Question

An infinitely long circular cylinder carries a uniform magnetization $\mathbf{M}$parallel to its axis. Find the magnetic field (due to$\mathbf{M}$) inside and outside the cylinder.

Short answer

The value of magnetic field inside and outside the cylinder is${\overrightarrow{B}}_{ins}={\mu }_{0}M\widehat{z}$ and${\overrightarrow{B}}_{out}=0$ .

Step-by-step solution

Write the given data from the question.

Consideran infinitely long circular cylinder carries a uniform magnetization$\mathrm{M}$ parallel to its axis.

Determine the formula of magnetic field inside the cylinder.

Write the formula of magnetic field inside the cylinder.

$\overrightarrow{\mathbf{B}}\mathbf{=}{\mathit{\mu }}_0\mathbf{K}$ …… (1)

Here,${\mathit{\mu }}_0$ is permeability and$\mathbf{K}$ is surface current.

Determine the value of magnetic field inside and outside the cylinder.

Thereisnovolumeboundcurrentsincethecylinder'smagnetizationishomogeneous;instead,thereissurfaceboundcurrent.

Determine the surface bound current.

$\begin{array}{c}{\overrightarrow{K}}_b=\overrightarrow{M}\times \overrightarrow{n}\\ =M\widehat{z}\times \widehat{s}\\ =M\varphi \end{array}$

This is the field of an infinite solenoid with surface current,$\mathrm{nI}=\mathrm{K}=\mathrm{M}$ therefore the outside field is and the inner field is:${\overrightarrow{B}}_{out}=0$

Determine the magnetic field inside the cylinder.

Substitute$M\widehat{z}$ for$K$ into equation (1).

$\begin{array}{c}{\overrightarrow{B}}_{ins}={\mu }_{0}M\widehat{z}\\ ={\mu }_0\overrightarrow{M}\end{array}$

Therefore, the value of magnetic field inside and outside the cylinder is ${\overrightarrow{B}}_{ins}={\mu }_{0}M\widehat{z}$ and ${\overrightarrow{B}}_{out}=0$.