Question

A current $I$ flows in a long thin-walled cylinder of radius $R$. What pressure do the walls of the cylinder experience?

Answer 1

Let $t=$ thickness of the wall of the cylinder. Then,
$J=\frac{I}{2\pi Rt}$ along z axis. The magnetic field due to this at a distance $r$
$(R-\frac{t}{2}<r<R+\frac{t}{2})$, is given by,
$B_{\phi }\left(2\pi r\right)={\mu }_0\frac{I}{2\pi Rt}\pi \{r^2-{(R-\frac{t}{2})}^2\}$
or, $B_{\phi }=\frac{{\mu }_{0}I}{4\pi Rrt}\{r^2-{(R-\frac{t}{2})}^2\}$
Now, $\overrightarrow{F}=\int \overrightarrow{J}\times \overrightarrow{B}dV$
and $p=\frac{F_r}{2\pi RL}=\frac{1}{2\pi RL}{\int }_{R-\frac{t}{2}}^{R+\frac{t}{2}}\frac{{\mu }_0{I}^2}{8{\pi }^2{R}^2{t}^{2}r}\{r^2-{(R-\frac{t}{2})}^2\}\times 2\pi rLdr$
$=\frac{{\mu }_0{I}^2}{8{\pi }^2{R}^3{t}^2}{\int }_{R-\frac{t}{2}}^{R+\frac{t}{2}}\{r^2-{(R-\frac{t}{2})}^2\}dr=\frac{{\mu }_0{I}^2}{8{\pi }^2{R}^3{t}^2}[\frac{{(R+\frac{t}{2})}^3-{(R-\frac{t}{2})}^3}{3}-{(R-\frac{t}{2})}^{2}t]$
$=\frac{{\mu }_0{I}^2}{8{\pi }^2{R}^{3}t}[Rt+0(t^2\left)\right]\approx \frac{{\mu }_0{I}^2}{8{\pi }^2{R}^2}$