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Is the condition $\dfrac{\partial\textbf{E}}{dt} = \boldsymbol{0}$ sufficient and necessary for the Biot-Savart law to hold? If it's sufficient but not necessary, under what other conditions does the Biot-Savart law hold? If it's necessary but not sufficient, what else is required for the Biot-Savart law to hold?

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It is necessary but not sufficient. For Biot-Savart to hold, you need your current carrying wires to be in infinite free space, such that there is a boundary condition for $\mathbf B$ to fall off to zero at infinity and more or less at the same rate for every direction.

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  • $\begingroup$ I'd appreciate it if you'd explain in more detail the "more or less at the same rate in every direction" thing. $\endgroup$ – user113773 Nov 27 '18 at 21:43

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