringe is formed at an angle of $0.01 \text{ radians}$ for light of wavelength $589 \text{ nm}$.

15.

A point $P$ on the screen is at a distance $y$ from the central maximum. If the path difference at $P$ is $\lambda/3$, what is the ratio of intensity at $P$ to the intensity at the central maximum?

Part III: Diffraction & Resolving Power

16.

For a single slit of width $a$, the first minimum of the diffraction pattern is at $30^\circ$. If the slit is illuminated by light of wavelength $650 \text{ nm}$, find the value of $a$.

17.

Compare the resolving power of a microscope using light of wavelength $400 \text{ nm}$ in air with its resolving power when oil ($n=1.5$) is used as the immersion medium.

18.

An astronomical telescope has an objective diameter of $2.5 \text{ m}$. What is its angular resolution for light of wavelength $500 \text{ nm}$? If two stars are at a distance of $10^{15} \text{ m}$, what is the minimum distance between them to be resolved?

19.

Draw a ray diagram for the diffraction of light by a circular aperture. Explain the concept of Airy's disc and how it limits the resolving power of a lens.

AI Prompt: Create a mathematically accurate physics diagram of Airy's disc. Show the diffraction pattern of a circular aperture: a bright central disc surrounded by concentric dark and bright rings of rapidly decreasing intensity. Label the central disc as Airy's Disc. Background fully white. Landscape mode.

File Name: Level3_Q19_AiryDiscDiffraction.png

20.

Derive the expression for the angular width of the central maximum in a single slit diffraction experiment and show that it is independent of the distance between the slit and the screen.

21.

The intensity of the first secondary maximum in single slit diffraction is approximately $4.5\%$ of the central maximum intensity. Provide a mathematical justification for this relative decrease.

22.

A slit of width $0.1 \text{ mm}$ is used in Fraunhofer diffraction. If the distance of the screen is $1 \text{ m}$ and wavelength is $500 \text{ nm}$, calculate the linear distance between the first and second minima.

23.

Calculate the limit of resolution of a microscope if the numerical aperture is $0.12$ and the wavelength used is $600 \text{ nm}$.

Part IV: Polarization & Brewster’s Law

24.

State and prove Brewster’s Law using Snell’s law. Explain why the reflected ray is perfectly plane polarized only at this specific angle.

25.

Unpolarized light of intensity $I_0$ is incident on a series of $N$ Polaroids. The transmission axis of each Polaroid is rotated by an angle $\theta$ with respect to the previous one. Derive the final transmitted intensity.

26.

An unpolarized light beam is incident on a glass surface at an angle $i$. If the reflected and refracted rays are found to be perpendicular, find the relationship between the refractive index and the angle $i$.

27.

A polarizer and an analyzer are oriented so that the maximum amount of light is transmitted. To what fraction of its maximum value does the transmitted intensity fall when the analyzer is rotated through (i) $30^\circ$, (ii) $45^\circ$, (iii) $60^\circ$, and (iv) $90^\circ$?

28.

Draw a diagram to illustrate the method of producing plane polarized light by double refraction in a Nicol prism.

AI Prompt: Create a mathematically accurate physics ray diagram showing a Nicol Prism. Show an incident unpolarized ray splitting into an Ordinary (O) ray and Extraordinary (E) ray. Show the O-ray undergoing Total Internal Reflection at the Canada Balsam layer, while the E-ray passes through. Background white. Landscape mode.

File Name: Level3_Q28_NicolPrismPolarization.png

29.

Calculate the angle between two Polaroids if the intensity of transmitted light is $1/3$ of the intensity of polarized light incident on the first Polaroid.

30.

If the refractive index of a medium is $1.5$, find the angle of refraction for light incident at the polarizing angle.

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