UPSC Physics Optional PYQs (CSE & IFoS)PYQs

Verify whether the electric potential V = 15 x^2 y z - 5 y^3 z satisfies Laplace's equation or not.

In a simple AC circuit involving only a resistor R = 50\,\Omega and a voltage source V, find the linear frequency of the generator if V = 0 \cdot 5\,V_m (V_m is peak e.m.f.) at time t = \frac{1}{720}\text{ s} (assuming V = 0 at t = 0).

A conducting sphere of radius 5\,\mathrm{cm} has a total charge of 12\,\mathrm{nC} uniformly distributed on its surface in free space. Determine the displacement vector \vec{D} on its surface and outside at a distance r from the centre of the sphere.

The electric potential of a grounded conducting sphere of radius a in a uniform electric field E = E_0\hat{z} is given as \phi(r, \theta) = -E_0 r \left[ 1 - \left( \frac{a}{r} \right)^3 \right] \cos\theta Find the expression for the surface charge density on the sphere.

Discuss the physical significance of Planck's radiation law in the context of emergence of New Physics. If the blackbody energy density u(\omega) has a functional dependence like u(\omega) \propto x^3 (e^x - 1)^{-1}, where x = \frac{\hbar\omega}{kT}, derive Wien's displacement law and comment on its importance.

A series RLC circuit has R=2\Omega. The energy stored in the circuit decreases by 1\% per period of oscillation. Its natural undamped frequency is 2\,\mathrm{kHz}. Determine the values of inductor L and the quality factor.

Calculate the strength of the magnetic field to bring a proton nucleus and a {}^{13}\text{C} nucleus to resonate at this frequency. Magnetic moment of \text{proton} = 2\cdot 7927\ \mu_\text{N} and magnetic moment of {}^{13}\text{C} = 0\cdot 7022\ \mu_\text{N}. The NMR instrument operates at 30\cdot 256\text{ MHz}.

Viewing ionosphere as a dielectric medium of refractive index \mu = \sqrt{1 - \omega_p^2 / \omega^2}, \omega_p being known as the plasma frequency, determine the group velocity of a radio wave of frequency \omega = \sqrt{2}\omega_p.

Under one-dimensional configuration, the charge density is given by \rho(x)=\dfrac{\rho_0x}{5}, where \rho_0 is a constant charge density. If the electric field \lvert\vec{E}\rvert=0 at x=0 and potential V=0 at x=5, determine V and \lvert\vec{E}\rvert.

In the circuit given below, find the values of currents I_1, I_2 and I.

Derive the equation that represents Poynting's theorem. What is its physical significance?

Using Planck's radiation law, deduce Wien's displacement law. How does this law enable one to estimate the surface temperature of the Sun or a star?

Using Ampere's Law and continuity equation, show that the divergence of the total current density is zero.

When connected in series, L_1, C_1 have the same resonant frequency as L_2, C_2 also connected in series. Prove that if all these circuit elements are connected in series, the new circuit will have the same resonant frequency as either of the circuits first mentioned.

For initial current conditions I=I_0 and \frac{dI}{dt}=0 at t=0, show that the time dependent current in the critical damping case for an LCR circuit is given by I=I_0\left(1+\frac{\gamma t}{2}\right)e^{-\gamma t/2} where \gamma=\frac{R}{L}, \omega_0^2=\frac{1}{LC}, \omega=\sqrt{\omega_0^2-\frac{\gamma^2}{4}} and \tan\delta=\frac{-\gamma}{2\omega}.

A plane electromagnetic wave of frequency w_1 travelling in z-direction and polarized in x-direction is incident on a dielectric medium separated by a plane boundary in the x-z plane. Show that the transmission and reflection coefficient are given by : R = \frac{(n_1 - n_2)^2}{(n_1 + n_2)^2} , \quad T = \frac{4 n_1 n_2}{(n_1 + n_2)^2} , where n_1 and n_2 are the refractive indices of the first and the second medium.

(i) Write Maxwell's equations in the absence of a medium. (ii) Explain the gauge transformation and gauge invariance. Show that the relation \text{div } \bar{A} + \frac{1}{c} \frac{\partial \phi}{\partial t} is Lorentz-covariant, where \bar{A} and \phi are the vector potential and the scalar potential respectively. (iii) Obtain Maxwell's wave equations satisfied by the scalar and vector potentials in the absence of charge and current.

An inductance L, capacitance C and resistance R are connected in series to form a circuit with AC source driving with E_{\text{rms}} = 120\text{ V} at f = 60\text{ Hz}. Compute the power factor and average power dissipated in the resistance if R = 200\ \Omega, X_L = 80\ \Omega and X_C = 150\ \Omega.

A parallel plate capacitor is connected to a battery. What is the electric current while the capacitor is being charged ? What is the displacement current between the plates of the capacitor ? Show that the rate of increase of the electric energy is equal to the surface integral of the Poynting vector over the surface enclosing the volume between the plates of the capacitor.

A sphere of homogeneous linear dielectric material is placed in an otherwise uniform electric field \bar{E}_o. Find the electric field inside the sphere.