A resistor of $5 . \Omega$, inductor of self inductance $\left(\frac{2}{\pi^2}\right) \mathrm{H}$ and a capacitor of unknown capacity are connected in series to an a.c. source of $100 \mathrm{~V}, 50 \mathrm{~Hz}$ supply. When the voltage and current are in phase, the value of capacitance is

A rigid body rotates about a fixed axis with variable angular velocity $(\alpha-\beta t)$ at time $t$, where $\alpha$ and $\beta$ are constants. The angle through which it rotates before it comes to rest is

The capacity of air filled parallel plate capacitor is $\mathrm{C}_0$. One-half of the space between the plates is filled with a dielectric constant ' K ' as shown in figure. The new capacity becomes $\mathrm{C}_{\mathrm{n}}$. The ratio $\mathrm{C}_{\mathrm{n}}$ to $\mathrm{C}_0$ is

Two black spheres $\mathrm{P} \& \mathrm{Q}$ have radii in the ratio $4: 3$. The wavelength of maximum intensity of radiation are in the ratio $4: 5$ respectively. The ratio of radiated power by P to Q is