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This question already has an answer here:

A magnetic field could not be induced unless there is a time varying electric flux associated with the amperial loop under consideration. But a well known field exists around a current carrying conductor. Then how can we say that electric field is absent around the conductor.

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marked as duplicate by Rob Jeffries, John Rennie electromagnetism Feb 26 at 16:16

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According to Ampère's circuital law, the magnetic field obeys the following equation:

$$\nabla \times \mathbf{B} = \mu_0\left(\mathbf{J} + \varepsilon_0 \frac{\partial \mathbf{E}} {\partial t} \right)$$

You could interpret the equation in the following way: the source of the magnetic field are a current density and a varying electric field. Although wires and cables conduct electricity, they are electriclly neutral. While it is true to say that there are electrons with negative charge travelling through the wire, it is also true that there are positive charged nuclei. Electrons and nuclei are uniformly distributed, so the effective electric field is equal to zero.

Therefore, the only contribution to the magnetic field comes from the current of the moving charges, the electrons.

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