Dielectrics play an important role in design of capacitors. The molecules of a dielectric may be polar or non-polar. When a dielectric slab is placed in an external electric field, opposite charges appear on the two surfaces of the slab perpendicular to electric field. Due to this an electric field is established inside the dielectric.

The capacitance of a capacitor is determined by the dielectric constant of the material that fills the space between the plates. Consequently, the energy storage capacity of a capacitor is also affected. Like resistors, capacitors can also be arranged in series and/or parallel.

(i) Which of the following is a polar molecule?

(A) $\mathrm{O}_2$

(B) $\mathrm{H}_2$

(C) $\mathrm{N}_2$

(D) HCl

(ii) Which of the following statements about dielectrics is correct?

(A) A polar dielectric has a net dipole moment in absence of an external electric field which gets modified due to the induced dipoles.

(B) The net dipole moments of induced dipoles is along the direction of the applied electric field.

(C) Dielectrics contain free charges.

(D) The electric field produced due to induced surface charges inside a dielectric is along the external electric field.

(iii) When a dielectric slab is inserted between the plates of an isolated charged capacitor, the energy stored in it

(A) increases and the electric field inside it also increases.

(B) decreases and the electric field also decreases.

(C) decreases and the electric field increases.

(D) increases and the electric field decreases.

(iv) (a) An air-filled capacitor with plate area A and plate separation $d$ has capacitance $\mathrm{C}_0$. A slab of dielectric constant K, area A and thickness $\left(\frac{d}{5}\right)$ is inserted between the plates. The capacitance of the capacitor will become

(A) $\left[\frac{4 \mathrm{~K}}{5 \mathrm{~K}+1}\right] \mathrm{C}_0$

(B) $\left[\frac{K+5}{4}\right] C_0$

(C) $\left[\frac{5 \mathrm{~K}}{4 \mathrm{~K}+1}\right] \mathrm{C}_0$

(D) $\left[\frac{K+4}{5 K}\right] C_0$

OR

(b) Two capacitors of capacitances $2 \mathrm{C}_0$ and $6 \mathrm{C}_0$ are first connected in series and then in parallel across the same battery. The ratio of energies stored in series combination to that in parallel is

(A) $\frac{1}{4}$

(B) $\frac{1}{6}$

(C) $\frac{2}{15}$

(D) $\frac{3}{16}$

A prism is an optical medium bounded by three refracting plane surfaces. A ray of light suffers successive refractions on passing through its two surfaces and deviates by a certain angle from its original path. The refractive index of the material of the prism is given by $\mu=\sin \left(\frac{A+\delta_m}{2}\right) / \sin \frac{A}{2}$. If the angle of incidence on the second surface is greater than an angle called critical angle, the ray will not be refracted from the second surface and is totally internally reflected.

(i) The critical angle for glass is $\theta_1$ and that for water is $\theta_2$. The critical angle for glass-water surface would be (given ${ }_a \mu_g=1.5,{ }_a \mu_w=1.33$ )

(A) less than $\theta_2$

(B) between $\theta_1$ and $\theta_2$

(C) greater than $\theta_2$

(D) less than $\theta_1$

(ii) When a ray of light of wavelength $\lambda$ and frequency $v$ is refracted into a denser medium

(A) $\lambda$ and $v$ both increase.

(B) $\lambda$ increases but $v$ is unchanged.

(C) $\lambda$ decreases but $v$ is unchanged.

(D) $\lambda$ and $v$ both decrease.

(iii) (a) The critical angle for a ray of light passing from glass to water is minimum for

(A) red colour

(B) blue colour

(C) yellow colour

(D) violet colour

OR

(b) Three beams of red, yellow and violet colours are passed through a prism, one by one under the same condition. When the prism is in the position of minimum deviation, the angles of refraction from the second surface are $r_{\mathrm{R}}, r_Y$ and $r_{\mathrm{V}}$ respectively. Then

(A) $r_V< r_Y

(B) $r_Y< r_R

(C) $r_R< r_Y

(D) $r_{\mathrm{R}}=r_{\mathrm{Y}}=r_{\mathrm{V}}$

(iv) A ray of light is incident normally on a prism ABC of refractive index $\sqrt{ } 2$, as shown in figure. After it strikes face AC, it will

(A) go straight undeviated

(B) just graze along the face AC

(C) refract and go out of the prism

(D) undergo total internal reflection

(a) (i) Draw equipotential surfaces for an electric dipole.

(ii) Two point charges $q_1$ and $q_2$ are located at $\overrightarrow{r_1}$ and $\vec{r}_2$ respectively in an external electric field $\vec{E}$. Obtain an expression for the potential energy of the system.

(iii) The dipole moment of a molecule is $10^{-30} \mathrm{Cm}$. It is placed in an electric field $\vec{E}$ of $10^5 \mathrm{~V} / \mathrm{m}$ such that its axis is along the electric field. The direction of $\vec{E}$ is suddenly changed by $60^{\circ}$ at an instant. Find the change in the potential energy of the dipole, at that instant.

OR

(b) (i) A thin spherical shell of radius R has a uniform surface charge density $\sigma$. Using Gauss' law, deduce an expression for electric field (i) outside and (ii) inside the shell.

(ii) Two long straight thin wires AB and CD have linear charge densities $10 \mu \mathrm{C} / \mathrm{m}$ and $-20 \mu \mathrm{C} / \mathrm{m}$, respectively. They are kept parallel to each other at a distance 1 m . Find magnitude and direction of the net electric field at a point midway between them.

(a) (i) You are given three circuit elements $\mathrm{X}, \mathrm{Y}$ and $Z$. They are connected one by one across a given ac source. It is found that V and I are in phase for element $X$. V leads I by $\left(\frac{\pi}{4}\right)$ for element $Y$ while I leads $V$ by $\left(\frac{\pi}{4}\right)$ for element $Z$. Identify elements X, Y and $Z$.

(ii) Establish the expression for impedance of circuit when elements $X, Y$ and $Z$ are connected in series to an ac source. Show the variation of current in the circuit with the frequency of the applied ac source.

(iii) In a series LCR circuit, obtain the conditions under which (i) impedance is minimum and (ii) wattless current flows in the circuit.

OR

(b) (i) Describe the construction and working of a transformer and hence obtain the relation for $\left(\frac{v_s}{v_p}\right)$ in terms of number of turns of primary and secondary.

(ii) Discuss four main causes of energy loss in a real transformer.