A compass needle oscillates 20 times per minute at a place where the dip is $$30°$$ and 30 times per minute where the dip is $$60°$$. The ratio of total magnetic field due to the earth at two places respectively is $$\frac{4}{\sqrt{x}}$$. The value of $$x$$ is _____.

$$T = 2\pi\sqrt{\frac{I}{M B_H}} \implies \nu = \frac{1}{2\pi}\sqrt{\frac{M B_H}{I}} \implies \nu \propto \sqrt{B_H} \implies \nu^2 \propto B \cos\theta$$

$$\frac{\nu_1^2}{\nu_2^2} = \frac{B_1 \cos\theta_1}{B_2 \cos\theta_2}$$

$$\left(\frac{20}{30}\right)^2 = \frac{B_1 \cos 30^\circ}{B_2 \cos 60^\circ} \implies \frac{4}{9} = \frac{B_1}{B_2} \times \frac{\sqrt{3}/2}{1/2} = \frac{B_1}{B_2}\sqrt{3}$$

$$\frac{B_1}{B_2} = \frac{4}{9\sqrt{3}}$$

$$\text{Comparing with the given ratio } \frac{4}{\sqrt{x}}: \sqrt{x} = 9\sqrt{3} = \sqrt{81 \times 3} = \sqrt{243} \implies x = 243$$