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Current is flow of charge. So if charge is moving, electric field is changing. So changing electric field should produce changing magnetic field, but I am not able to see how magnetic field is changing around a wire carrying current. For a given radius magnetic field is constant around a wire. Please let me know what am I missing in my understanding.

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    $\begingroup$ Electric field that changes with time creates a magnetic field which changes in space, not in time. The magnetic field generated at a particular position will be constant. $\endgroup$ – Kalpak Gupta Oct 3 '17 at 11:57
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Current is the rate of flow of charge. Which means that it is the amount of charge passing through the cross section of the wire per unit time. The current therefore remains constant as the amount of charge passing through the cross section of the wire per unit time remains constant. Electric field changes with the change in current. As there is no change in the current in the wire, it produces a constant magnetic field. If however the current in the wire would vary with time, it would produce a varying magnetic field.

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