A parallel plate air capacitor having area of each plate '$A$' and the distance between the plates '$d$' has uniform electric field E in the space between the plates. The energy stored in the capacitor is ($\epsilon_0$ = permittivity of free space.)

In meter bridge experiment, two resistances X and Y in the two gaps give a null point dividing the wire in the ratio $2 : 3$. When each resistance is increased by $30\,\Omega$, the null point divides the wire in the ratio $5 : 6$. The resistance X and Y are respectively

Two cells of e.m.f. $E_1$ and $E_2$ ($E_1 > E_2$) are connected as shown in figure.

When a potentiometer is connected between points A and B the balancing length of potentiometer wire is $412$ cm. When same potentiometer wire is connected between points A and C the balancing length is $103$ cm. The ratio $E_1 : E_2$ is

The periodic time of a magnet in vibration magnetometer is $T_0$. If a magnet is replaced by another magnet whose moment of inertia is three times and magnetic moment is one-third of the original magnet, then time period of new magnet will be