Question

The lateral surface of a long straight solenoid with $n$ turns per unit length experiences a pressure of $p=\frac{1}{x}{\mu }_0{n}^2{I}^2$ when a current $I$ flows through it. Find $x$.

Answer 1

When self-forces are involved, a typical factor of $\frac{1}{2}$ comes into play. For example, the force on a current carrying straight wire in a magnetic induction $B$ is $BIl$. If the magnetic induction $B$ is due to the current itself then the force can be written as,
$F={\int }_0^{I}B\left(I^{\prime }\right)d{I}^{\prime }l$
If $B\left(I^{\prime }\right)\propto I^{\prime }$, then this becomes, $F=\frac{1}{2}B\left(I\right)Il$.
In the present case, $B\left(I\right)={\mu }_{0}nI$ and this acts on $nI$ ampere turns per unit length, so, pressure $p=\frac{F}{Area}=\frac{1}{2}{\mu }_{0}n\frac{I\times nI\times 1\times l}{1\times l}=\frac{1}{2}{\mu }_0{n}^2{I}^2$