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Question 5.30: Use the results of Ex. 5.11 to find the magnetic field insid...

Use the results of Ex. 5.11 to find the magnetic field inside a solidsphere, of uniform charge density ρ and radius R, that is rotating at a constant angular velocity ω.

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Use Eq. 5.69, with  R \rightarrow \bar{r} \text { and } \sigma \rightarrow \rho d \bar{r} :

A (r, \theta, \phi)= \begin{cases}\frac{\mu_{0} R \omega \sigma}{3} r \sin \theta \hat{ \phi }, & (r \leq R), \\ \frac{\mu_{0} R^{4} \omega \sigma}{3} \frac{\sin \theta}{r^{2}} \hat{ \phi }, & (r \geq R).\end{cases}                              (5.69)

A =\frac{\mu_{0} \omega \rho}{3} \frac{\sin \theta}{r^{2}} \hat{\phi} \int_{0}^{r} \bar{r}^{4} d \bar{r}+\frac{\mu_{0} \omega \rho}{3} r \sin \theta \hat{\phi} \int_{r}^{R} \bar{r} d \bar{r}

 

=\left(\frac{\mu_{0} \omega \rho}{3}\right) \sin \theta\left[\frac{1}{r^{2}}\left(\frac{r^{5}}{5}\right)+\frac{r}{2}\left(R^{2}-r^{2}\right)\right] \hat{\phi}=\frac{\mu_{0} \omega \rho}{2} r \sin \theta\left(\frac{R^{2}}{3}-\frac{r^{2}}{5}\right) \hat{\phi} .

 

B = \nabla \times A =\frac{\mu_{0} \omega \rho}{2}\left\{\frac{1}{r \sin \theta} \frac{\partial}{\partial \theta}\left[\sin \theta r \sin \theta\left(\frac{R^{2}}{3}-\frac{r^{2}}{5}\right)\right] \hat{ r }-\frac{1}{r} \frac{\partial}{\partial r}\left[r^{2} \sin \theta\left(\frac{R^{2}}{3}-\frac{r^{2}}{5}\right)\right] \hat{ \theta }\right\}

 

=\mu_{0} \omega \rho\left[\left(\frac{R^{2}}{3}-\frac{r^{2}}{5}\right) \cos \theta \hat{ r }-\left(\frac{R^{2}}{3}-\frac{2 r^{2}}{5}\right) \sin \theta \hat{ \theta }\right] .

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