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

Explanation

Let `t` = thickness of the wall of the cylinder. Then,
`J = I//2 pi R t` along `z` axis. The magnetic field due to this at a distance `r`
`(R - (t)/(2) lt r lt R + (t)/(2))`, is given by,
`B_(varphi) (2pi r) = mu_(0) (I)/(2pi R t) pi {r^(2) - (R - (t)/(2))^(2)}`
or, `B_(varphi) = (mu_(0) I)/(4pi R r t) [r^(2) - (R - t//2)^(2)]`
Now, `vec(F) = int vec(J) xx vec(B) dV`
and `p = (F_(r))/(2pi R L) = (1)/(2pi R L) int_(R - (t)/(2))^(R + (t)/(2)) (mu_(0) I^(2))/(8 pi R^(2) t^(2) r) {r^(2) - (R - (t)/(2))^(2)} xx 2pi r L dr`
`= (mu_(0) I^(2))/(8pi^(2) R^(3) t^(2)) int_(R - (t)/(2))^(R + (t)/(2)) {R^(2) - (R - (t)/(2))^(2)} dr = (mu_(0) I^(2))/(8pi R^(3) t^(2)) [((R + (t)/(2))^(3) - (R - (t)/(2))^(3) )/(3) - (R - (t)/(2))^(2) t]`
`= (mu_(0) I^(2))/(8pi^(2) R^(3) t) [Rt + 0 (t^(2))] = (mu_(0) I^(2))/(8pi^(2) R^(2))`