91Ó°ÊÓ

There is a sphere of radius $R$ of steady current density $J$.

The magnetic field as a function of the magnetic vector potential is

$\mathbf{B}=∇×\hat{A}$

The magnetic vector potential corresponding to a current density $\stackrel{→}{j}$is

Here, ${\mathrm{μ}}_0$is the permeability of free space.

The volume integral of the curl of a vector function

The magnetic moment of a current distribution is

$m=\frac{1}{2}∫x×Jd$

(a)

The average magnetic field over a sphere of radius is

Apply equation (1)

$B_{\text{ave}}=\frac{1}{4}∫R^3∫∇×Ad\mathrm{τ}$

Use equation (3) to get,

Use equation (2) to get,

$B_{\mathrm{ave}}=-\frac{1}{\frac{4}{3}\mathrm{Ï€}{R}^3}\frac{{\mathrm{μ}}_0}{4\mathrm{Ï€}}âˆ\circledR ∫\frac{\stackrel{→}{J}}{r}d{\mathrm{Ï„}}^{\text{'}}×da$

The point ${\stackrel{→}{r}}^{\text{'}\text{'}}$is chosen to be on the $z$axis. Therefore,

$\backslash begin\left\{aligned\right\}r\&=\backslash sqrt\{R^\{2\}+z^\{\backslash prime2\}-2Rz^\{\backslash prime\}\backslash \cos \backslash theta\}\backslash \backslash d^\{\backslash lambda\}\&=R^\left\{2\right\}\backslash \sin \backslash thetad\backslash thetad\backslash phi\backslash hat\left\{r\right\}\backslash end\left\{aligned\right\}$

The $x$and $y$are components of the surface, integration is thus zero. The z component is

$âˆ\circledR \frac{da^}{r}=âˆ\circledR \frac{R^2\sin \mathrm{θ}d\mathrm{θ}d\mathrm{Ï•}\hat{z}\cos \mathrm{θ}}{\sqrt{R^2+z^{\text{'}2}-2R{z}^{\text{'}}\cos \mathrm{θ}}}$

$=2\mathrm{π}{R}^2\hat{z}{∫}_0^{\mathrm{π}}\frac{\sin \mathrm{θ}\cos \mathrm{θ}d\mathrm{θ}}{\sqrt{R^2+z^{\text{'}2}-2R{z}^{\text{'}}\cos \mathrm{θ}}}$

Convert

$u=\cos \mathrm{θ}$

$du=-\sin \mathrm{θ}d\mathrm{θ}$

Solve further as,

$âˆ\circledR \frac{da^}{r}=-2\mathrm{Ï€}{R}^2\hat{z}{∫}_1^{-1}\frac{udu}{\sqrt{R^2+z^{\text{'}2}-2R{z}^{\text{'}}u}}$

$=2\mathrm{Ï€}{R}^2\hat{z}{\left[-\frac{2\left\{2\left(R^2+z^{\text{'}2}\right)+2R{z}^{\text{'}}u\right\}}{3{\left(2R{z}^{\text{'}}\right)}^2}\sqrt{R^2+z^{\text{'}2}-2R{z}^{\text{'}}u}\right]}_{-1}^1$

$=\frac{2\mathrm{Ï€}\hat{z}}{3z^{\text{'}2}}\left[-\left(R^2+z^{\text{'}2}+R{z}^{\text{'}}\right)\left|R-z^{\text{'}}\right|+\left(R^2+z^{\text{'}2}-R{z}^{\text{'}}\right)\left(R+z^{\text{'}}\right)\right]$

Inside the sphere, $R>z^{\text{'}}$. Therefore,

$\backslash begin\left\{aligned\right\}\backslash prod\backslash frac\{da\mathrm{Ì‚}\}\left\{r\right\}\&=\backslash frac\{2\backslash pi\backslash hat\{z\left\}\right\}\{3z^\{\backslash prime2\left\}\right\}\backslash left[-\backslash left(R^\left\{2\right\}+z^\{\backslash prime2\}+Rz^\{\backslash prime\}\backslash right)\backslash left(R-z^\{\backslash prime\}\backslash right)+\backslash left(R^\left\{2\right\}+z^\{\backslash prime2\}-Rz^\{\backslash prime\}\backslash right)\backslash left(R+z^\{\backslash prime\}\backslash right)\backslash right]\backslash \backslash \&=\backslash frac\{2\backslash pi\backslash hat\{z\left\}\right\}\{3z^\{\backslash prime2\left\}\right\}\backslash left[-R^\{3\}-Rz^\{\backslash prime2\}-R^\{2\}z^\{\backslash prime\}+R^\{2\}z^\{\backslash prime\}+z^\{\backslash prime3\}+Rz^\{\backslash prime2\}+R^\{3\}+Rz^\{\backslash prime2\}-R^\{2\}z^\{\backslash prime\}+R^\{2\}z^\{\backslash prime\}+z^\{\backslash prime3\}-Rz^\{\backslash prime2\}\backslash right]\backslash \backslash \&=\backslash frac\{2\backslash pi\backslash hat\{z\left\}\right\}\{3z^\{\backslash prime2\left\}\right\}\backslash times2z^\{\backslash prime3\}\backslash \backslash \&=\backslash frac\{4\backslash piz^\{\backslash prime\}\backslash hat\{z\left\}\right\}\left\{3\right\}\backslash end\left\{aligned\right\}$

