Vardaan Institute - Physics Worksheet (Level 3)

Topic 1: Advanced Reflection & Spherical Mirrors

A point object is moving on the principal axis of a concave mirror of focal length $f$. If the object's velocity is $v_o$, derive an expression for the velocity of the image $v_i$ in terms of the magnification $m$.

A short linear object of length $L$ lies along the principal axis of a concave mirror of focal length $f$ at a distance $u$ from the pole. Prove that its longitudinal magnification is equal to the square of its transverse magnification (for a very short object).

Two concave mirrors of equal focal length $f$ are placed facing each other coaxially at a distance $d = 3f$. A point object is placed exactly midway between them. Find the distance of the second image formed by the system from the first mirror.

A point object is placed at a distance of $30\text{ cm}$ from a concave mirror of focal length $20\text{ cm}$. The mirror is cut into two equal halves horizontally, and the two halves are pulled apart such that the gap between them is $1\text{ cm}$. Find the distance between the two images formed.

Draw a ray diagram illustrating spherical aberration in a concave mirror of large aperture. Show marginal rays converging closer to the mirror than paraxial rays.

AI Prompt: Create a mathematically correct physics ray diagram showing spherical aberration in a concave mirror of large aperture. Show parallel incident rays. The paraxial rays (close to axis) converge at the true focus F. The marginal rays (far from axis) reflect and converge at a point closer to the mirror pole than F. Fully white background, landscape mode, precise clean lines.

File Name: Level3_Q5_SphericalAberration.png

A concave mirror of radius $R$ is kept horizontally with its opening facing upwards and it is filled with water (refractive index $\mu$). A point object is placed on the principal axis. Find the equivalent focal length of this optical system.

Derive Newton's formula $x_1 x_2 = f^2$ for a spherical mirror, where $x_1$ and $x_2$ are the distances of the object and the real image from the principal focus, respectively.

An object is placed in front of a convex mirror. A graph is plotted between $1/v$ on the y-axis and $1/u$ on the x-axis. Find the intercepts on both axes and the slope of the graph.

In an experiment to determine the focal length of a concave mirror using the $u-v$ method, the measured values are $u = 30.0 \pm 0.1\text{ cm}$ and $v = 60.0 \pm 0.2\text{ cm}$. Calculate the focal length with its maximum percentage error.

10.

A luminous point object is placed at a distance $u$ ($u > f$) on the principal axis of a concave mirror of focal length $f$ and aperture diameter $D$. Find the diameter of the reflected beam at a distance $x$ from the mirror ($x > v$).

Topic 2: Complex Refraction, Apparent Depth & TIR

11.

A light ray enters a medium whose refractive index varies with the y-coordinate as $n(y) = n_0 \sqrt{1+ky}$, where $n_0$ and $k$ are positive constants. If the ray is incident from air at the origin at an angle $\theta$ with the y-axis, derive the equation of the trajectory of the ray.

12.

A tank contains three immiscible liquids of refractive indices $\mu_1, \mu_2, \mu_3$ and depths $d_1, d_2, d_3$ respectively. Derive the expression for the apparent depth of the bottom of the tank when viewed normally from the top.

13.

A bird is flying vertically downwards towards the surface of water ($n = 4/3$) with a speed of $2\text{ m/s}$. A fish inside the water is rising vertically upwards along the same line with a speed of $1\text{ m/s}$. Find the velocity of the bird as observed by the fish.

14.

A ray of light enters a right-angled prism ($A=90^\circ, B=C=45^\circ$) made of glass of refractive index $1.5$ normally through one of its shorter faces. What should be the maximum refractive index of a liquid placed on the hypotenuse face so that the ray suffers total internal reflection?

15.

For an optical fiber with core refractive index $n_1$ and cladding refractive index $n_2$ ($n_1 > n_2$), placed in air, derive the expression for the Numerical Aperture (NA) and the maximum acceptance angle.

16.

Draw a ray diagram showing the acceptance angle cone for an optical fiber.

AI Prompt: Create a mathematically correct physics ray diagram showing the acceptance cone of an optical fiber. Show the core and cladding. Draw a 3D-like cone at the entrance face representing the maximum angle of incidence (acceptance angle) for which rays will undergo Total Internal Reflection inside the core. Label n_1, n_2, and the acceptance angle theta_a. Fully white background, landscape mode.

