(a) An electric dipole consists of two point charges $q$ and $-q$ separated by a distance $2a$. Derive an expression for the electric field $\vec{\text{E}}$ due to this dipole at a point distant $r$ from the centre of the dipole on the equatorial plane. Write the expression for the electric field at a far off point, i.e. $r \gg a$. (b) A dipole is placed in $x\text{-}y$ plane such that charges $q$ and $-q$ are located at $x = a$ and $x = b$ respectively. There exists an electric field $\vec{\text{E}} = 2\hat{\text{i}} \frac{\text{N}}{\text{C}}$ in the region. Calculate the force $\vec{\text{F}}$ and torque $\vec{\tau}$ experienced by the dipole. OR (a) Two cells of emf $E_1$ and $E_2$ with internal resistances $r_1$ and $r_2$ respectively, are connected in parallel by connecting their positive terminals together and negative terminals together. Deduce an expression for equivalent emf and equivalent internal resistance of the combination. (b) A parallel combination, as stated in (a) above, of two cells of emfs $E$ and $3E$ and internal resistances $R$ each is connected across a resistance $2R$. Find the current that flows through resistance $2R$.