A uniformly charged solid sphere of radius R carries a total charge Q, and is set spinning with angular velocity ω about the z axis. (a) What is the magnetic dipole moment of the sphere? (b) Find the average magnetic field within the sphere (see Prob. 5.59). (c) Find the approximate vector

Question

A uniformly charged solid sphere of radius R carries a total charge Q, and is set spinning with angular velocity ω about the z axis.

(a) What is the magnetic dipole moment of the sphere?

(b) Find the average magnetic field within the sphere (see Prob. 5.59).

(c) Find the approximate vector potential at a point (r, θ) where [latex]r \gg R [/latex].

(d) Find the exact potential at a point (r, θ) outside the sphere, and check that it is consistent with (c). [Hint: refer to Ex. 5.11.]

(e) Find the magnetic field at a point (r, θ) inside the sphere (Prob. 5.30), and check that it is consistent with (b).

Accepted Answer

(a) Problem 5.53 gives the dipole moment of a shell: [latex]m =\frac{4 \pi}{3} \sigma \omega R^{4} \hat{ z } . ext { Let } R ightarrow r, \sigma ightarrow ho d r[/latex] and integrate:

[latex]m =\frac{4 \pi}{3} \omega ho \hat{ z } \int_{0}^{R} r^{4} d r=\frac{4 \pi}{3} \omega ho \frac{R^{5}}{5} \hat{ z } . \quad ext { But } ho=\frac{Q}{(4 / 3) \pi R^{3}}, ext { so } m =\frac{1}{5} Q \omega R^{2} \hat{ z }[/latex] .

(b) [latex]B _{ ave }=\frac{\mu_{0}}{4 \pi} \frac{2 m }{R^{3}}=\frac{\mu_{0}}{4 \pi} \frac{2 Q \omega}{5 R} \hat{ z }[/latex] .

(c) [latex]A \cong \frac{\mu_{0}}{4 \pi} \frac{m \sin heta}{r^{2}} \hat{\phi}=\frac{\mu_{0}}{4 \pi} \frac{Q \omega R^{2}}{5} \frac{\sin heta}{r^{2}} \hat{\phi}[/latex] .

(d) Use Eq. 5.69, with [latex]R ightarrow \bar{r}, \sigma ightarrow ho d \bar{r}[/latex] , and integrate:

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

[latex]A =\frac{\mu_{0} \omega ho}{3} \frac{\sin heta}{r^{2}} \hat{\phi} \int_{0}^{R} \bar{r}^{4} d \bar{r}=\frac{\mu_{0} \omega}{3} \frac{3 Q}{4 \pi R^{3}} \frac{\sin heta}{r^{2}} \frac{R^{5}}{5} \hat{\phi}=\frac{\mu_{0}}{4 \pi} \frac{Q \omega R^{2}}{5} \frac{\sin heta}{r^{2}} \hat{\phi}[/latex] .
This is identical to (c); evidently the field is pure dipole, for points outside the sphere.

(e) According to Prob. 5.30, the field is [latex]B =\frac{\mu_{0} \omega Q}{4 \pi R}\left[\left(1-\frac{3 r^{2}}{5 R^{2}} ight) \cos heta \hat{ r }-\left(1-\frac{6 r^{2}}{5 R^{2}} ight) \sin heta \hat{ heta } ight][/latex] .
The average obviously points in the z direction, so take the z component of [latex]\hat{ r }(\cos heta) ext { and } \hat{ heta }(-\sin heta)[/latex] :

[latex]B_{ ave }=\frac{\mu_{0} \omega Q}{4 \pi R} \frac{1}{(4 / 3) \pi R^{3}} \int\left[\left(1-\frac{3 r^{2}}{5 R^{2}} ight) \cos ^{2} heta+\left(1-\frac{6 r^{2}}{5 R^{2}} ight) \sin ^{2} heta ight] r^{2} \sin heta d r d heta d \phi[/latex]

[latex]=\frac{3 \mu_{0} \omega Q}{\left(4 \pi R^{2} ight)^{2}} 2 \pi \int_{0}^{\pi}\left[\left(\frac{r^{3}}{3}-\frac{3}{5} \frac{R^{5}}{5 R^{2}} ight) \cos ^{2} heta+\left(\frac{R^{3}}{3}-\frac{6}{5} \frac{R^{5}}{5 R^{2}} ight) \sin ^{2} heta ight] \sin heta d heta [/latex]

[latex]=\frac{3 \mu_{0} \omega Q}{8 \pi R^{4}} R^{3} \int_{0}^{\pi}\left(\frac{16}{75} \cos ^{2} heta+\frac{7}{75} \sin ^{2} heta ight) \sin heta d heta=\frac{3 \mu_{0} \omega Q}{8 \pi R} \frac{1}{75} \int_{0}^{\pi}\left(7+9 \cos ^{2} heta ight) \sin heta d heta [/latex]

[latex]=\left.\frac{\mu_{0} \omega Q}{200 \pi R}\left(-7 \cos heta-3 \cos ^{3} heta ight) ight|_{0} ^{\pi}=\frac{\mu_{0} \omega Q}{200 \pi R}(20)=\frac{\mu_{0} \omega Q}{10 \pi R}( ext { same as (b) })[/latex] .

Question 5.60: A uniformly charged solid sphere of radius R carries a total...

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