File Name: Level3_Q16_AcceptanceCone.png

17.

A light ray passes through a rectangular glass slab of thickness $t$ and refractive index $\mu$. For what angle of incidence will the lateral shift be maximum? Calculate this maximum lateral shift.

18.

A small air bubble is trapped inside a solid glass sphere ($n=1.5$) of radius $R$. The bubble is at a distance $R/2$ from the center. Find its apparent position when viewed from the diametrically opposite side of the sphere.

19.

A rectangular glass slab ($n=1.5$) of thickness $10\text{ cm}$ is silvered at its back surface. An object is placed $15\text{ cm}$ in front of the unsilvered face. Find the position of the final image.

20.

How does the critical angle for a given pair of media vary with the wavelength of incident light? Use Cauchy's formula to justify your answer.

Topic 3: Advanced Lens Systems & Combinations

21.

A biconvex lens of focal length $f$ and refractive index $\mu$ has one of its surfaces silvered. It behaves like a curved mirror. Derive the expression for the equivalent focal length of this silvered lens system.

22.

Two thin convex lenses of focal lengths $f_1$ and $f_2$ are placed coaxially separated by a distance $d$. Derive the expression for the equivalent focal length of the combination: $\frac{1}{F} = \frac{1}{f_1} + \frac{1}{f_2} - \frac{d}{f_1 f_2}$.

23.

Write the condition for achromatism for two thin lenses of dispersive powers $\omega_1$ and $\omega_2$ and focal lengths $f_1$ and $f_2$ placed in contact. Why must one lens be convex and the other concave?

24.

In the displacement method, a convex lens is moved between a fixed object and a screen. If $I_1$ and $I_2$ are the heights of the images obtained in the two conjugate positions, prove that the height of the object is $O = \sqrt{I_1 I_2}$.

25.

An equiconvex lens of focal length $f$ is cut into two identical plano-convex halves horizontally. One half is shifted upwards by a distance $a$ ($a \ll f$). A point source is placed at distance $u$ ($u > f$) on the principal axis of the original lens. Find the distance between the two images formed.

26.

Draw a ray diagram showing an achromatic doublet (convex crown glass and concave flint glass) forming a focused image free from longitudinal chromatic aberration.

AI Prompt: Create a mathematically correct physics ray diagram showing an achromatic doublet. Show a biconvex lens (Crown glass) cemented to a plano-concave lens (Flint glass). Draw parallel rays of white light incident on the doublet, passing through, and converging precisely at a single focal point, demonstrating the elimination of longitudinal chromatic aberration. Fully white background, landscape mode.

File Name: Level3_Q26_AchromaticDoublet.png

27.

A glass lens is immersed in a liquid. The temperature of the liquid is slowly increased. If the refractive index of the liquid decreases faster with temperature than that of glass, what will happen to the focal length of the lens over time?

28.

Two identical concavo-convex lenses (watch glasses) of refractive index $1.5$ have radii of curvature $10\text{ cm}$ and $20\text{ cm}$. They are placed with their convex faces in contact, and the space between them is filled with water ($n=4/3$). Find the focal length of the system.

29.

A point object is moving towards a convex lens of focal length $f$ with a constant velocity $v_o$ along the principal axis. Find the velocity of the image as a function of the object distance $u$.

30.

A biconvex thick lens of thickness $t$ and refractive index $n$ has radii of curvature $R_1$ and $R_2$. Write the generalized Lens Maker's formula for the effective focal length of this thick lens.

Topic 4: Prism, Dispersion & Spectrometry

31.

A ray of light is incident on one face of a prism of refracting angle $A$ and refractive index $\mu$. Derive the condition for the ray to suffer total internal reflection at the second face (no emergence) irrespective of the angle of incidence.

32.

Two thin prisms of angles $A_1$ and $A_2$, and refractive indices $\mu_1$ and $\mu_2$, are combined to produce dispersion without deviation. Derive the ratio $A_1/A_2$ and the expression for the net angular dispersion.

33.

