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Suppose you have a uniform ring charge rotating at constant angular velocity so that you also have a uniform ring of steady current, and thus you can use the Biot-Savart Law to compute the magnetic field. But I also remember from college that to good approximation you can use electrostatics to compute the electric field due to the charge if the current is steady, in spite of the fact that charges are moving. I never understood how this works. Can someone offer insight?

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In the stationary situation all partial derivatives with respect to time vanish (this is in a sense the definition of a stationary state). Looking at the Maxwell equation relevant to determine the electric field, you note that they are given by $$\nabla \cdot \mathbf E = \rho, \qquad \text{and} \qquad \nabla\times\mathbf{E} = -\frac1c\partial_t\mathbf B=0,$$ i.e., the same equations as in the steady state, so you can introduce a potential and reduce the set of equation to Poisson's equation.

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  • $\begingroup$ That's crystal clear. I was anticipating some complex cancellation of first order contributions. It would also seem I was somewhat mistaken in that given ideal sources, the reliability of electrostatics is in fact exact, not just an approximation. That's good to know. $\endgroup$ – David H Sep 5 '12 at 18:41

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