Thus, from equation (5),

$\backslash begin\left\{aligned\right\}B\_\{\backslash text\{ave\left\}\right\}^\left\{R\right\}\&=-\backslash frac\{3\backslash mu\_\{0\left\}\right\}\{16\backslash pi^\{2\}R^\{3\left\}\right\}\backslash frac\{4\backslash piR^\{3\left\}\right\}\left\{3\right\}\backslash intJ\backslash times\backslash frac\{\backslash hat\{r\}^\{\backslash prime\left\}\right\}\{r^\{\backslash prime2\left\}\right\}d\backslash tau^\{\backslash prime\}\backslash \backslash \&=\backslash frac\{\backslash mu\_\{0\left\}\right\}\{4\backslash pi\}\backslash int\backslash stackrel\left\{D\right\}\left\{J\right\}\backslash times\backslash frac\{\backslash hat\{r\}^\{\backslash prime\left\}\right\}\{r^\{\backslash prime2\left\}\right\}d\backslash tau^\{\backslash prime\}\backslash end\left\{aligned\right\}$

Use equation (4) to get,

$B_{\mathrm{ave}}=\frac{2{\mathrm{μ}}_0\tilde{m}}{4\mathrm{π}{R}^3}$

Thus, the average field inside the sphere is$\frac{2{\mathrm{μ}}_0\stackrel{m}{m}}{4\mathrm{π}{R}^3}$

(b)

Outside the sphere, $R<z^{\text{'}}$. Therefore from equation (6),

$\backslash begin\left\{aligned\right\}\backslash dot\left\{y\right\}\backslash frac\{da\}\left\{r\right\}\&=\backslash frac\{2\backslash pi\backslash hat\{z\left\}\right\}\{3z^\{\backslash prime2\left\}\right\}\backslash left[-\backslash left(R^\left\{2\right\}+z^\{\backslash prime2\}+Rz^\{\backslash prime\}\backslash right)\backslash left(-R+z^\{\backslash prime\}\backslash right)+\backslash left(R^\left\{2\right\}+z^\{\backslash prime2\}-Rz^\{\backslash prime\}\backslash right)\backslash left(R+z^\{\backslash prime\}\backslash right)\backslash right]\backslash \backslash \&=\backslash frac\{2\backslash pi\backslash hat\{z\left\}\right\}\{3z^\{\backslash prime2\left\}\right\}\backslash left[R^\{3\}+Rz^\{\backslash prime2\}+R^\{2\}z^\{\backslash prime\}-R^\{2\}z^\{\backslash prime\}-z^\{\backslash prime3\}-Rz^\{\backslash prime2\}+R^\{3\}+Rz^\{\backslash prime2\}-R^\{2\}z^\{\backslash prime\}+R^\{2\}z^\{\backslash prime\}+z^\{\backslash prime3\}-Rz^\{\backslash prime2\}\backslash right]\backslash \backslash \&=\backslash frac\{2\backslash pi\backslash hat\{z\left\}\right\}\{3z^\{\backslash prime2\left\}\right\}\backslash times2R^\left\{3\right\}\backslash \backslash \&=\backslash frac\{4\backslash piR^\{3\}\backslash hat\{z\left\}\right\}\{3z^\{\backslash prime2\left\}\right\}\backslash end\left\{aligned\right\}$

Thus, from equation (5),

$\backslash begin\left\{aligned\right\}B\_\{\backslash text\{ave\left\}\right\}^\left\{R\right\}\&=-\backslash frac\{3\backslash mu\_\{0\left\}\right\}\{16\backslash pi^\{2\}R^\{3\left\}\right\}\backslash frac\{4\backslash piR^\{3\left\}\right\}\left\{3\right\}\backslash intJ\backslash times\backslash frac\{\backslash hat\{r\}^\{\backslash prime\left\}\right\}\{r^\{\backslash prime2\left\}\right\}d\backslash tau^\{\backslash prime\}\backslash \backslash \&=\backslash frac\{\backslash mu\_\{0\left\}\right\}\{4\backslash pi\}\backslash int\backslash stackrel\left\{D\right\}\left\{J\right\}\backslash times\backslash frac\{\backslash hat\{r\}^\{\backslash prime\left\}\right\}\{r^\{\backslash prime2\left\}\right\}d\backslash tau^\{\backslash prime\}\backslash end\left\{aligned\right\}$

Thus, the average field outside the sphere is $\frac{{\mathrm{μ}}_0}{4\mathrm{π}}∫\sqrt{J}×\frac{{\hat{r}}^{\text{'}}}{r^{\text{'}2}}d{\mathrm{τ}}^{\text{'}}$which is also the field at the center..