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Maybe I am being confused. It was some years ago I did this. An electric current changes charge distribution which creates rotation in $\bf B$. So in Ampères / Biot-Savarts law what is the difference between $\bf J$ and the time derivative of $\bf E$ field? Is one of them change of $\bf E$ field which could be explained except for a current or other way around?

Here is the version of equation which I am looking at: $$\nabla\times {\bf B} = \mu_0 \left( {\bf J}+\epsilon_0\frac{\partial {\bf E}}{\partial t} \right)$$

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Think of a parallel plate capacitor as it charges.

The electric field across the capacitor changes but there is no current between the parallel plates. The only way to explain the magnetic field between the plates is using the $\frac{\partial{E}}{\partial t}$ term.

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  • $\begingroup$ @mathreadler, you're welcome. Please accept the answer if you are satisfied with it. $\endgroup$ Jan 27, 2019 at 16:01
  • $\begingroup$ I have a principle of never accepting answer until after 24 hours. To not discourage other people from answering. $\endgroup$ Jan 27, 2019 at 16:12
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There is also a case in which $\nabla \times B =0$, in which case $J$ and $\frac{\partial {\bf E}}{\partial t}$ are proportional.

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  • $\begingroup$ But the OP is asking about the difference between these two terms. In classical mechanics force and acceleration are proportional, but saying this doesn't tell me what differentiates the two things. $\endgroup$ Jan 29, 2019 at 5:27
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$\partial \mathbf E / \partial t$ represents the rate of change of the electric field $\mathbf E$ at that point. Electric fields do not need to change at a point due to movement of charges at that point. For example, consider the space between the plates of a charging capacitor (moving charges are nearby, but not at that point) or consider electromagnetic waves travelling through a vacuum (no moving charges are present at all).

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