(a) Two batteries of emf's 3 V and 6 V and internal resistances $0.2 \Omega$ and $0.4 \Omega$ are connected in parallel. This combination is connected to a $4 \Omega$ resistor. Find:

(i) the equivalent emf of the combination

(ii) the equivalent internal resistance of the combination

(iii) the current drawn from the combination

OR

(b) (i) A conductor of length $l$ is connected across an ideal cell of emf $E$. Keeping the cell connected, the length of the conductor is increased to $2 l$ by gradually stretching it. If $R$ and $R^{\prime}$ are initial and final values of resistance and $v_d$ and $v_d{ }^{\prime}$ are initial and final values of drift velocity, find the relation between

(1) $R^{\prime}$ and $R$ and

(2) $v_d{ }^{\prime}$ and $v_d$.

(ii) When electrons drift in a conductor from lower to higher potential, does it mean that all the 'free electrons' of the conductor are moving in the same direction?

Define current density and relaxation time. Derive an expression for resistivity of a conductor in terms of number density of charge carriers in the conductor and relaxation time.

(a) Briefly describe how the current sensitivity of a moving coil galvanometer can be increased.

(b) A galvanometer shows full scale deflection for current $$I_g$$. A resistance $$R_1$$ is required to convert it into a voltmeter of range $$(0-V)$$ and a resistance $$R_2$$ to convert it into a voltmeter of range $$(0-2 \mathrm{~V})$$. Find the resistance of the galvanometer.