PYQ Bank

N particles are distributed among three states having energies E = 0, E = kT and E = 2kT. If the total equilibrium energy of the system is 1000 kT, what is the value of N?

Consider the following statement: ``The Fermi energy of a given material is the energy of that quantum state which has the probability equal to \frac{1}{2} of being occupied by the conduction electrons.'' Is the above statement correct? Give reasons for your answer.

Calculate the number of different arrangements of 10 indistinguishable particles in 15 cells of equal a priori probability, considering that one cell contains only one particle.

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.

(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.

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.

A wire of length 2 m is perpendicular to X-Y plane. It is moved with a velocity \vec{V}=(2\hat{i}+3\hat{j}+\hat{k})\,\mathrm{ms}^{-1} through a region of uniform induction \vec{B}=(\hat{i}+2\hat{j})\mathrm{Wm}^{-2}. Compute the potential difference between the ends of the wire.

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

A series circuit has an inductance of 200 microhenries, a capacitance of 0.0005 microfarad and a resistance of 10 ohms. Find the resonant frequency and quality factor of the circuit.

Discuss the growth of current when an e.m.f. is suddenly applied to a circuit containing resistance, inductance and capacitance in series. What is the time constant of the circuit?

What happens if the primary winding of a transformer is connected to a battery?

What is meant by a dielectric? Define polarization vector P and relate it with the average molecular dipole moment. Obtain expression for the potential due to a polarized dielectric in terms of the polarization vector.

Obtain Poisson's equation in electrostatics from Gauss' law. What form does it take when the charge density is zero?

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.

Explain the term 'Poynting vector' and state the significance of Poynting theorem.

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.

Calculate the skin depth for radio waves in free space of wavelength 3 m in copper, given that electrical conductivity for copper is 6\times10^{7}\,\Omega^{-1}\,\mathrm{m}^{-1}.

Using Maxwell's field equations for a homogeneous non-conducting medium, derive the wave equation for the electric field. Calculate the velocity of EM wave in free space.