91Ó°ÊÓ

Consider a long circular cylinder of radius $R$carries a magnetization $\text{M}={\text{ks}}^{\text{2}}\widehat{\mathrm{Ï•}}$. Where $k$is a constant, $s$is the distance from the axis, and $\widehat{\mathrm{Ï•}}$ is the usual azimuthal unit vector.

Write the formula of magnetic field outside the cylinder.

${\stackrel{\mathbf{→}}{\mathbf{B}}}_{\mathrm{out}}=\frac{{\mathrm{μ}}_0{I}_{\mathrm{enc}}}{\mathbf{2}\mathrm{π}\mathbf{s}}$ …… (1)

Here,${\mathrm{μ}}_0$ is permeability, $I_{\mathrm{enc}}$is current enclosed the cylinder and $s$ is the distance from the axis.

Write the formula of magnetic field inside the cylinder.

${\stackrel{\mathbf{→}}{\mathbf{B}}}_{\mathrm{ins}}=\frac{{\mathit{μ}}_0}{\mathbf{2}\mathrm{π}\mathbf{s}}{∫}_S{\mathbf{J}}_b\mathbf{dS}$ …… (2)

Here, ${\mathit{μ}}_0$ is permeability, $\mathbf{s}$is the distance from the axis and ${\mathit{J}}_{\mathbf{b}}$is surface bound current.

First we determine the bound currents:

$\begin{array}{c}{\stackrel{→}{K}}_b=\stackrel{→}{M}×\widehat{s}\\ =−k{R}^2\widehat{z}\end{array}$

Determine the surface bound currents:

$\begin{array}{c}{\stackrel{→}{J}}_b=∇×\stackrel{→}{M}\\ =\frac{1}{s}\frac{∂}{∂s}\left(sk{s}^2\right)\widehat{z}\\ =3ks\widehat{z}\end{array}$

Now, determine the magnetic field outside the cylinder. Use a circular Amperian loop:

$\begin{array}{c}{\stackrel{→}{B}}_{out}=2\mathrm{π}{k}_b+{∫}_0^R{∫}_0^{2\mathrm{π}}{J}_{b}sdsd\mathrm{ϕ}\\ =\left(2\mathrm{π}R\right)(−k{R}^2)+3k{∫}_0^R{∫}_0^{2\mathrm{π}}{s}^{2}ds\\ {\stackrel{→}{B}}_{out}=0\end{array}$

Determine the magnetic field inside the cylinder. Use a circular Amperian loop (this time of radius $s<R$):

$\begin{array}{c}{\stackrel{→}{B}}_{ins}=\frac{{\mathrm{μ}}_0}{2\mathrm{π}s}{∫}_S{J}_{b}dS\\ ={\mathrm{μ}}_{0}3k{∫}_0^s{∫}_0^{2\mathrm{π}}{s}^{2}dsd\mathrm{ϕ}\\ ={\mathrm{μ}}_{0}2\mathrm{π}k{s}^2\\ {\stackrel{→}{B}}_{ins}={\mathrm{μ}}_{0}k{s}^2\widehat{\mathrm{ϕ}}\end{array}$

Therefore, the value of magnetic field outside and inside the cylinder is${\stackrel{→}{B}}_{out}=0$ and ${\stackrel{→}{B}}_{ins}={\mathrm{μ}}_{0}k{s}^2\widehat{\mathrm{ϕ}}$.