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Question

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

Solution
Verified by Toppr

Let t= thickness of the wall of the cylinder. Then,
J=I2πRt along z axis. The magnetic field due to this at a distance r
(Rt2<r<R+t2), is given by,
Bφ(2πr)=μ0I2πRtπ{r2(Rt2)2}
or, Bφ=μ0I4πRrt{r2(Rt2)2}
Now, F=J×BdV
and p=Fr2πRL=12πRLR+t2Rt2μ0I28π2R2t2r{r2(Rt2)2}×2πrLdr
=μ0I28π2R3t2R+t2Rt2{r2(Rt2)2}dr=μ0I28π2R3t2⎢ ⎢ ⎢ ⎢ ⎢(R+t2)3(Rt2)33(Rt2)2t⎥ ⎥ ⎥ ⎥ ⎥
=μ0I28π2R3t[Rt+0(t2)]μ0I28π2R2

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