A battery of emf $E$ and internal resistance $r$ is connected to a rheostat. When a current of 2 A is drawn from the battery, the potential difference across the rheostat is $\mathbf{5 ~ V}$. The potential difference becomes 4 V when a current of 4 A is drawn from the battery. Calculate the value of $E$ and $r$.

(a) In a diffraction experiment, the slit is illuminated by light of wavelength 600 nm . The first minimum of the pattern falls at $\theta=30^{\circ}$. Calculate the width of the slit.

OR

(b) In a Young's double-slit experiment, two light waves, each of intensity $I_0$, interfere at a point, having a path difference $\frac{\lambda}{8}$ on the screen. Find the intensity at this point.

A transparent solid cylindrical rod (refractive index $\frac{2}{\sqrt{3}}$ ) is kept in air. A ray of light incident on its face travels along the surface of the rod, as shown in figure. Calculate the angle $\theta$.

Prove that, in Bohr model of hydrogen atom, the time period of revolution of an electron in $n^{\text {th }}$ orbit is proportional to $n^3$.