Question 12

1. Identify the system: We have a rotating hollow cone with surface charge $Q$. This system acts as a magnetic dipole for far-off points ($z \gg R, h$).

2. Calculate the Magnetic Dipole Moment ($M$): Consider an elemental ring on the curved surface at a distance $s$ along the slant height from the tip.

Let $L = \sqrt{R^2 + h^2}$ be the total slant height.

The radius of the ring is $r = \frac{R}{L}s$.

The charge on the ring element $dq = \sigma (2\pi r ds) = \frac{Q}{\pi R L} (2\pi \frac{R}{L}s) ds = \frac{2Q}{L^2} s ds$.

The magnetic moment of this rotating ring is $dM = I \cdot Area = (\frac{dq \omega}{2\pi}) (\pi r^2) = \frac{1}{2} \omega r^2 dq$.

Substituting $r$ and $dq$:

$dM = \frac{1}{2} \omega (\frac{R^2}{L^2} s^2) (\frac{2Q}{L^2} s) ds = \frac{Q \omega R^2}{L^4} s^3 ds$

Integrating from $s = 0$ to $s = L$:

$M = \int_0^L \frac{Q \omega R^2}{L^4} s^3 ds = \frac{Q \omega R^2}{L^4} \left[ \frac{s^4}{4} \right]_0^L = \frac{1}{4} Q \omega R^2$

3. Calculate Magnetic Field on Axis: For a dipole at a distance $z \gg$ dimensions of the source, the magnetic field on the axis is given by: $B = \frac{\mu_0}{4\pi} \frac{2M}{z^3}$

Substituting the value of $M$:

$B = \frac{\mu_0}{4\pi} \frac{2(\frac{1}{4} Q \omega R^2)}{z^3} = \frac{0.5 \mu_0 Q R^2 \omega}{4\pi z^3}$

4. Determine $n$: Comparing with the given expression $B = \frac{n \mu_0 Q R^2 \omega}{4 \pi z^3}$, we find $n = 0.5$.