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I am studying Electromagnetism from Griffiths and in the book it is stated that diagonal elements of Maxwell's tensor represent pressure. I want to calculate pressure on the wirings of an infinitely long solenoid with constant current. I found $$T_{xx} = T_{yy} = -\frac{\mu_0n^2I^2}{2} $$ and $$T_{zz}= \frac{\mu_0n^2I^2}{2} $$ I also know the actual answer from lorrentz force per area: $$Pressure = \frac{\mu_0n^2I^2}{2} $$ My question is how do we get the answer from those three components of stress tensor ? They are the same (except the minus sign of the $ T_{zz}$) but there are three of them ?

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  • $\begingroup$ Is this a homework question? $\endgroup$
    – hft
    Apr 18, 2022 at 19:34

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It seems like you're curious as to what the three different components of the stress tensor mean. Roughly speaking, $T_{xx}$, $T_{yy}$, and $T_{zz}$ tell you how much force per area is being exerted across a plane perpendicular to the $x$-axis, $y$-axis, and $z$-axis respectively.

So if your solenoid was parallel to the $z$-axis, and you imagined a plane bisecting it perpendicular to its length, then there would be a net repulsive force of $T_zz$ per unit of area inside the solenoid. Conversely, if you bisected this solenoid along a plane parallel to the axis, the force between the halves would be attractive but have the same magnitude (since $T_{xx} = T_{yy}$ is negative.)

In general, the force $d\vec{F}$ exerted on a unit area $d\vec{a}$ is given by $dF_i = \sum_j T_{ij} da_j$, where $i$ and $j$ run over the index values $\{x, y, z\}$.

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My question is how do we get the answer from those three components of stress tensor ?

What you are asking is not clear. Are you asking how to get the off-diagonal components? Or are you asking at how you arrived at the three expression for the diagonal components that you wrote in your post?

I will assume the former, since you wrote "I found" in your post, and that seems to imply that you already know how you got the answer for the diagonal components.

For the off-diagonal components you use the same general expression: $$ T_{ij} = \frac{-1}{\mu_0}\left(B_iB_j - \frac{B^2}{2}\right) $$

And you use whatever expression for the B field you seem to have already used, but did not state in the question. Presumably something like $\mathbf B = \hat z \mu_0 n I$.

My question is how do we get the answer from those three components of stress tensor ?

Here, I think what you mean by "get the answer" is something like "relate the components of the stress tensor to the Pressure."

To do this, use the definition of the stress tensor. In particular, the pressure that would be exerted on a surface with normal in the $i$ direction is $T_{ii}$.

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