UPSC Physics Optional PYQs (CSE & IFoS)PYQs

(ii) The earth may be modeled as a spherical capacitor with a = 6\cdot 5 \times 10^6\text{ m} and b \rightarrow \infty. Determine C, if the medium surrounding the earth is free space.

A spherical capacitor is made of concentric conductors of radii a and b (b > a). The total charge on the inner sphere of radius a is Q. (i) Derive an expression for the capacitance C.

Consider a long, line charge with charge density \rho_l = 10^{-6}\text{ coulomb/m}. Find the force acting on a dust particle carrying -10^{-9}\text{ coulomb}, 1\text{ metre} away from the line charge in free space.

A radiation gas of temperature T fills a cavity of volume V. The system expands adiabatically and reversibly to a volume equal to 8V. By what factor does the temperature change?

Consider the L\text{-}C\text{-}R circuit shown below :

\begin{align*} R &= 0\cdot 1\ \Omega \\ L &= 1\text{ nH} \\ C &= 1\text{ nF} \end{align*}

(i) Determine its resonance frequency f_0.

For two isotropic media with \mu_1 \neq \mu_2 and \varepsilon_1 \neq \varepsilon_2, find an expression for the Brewster angle \theta_b for parallel polarization.

If the magnetic moment of proton is 2.793\ \mu_N calculate, giving necessary steps, the radio frequency at which nuclear magnetic resonance occurs in water kept in a uniform magnetic field of 2.4\text{ T}.

Consider a perfectly conducting half-space as shown below :

A uniform plane wave given by

\begin{align*} \vec{E}^i &= \hat{x} E_0 e^{-j k z} \\ \vec{H}^i &= \hat{y} \frac{E_0}{\eta_0} e^{-j k z} \quad \eta_0 = 120\pi\text{ ohm} \end{align*}

is incident normally on the boundary. Write down the expressions for the reflected electric and magnetic fields.

The electric field for a uniform plane wave in free space is given by \vec{E} = (\hat{x} + \hat{y})10 e^{j 10 z}. Determine the corresponding magnetic field vector.

The following inputs are given :

\begin{align*} T_0 &= 5500\text{ K} && \text{(Sun surface temperature)}, \\ R &= 7 \times 10^{10}\text{ cm} && \text{(Sun radius)}, \\ r &= 6\cdot 4 \times 10^8\text{ cm} && \text{(Earth radius)}, \\ D &= 1\cdot 5 \times 10^{13}\text{ cm} && \text{(Sun--Earth distance)}. \end{align*}

Assume that the earth and the sun both absorb all electromagnetic radiations incident on them, and that the earth is at a constant temperature T over the day-night cycle. Calculate T.

Show that when the temperature T of a radiating object is not too different from the surrounding temperature T_0, the object obeys Newton's law of cooling.

A long wire of radius a carries I amperes of current. The magnetic field surrounding it is given by H_\phi = \dfrac{I}{2\pi\rho} for \rho > a. Obtain expressions for (i) the magnetic energy stored per unit length in the region b \ge \rho \ge a and (ii) the equivalent inductance L per unit length.

The electromagnetic field inside a device is given by \vec{E} = \hat{y} E_0 \sin(k_x x) e^{-j k_z z}, \quad\text{where } k_x = \frac{n\pi}{a} \vec{H} = E_0 [\hat{x} \frac{-k_z}{\omega\mu} \sin(k_x x) + \hat{z} \frac{j k_x}{\omega\mu} \cos(k_x x)] e^{-j k_z z} Obtain an expression for the z-component of time-averaged Poynting vector \vec{S}.

A point charge +q is located near the corner of a horizontal and a vertical plate as shown below :

Obtain an expression for the electrostatic potential \phi_P using the image method.

Consider a system of two thin lenses of focal lengths +15\text{ cm} and -20\text{ cm} separated by a distance of 25\text{ cm} in air. Determine the system matrix. For an object of height 1\text{ cm} placed at a distance of 27\cdot 5\text{ cm} in front of the convex lens, find the size and position of the image.

Why is the NA of the single-mode optical fiber low as compared to multimode fiber?

What do you mean by underfilled and overfilled conditions with reference to numerical aperture in exciting light in fiber?

With a suitable diagram, deduce an expression for numerical aperture (NA) for an optical fiber having refractive indices of core and cladding n_1 and n_2, respectively and being placed in a medium of index n_0.

(ii) A left circularly polarized beam of light having \lambda_0 = 5893\text{ \AA} is incident on a calcite crystal with its optic axis cut parallel to the surface. The crystal has thickness d = 0\cdot 005141\text{ mm}, n_o = 1\cdot 65836 and n_e = 1\cdot 48641 at this \lambda_0. What will be the state of polarization of the incident beam?

(i) What is birefringence? Indicate how it can be used to obtain plane and circularly polarized light.