- #1
latentcorpse
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A small sphere is uniformly charged throughout its' volume and rotating with constant angular velocity [itex]\omega[/itex]. Show it's magnetic moment is given by
[itex]m=\frac{1}{5}Q \omega a^2[/itex]. the question doesn't say but I'm assuming a is the radius.
Anyway, so far I have:
[itex]m=\frac{1}{2} \int_V dV (\mathbf{r \wedge J})[/itex]
but [itex]\mathbf{J}= \rho \mathbf{v} = \rho \mathbf{\omega \wedge r}[/itex]
and so [itex]m=\frac{\rho}{2} \int_V r^2 \mathbf{\omega} - (\mathbf{r \cdot \omega})\mathbf{r}[/itex]
but i don't really know where to go from here?
[itex]m=\frac{1}{5}Q \omega a^2[/itex]. the question doesn't say but I'm assuming a is the radius.
Anyway, so far I have:
[itex]m=\frac{1}{2} \int_V dV (\mathbf{r \wedge J})[/itex]
but [itex]\mathbf{J}= \rho \mathbf{v} = \rho \mathbf{\omega \wedge r}[/itex]
and so [itex]m=\frac{\rho}{2} \int_V r^2 \mathbf{\omega} - (\mathbf{r \cdot \omega})\mathbf{r}[/itex]
but i don't really know where to go from here?