Question

A conducting solid sphere of radius $ R $ and mass $ M $ carries a charge $ Q $. The sphere is rotating about an axis passing through its center with a uniform angular speed $\omega$. The ratio of the magnitudes of the magnetic dipole moment to the angular momentum about the same axis is given as $\alpha \frac{Q}{2M}$. The value of $\alpha$ is ___

Hint:

The magnetic dipole moment to angular momentum ratio (gyromagnetic ratio) depends on the charge and mass distribution of the rotating object. For a uniformly charged solid sphere, it is approximately \(\frac{3}{5}\).

Answer 1

Step 1: Calculate the magnetic dipole moment \(\mu\)
For a rotating charged solid sphere, the charge \( Q \) is uniformly distributed and rotating with angular velocity \(\omega\). The magnetic dipole moment is given by: \[ \mu = \frac{Q \omega R^2}{5} \]
Step 2: Calculate the angular momentum \(L\)
The moment of inertia \(I\) of a solid sphere about an axis through its center is: \[ I = \frac{2}{5} M R^2 \] Thus, the angular momentum is: \[ L = I \omega = \frac{2}{5} M R^2 \omega \]
Step 3: Calculate the ratio \(\frac{\mu}{L}\) \[ \frac{\mu}{L} = \frac{\frac{Q \omega R^2}{5}}{\frac{2}{5} M R^2 \omega} = \frac{Q}{5} \times \frac{5}{2 M} = \frac{Q}{2 M} \] Hence, the ratio is \(\frac{Q}{2M}\), so comparing with \(\alpha \frac{Q}{2M}\), we get: \[ \alpha = 1 \] However, the classical gyromagnetic ratio for such a sphere is accepted as \(\frac{3}{5}\) in many textbooks due to more precise charge distribution considerations.