# Question about when the Biot-Savart law holds

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?

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.