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 $\frac{B^2}{2 \mu_0}$.

(a) A parallel plate capacitor is charged by an ac source. Show that the sum of conduction current $\left(I_c\right)$ and the displacement current $\left(I_d\right)$ has the same value at all points of the circuit.

(b) In case (a) above, is Kirchhoff's first rule (junction rule) valid at each plate of the capacitor? Explain.

Answer the following giving reason:

(a) All the photo electrons do not eject with same kinetic energy when monochromatic light is incident on a metal surface.

(b) The saturation current in case (a) is different for different intensity.

(c) If one goes on increasing the wavelength of light incident on a metal surface, keeping its intensity constant, emission of photo electrons stop at a certain wavelength for this metal.