(a) (i) Derive an expression for the resistivity of a conductor in terms of number density of free electrons and relaxation time. (ii) The figure shows the plot of current through a cross-section of wire over two different time intervals. Compare the charges ($Q_1$ and $Q_2$) that pass through the cross-section during these time intervals. [Diagram showing a plot of I versus t with pulses Q1 and Q2] OR (b) (i) A battery of emf E and internal resistance r is connected to a variable external resistance R. (I) Obtain the expression for current I in the circuit and the value of maximum current the battery can supply. (II) Obtain the terminal voltage V across the battery and its maximum possible value. (ii) The above battery sends a current $I_1$ when $R = R_1$ and a current $I_2$ when $R = R_2$. Obtain the internal resistance of the battery in terms of $I_1$, $I_2$, $R_1$ and $R_2$.