A circuit consisting of a capacitor $C$, a resistor of resistance $R$ and an ideal battery of emf $V$, as shown in figure is known as $R C$ series circuit.

As soon as the circuit is completed by closing key $S_1$ (keeping $S_2$ open) charges begin to flow between the

capacitor plates and the battery terminals. The charge on the capacitor increases and consequently the potential difference $V_c(=q / C)$ across the capacitor also increases with time. When this potential difference equals the potential difference across the battery, the capacitor is fully charged $(Q=V C)$. During this process of charging, the charge $q$ on the capacitor changes with time $t$ as $q=Q \left[1-e^{-t / R C}\right]$

The charging current can be obtained by differentiating it and using $\frac{d}{d x}\left(e^{m x}\right)=m e^{m x}$

Consider the case when $R=20 \mathrm{k} \Omega, C=500 \mu \mathrm{~F}$ and $V=10$ V.

(i) The final charge on the capacitor, when key $S_1$ is closed and $S_2$ is open, is

(A) $5 \mu \mathrm{C}$

(B) 5 mC

(C) 25 mC

(D) 0.1 C

(ii) For sufficient time the key $S_1$ is closed and $S_2$ is open. Now key $S_2$ is closed and $S_1$ is open. What is the final charge on the capacitor?

(A) Zero

(B) 5 mC

(C) 2.5 mC

(D) $5 \mu \mathrm{C}$

(iii) The dimensional formula for $R C$ is

(A) $\left[M L^2 T^{-3} A^{-2}\right]$

(B) $\left[M^0 L^0 T^{-1} A^0\right]$

(C) $\left[M^{-1} L^{-2} T^4 A^2\right]$

(D) $\left[M^0 L^0 T A^0\right]$

(iv) The key $S_1$ is closed and $S_2$ is open. The value of current in the resistor after 5 seconds, is

(A) $\frac{1}{2 \sqrt{e}} \mathrm{~mA}$

(B) $\sqrt{e} \mathrm{~mA}$

(C) $\frac{1}{\sqrt{e}} \mathrm{~mA}$

(D) $\frac{1}{2 e} \mathrm{~mA}$

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 capacitor is a system of two conductors separated by an insulator. The two conductors have equal and opposite charges with a potential difference between them. The capacitance of a capacitor depends on the geometrical configuration (shape, size and separation) of the system and also on the nature of the insulator separating the two conductors. They are used to store charges. Like resistors, capacitors can be arranged in series or parallel or a combination of both to obtain desired value of capacitance.

(i) Find the equivalent capacitance between points A and B in the given diagram.

(ii) A dielectric slab is inserted between the plates of a parallel plate capacitor. The electric field between the plates decreases. Explain.

(iii) A capacitor A of capacitance C, having charge Q is connected across another uncharged capacitor B of capacitance $$2 C$$. Find an expression for (a) the potential difference across the combination and (b) the charge lost by capacitor A.

OR

(iii) Two slabs of dielectric constants $$2 \mathrm{~K}$$ and $$\mathrm{K}$$ fill the space between the plates of a parallel plate capacitor of plate area A and plate separation $$d$$ as shown in figure. Find an expression for capacitance of the system.