A $p$-type Si semiconductor is made by doping an average of one dopant atom per $5 \times 10^7$ silicon atoms. If the number density of silicon atoms in the specimen is $5 \times 10^{28}$ atoms $\mathrm{m}^{-3}$, find the number of holes created per cubic centimetre in the specimen due to doping. Also give one example of such dopants.

(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?

Using Biot-Savart law, derive expression for the magnetic field $(B)$ due to a circular current carrying loop at a point on its axis and hence at its centre.

(a) Show that the energy required to build up the current $I$ in a coil of inductance $L$ is $\frac{1}{2} L I^2$.

(b) Considering the case of magnetic field produced by air-filled current carrying solenoid, show that the magnetic energy density of a magnetic field $B$ is $\frac{B^2}{2 \mu_0}$.