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athrun200
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Homework Statement
In fact it is problem 8.5 in Griffiths 3rd ed p357
Consider an infinite parallel plate capacitor with the lower plate carrying the charger density [itex] - \sigma [/itex], and the upper plate carrying the charge density [itex] + \sigma [/itex].
(a) Determine all nine elements of the stress tensor, in the region between the plates. Display your answer as a 3x3 matrix
(b)Use Eq 8.22,[itex]\overrightarrow F = \oint\limits_S {T \cdot d\overrightarrow a } [/itex] for [itex]\overrightarrow S = 0[/itex], to determine the force per unit area on the top plate.
(c) What is the momentum per unit area, per unit time, crossing the xy plane?
(d) Find the recoil force per unit area on the top plate.
Homework Equations
Maxwell's Stress Tensor
[itex]\overrightarrow F = \oint\limits_S {T \cdot d\overrightarrow a } - {\varepsilon _0}{{\bar \mu }_0}\frac{d}{{dt}}\int {Sd\tau } [/itex]
Momentum density
[itex]{\wp _{em}} = {\varepsilon _0}{\mu _0}\overrightarrow S [/itex]
[itex]\frac{d}{{dt}}({\wp _{em}} + {\wp _{mech}}) = \nabla \cdot T[/itex]
The Attempt at a Solution
Part (a) and (b) are easy for me.
(a) [itex]{T_{xy}} = {T_{xz}} = {T_{yz}} = ... = 0[/itex] and by using some equations we can find [itex]{T_{xx}},{T_{yy}}{\rm{ and }}{T_{zz}}[/itex]
So the answer is [itex]T = \frac{{{\sigma ^2}}}{{2{\varepsilon _0}}}\left( {\begin{array}{*{20}{c}}
{ - 1}&0&0\\
0&{ - 1}&0\\
0&0&1
\end{array}} \right)[/itex]
(b) is also easy, use that equation provided we can find the answer [itex]\overrightarrow f = \frac{{{\sigma ^2}}}{{2{\varepsilon _0}}}\widehat z[/itex]
The problem is (c)
I think the equation [itex]\frac{d}{{dt}}({\wp _{em}} + {\wp _{mech}}) = \nabla \cdot T[/itex] is useful to solve it with [itex]{\wp _{mech}}=0[/itex]
However on the right hand side we have [itex]\nabla \cdot T[/itex]. The div of a tensor would be? I learned only the div over a vector, but not a tensor.