A particle of charge ${}^{\text{'}}q^{\text{'}}\mathrm{and} \mathrm{mass} {}^{\text{'}}m^{\text{'}}$ moves in a circular orbit of radius ${}^{\text{'}}r^{\text{'}} \mathrm{with} \mathrm{angular} \mathrm{speed} {}^{\text{'}}\mathrm{\omega }^{\text{'}}$. The ratio of the magnitude of its magnetic movement to that of its angular momentum depends on

Given data: 

charge q, mass m, radius r, angular speed \omega

Concept used: dipole in a uniform magnetic field

Detailed solution:

The angular momentum L of the particle is given by $\mathrm{L}={\mathrm{mr}}^2\mathrm{\omega }$

$\therefore \mathrm{Frequency} \mathrm{n}=\frac{\mathrm{\omega }}{2\mathrm{\pi }}$

Further $\mathrm{i}=\mathrm{q}\times \mathrm{n}=\frac{\mathrm{\omega }q}{2\pi }$

Magnetic Moment, $\mathrm{M}=\mathrm{iA}=\frac{\mathrm{\omega }q}{2\pi }\times \pi {\mathrm{r}}^2;$

$\therefore M=\frac{{\mathrm{\omega qr}}^2}{2}$

So, $\frac{\mathrm{M}}{\mathrm{L}}=\frac{{\mathrm{\omega qr}}^2}{2{\mathrm{mr}}^2\mathrm{\omega }}=\frac{\mathrm{q}}{2 \mathrm{m}}$

Thus the required ratio depends upon both $\mathrm{q} and \mathrm{m}.$