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At the centre of a spherical continuous charge distribution with no external electric fields, the electric field is zero from symmetry arguments. But does the stress at the centre remain finite?

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Stress from what? From the electromagnatic field, or from the other stuff? –  Ron Maimon Jul 26 '12 at 1:46
    
@RonMaimon from the electric field of the charges acting on the centre. I thought someone might come up with a nice simple argument on the limiting process of $dr^3$ and how it affects the answer. –  Physiks lover Jul 26 '12 at 11:54
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The electromagnetic stress would still be zero, yes, because you can calculate it from

$$\sigma_{i j} \equiv \epsilon_0 \left(E_i E_j - \frac{1}{2} \delta_{ij} E^2\right) + \frac{1}{\mu_0} \left(B_i B_j - \frac{1}{2} \delta_{ij} B^2\right)$$

At a point where $\vec{E} = 0$, and given that this is an electrostatic situation so $\vec{B} = 0$, all components of the stress tensor will be zero.

Quantities which might not be zero would be those which involve integrals of the electric field, for example. (Also the mechanical pressure - the diagonal components of the mechanical stress tensor - would be nonzero in practice, but I'm guessing that's not what you are asking about.)

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