A thin prism of angle $A=6^\circ$ has a refractive index $1.5$. One of its refracting faces is silvered. A ray of light incident at an angle $i$ on the unsilvered face retraces its path after reflection from the silvered face. Find the angle $i$.

34.

Draw a ray diagram illustrating the combination of two prisms of different materials to achieve dispersion without deviation (Direct Vision Spectroscope principle).

AI Prompt: Create a mathematically correct physics ray diagram showing two thin prisms inverted relative to each other (e.g., Crown glass followed by Flint glass). A white light beam enters the first prism, disperses, then enters the second prism. The second prism cancels the mean deviation so the emergent yellow ray is parallel to the incident ray, but the red and violet rays remain angularly separated (dispersion without deviation). Fully white background, landscape mode.

File Name: Level3_Q34_DispersionNoDeviation.png

35.

Derive an expression for the maximum angle of deviation $\delta_{max}$ for a ray passing through a prism of angle $A$ and refractive index $\mu$. Under what incidence condition does this occur?

36.

Plot the variation of the angle of minimum deviation ($\delta_m$) with the wavelength ($\lambda$) of incident light for a given glass prism.

37.

A ray passes symmetrically through a glass prism of angle $A$ such that the angle of incidence is equal to $3/4$ of the angle of the prism. Calculate the refractive index of the prism material in terms of $A$.

38.

A hollow equilateral glass prism is filled with a liquid. The angle of minimum deviation is found to be $30^\circ$. What is the refractive index of the liquid? (Assume the glass walls are infinitely thin).

39.

Using Cauchy's dispersion formula $n(\lambda) = a + b/\lambda^2$, explain why the dispersive power of a material is not a single constant but varies slightly across different regions of the spectrum.

40.

In a spectrometer experiment, the angle of the prism is measured as $60^\circ$ and the angle of minimum deviation for sodium light is $39^\circ 30'$. Calculate the refractive index of the prism material. ($\sin 49^\circ 45' \approx 0.763$)

Topic 5: Optical Instruments & Resolving Power

41.

Define the resolving power of a microscope. Write its formula and explain how Numerical Aperture (NA) affects it.

42.

Define the resolving power of an astronomical telescope. State the Rayleigh criterion for resolution and derive the limit of resolution $\Delta\theta = 1.22\lambda / D$.

43.

Draw a labelled ray diagram of a Galilean telescope in normal adjustment.

AI Prompt: Create a mathematically correct physics ray diagram of a Galilean telescope in normal adjustment. Show a convex objective lens and a concave eyepiece lens. Parallel rays from infinity enter the objective, converge towards its focus, but are intercepted by the concave eyepiece placed before the focal point. The rays exit the concave eyepiece as parallel rays. The background of the whole image should be fully white. It should be in landscape mode. High quality, precise clean lines.

File Name: Level3_Q43_GalileanTelescope.png

44.

A compound microscope has a magnifying power $M_1$ when the final image is at infinity, and $M_2$ when the final image is at the least distance of distinct vision $D$. Find the ratio $M_2/M_1$ in terms of $D$ and $f_e$.

45.

A terrestrial telescope has an objective of focal length $f_o$, an erecting lens of focal length $f$, and an eyepiece of focal length $f_e$. Write the expressions for its magnifying power and tube length in normal adjustment.

46.

An astronomical telescope has a magnifying power of $10$ in normal adjustment. If the length of the tube is $44\text{ cm}$, find the focal lengths of the objective and the eyepiece.

47.

A person suffers from both myopia and hypermetropia. His near point is $50\text{ cm}$ and far point is $300\text{ cm}$. Calculate the power of the lenses required for the upper and lower parts of his bifocal spectacles to see distant objects and read a book at $25\text{ cm}$, respectively.

48.

Derive the magnifying power of an astronomical telescope when the object being viewed is at a finite distance $u_o$ (not infinity) and the final image is at $D$.

49.

What is the 'exit pupil' of a telescope? Prove that the magnifying power of a telescope in normal adjustment is equal to the ratio of the diameter of the objective lens to the diameter of the exit pupil.

50.

To overcome chromatic and spherical aberrations, modern large telescopes do not use a single convex lens as an objective. Explain how an achromatic doublet minimizes chromatic aberration, and how parabolic mirrors eliminate spherical aberration.

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