Can a magnetic field be induced without an electric field? Because, as far as I know, a time varying electric field induces a magnetic field an vice versa. But in the case of conductors carrying currennt, it doesn't seem that electric field varies with time, then how is a magnetic field induced?
Answer
One of Maxwell’s four equations for electromagnetism in a vacuum shows how magnetic fields are produced:
$$\nabla\times\mathbf{B}=\frac{1}{c}\left(4\pi\mathbf{J}+\frac{\partial\mathbf{E}}{\partial t}\right).$$
(I’ve written it in Gaussian units.)
From this equation you can see that there are two different sources for magnetic fields: the first is a current density, and the second is a changing electric field.
So to have a magnetic field you do not need to have a time-varying electric field. You can just have moving charge. But when a magnetic field is produced by moving charge, physicists don’t call it “induced”.
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