[Physics] Magnetic Dipole Moment

Question

I've been trying to tackle this problem for more than a day now, and would appreciate some help. The problem is to prove that the magnetic dipole moment of a spherical ball of mass $m$ and charge $Q$, whose charge is distributed uniformly only on its surface, rotating about its center, is equal to $$ \vec{\mu} = \frac{5Q}{6m} \vec{L}.$$
My reasoning is this: We know the formula for the dipole moment in terms of the surface charge density $K$, which is $$\vec{\mu} = \frac{1}{2} \oint \vec{r} \, ' \times \vec{K} \, dA.$$ Ok, next we have to figure out these quantities. Well, by definition $\vec{K} \equiv \sigma \vec{v}$, where $\sigma$ is the surface charge density. Using this we can write it as
$$
\begin{align}
\vec{K} &= \sigma \vec{v} \\
&= \sigma \vec{\omega} \times \vec{r} \\
&= \sigma \omega R \sin \theta \hat{\phi}.
\end{align}
$$
Next, we know that $\vec{r} \, ' = R\hat{r},$ since what we are integrating only lies on the surface. Now, all we do is take the cross product:
$$
\begin{align}
\vec{\mu} &= \frac{1}{2} \oint R\hat{r} \times \sigma \omega R \sin \theta \hat{\phi} \, dA \\
&= \frac{\sigma \omega R^2}{2} \oint \sin \theta (-\hat{\theta}) \, dA.\\
\end{align}
$$
We know that the dipole moment will be in the $z$ direction because that is the direction of the angular momentum, so we can rewrite $\hat{\theta}$ in our equation above, taking only the $z$ component. $\hat{\theta} = \cos \theta \cos \phi \hat{i} + \cos \theta \sin \phi \hat{j} – \sin \theta \hat{k}$ and so we get $$\vec{\mu} = \frac{\sigma \omega R^2}{2} \oint \sin^2 \theta \hat{k} \, dA$$
We know that $dA = R^2 \sin \theta \, d\theta \, d\phi$, and our limits of integration are from $0$ to $2 \pi$ for $\pi$ and from $0$ to $\phi$ for $\theta$, so our final result comes out to:
$$
\begin{align}
\vec{\mu} &= \frac{\sigma \omega R^2}{2} \int_{\phi = 0}^{2 \pi} \int_{\theta = 0}^{\pi} \sin^2 \theta \hat{k} (R^2 \sin \theta \, d\theta \, d\phi) \\
&= \frac{\sigma \omega R^4}{2} \int_{\phi = 0}^{2\pi} \int_{\theta = 0}^{\pi} \sin^3 \theta \hat{k} \, d\theta \, d\phi \\
&= \frac{\sigma \omega R^4}{2} \frac{4}{3} 2 \pi \hat{k} \\
&= \frac{4}{3} \sigma \omega \pi R^4 \hat{k}.
\end{align}
$$
Last but not least, we need to put this into the correct form as the problem asks us for. We know that $\sigma = \frac{Q}{4\pi R^2}$ and $\vec{L} = I_{sphere} \vec{\omega} = \frac{2}{3} mR^2 \omega \hat{k},$ since the angular velocity is directed in the $z$ direction. To finish it off, we plug in these values and get:
$$
\begin{align}
\vec{\mu} &= \frac{4}{3} \frac{Q}{4\pi R^2} \omega R^4 \pi \hat{k} \\
&= \frac{Q\omega R^2}{3} \hat{k} \\
&= \frac{Q}{3} \frac{3\vec{L}}{2m} \\
&= \boxed{\frac{Q}{2m} \vec{L}}
\end{align}
$$
oops… Once again, this is not the correct answer and I cannot find any faults, so help would be much appreciated! Thank you in advance as usual.

Answer 1

The moment of inertia is supposed to be $$I = \frac{2}{5} mR^2,$$ because it is a solid sphere, not a thing spherical shell. Plugging the correct moment of inertia yields the correct answer!