# Can a magnetic field be induced without an electric field?

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?

• If you hold a permanent magnet over a peace of metal, will that induce a magnetic field? Commented Feb 26, 2019 at 6:04

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”.

• But aren't moving charges producing an electric field? And what about spin? Can a neutron for instance create a B field while no electric field? Commented Feb 26, 2019 at 10:24
• Moving charges don’t necessarily produce an electric field. For example you can have a positive charge density moving to the right and a negative charge density moving to the left at the same speed. There would be no net charge density but there would be a net current density. Quantum spin is another source of megnetic field. It is not in Maxwell’s equations because they are classical, not quantum. Commented Feb 26, 2019 at 17:03

From Griffiths, Electrodynamics, Jefimenko’s equations are given as $${\bf E}({\bf r},t) = \frac{1}{4 \pi \epsilon_0} \int [ \frac{\rho ({\bf r}',t_r)}{{\mathfrak r}^2} {\bf \hat{\mathfrak r}} + \frac{\dot{\rho} ({\bf r}',t_r)}{c {\mathfrak r}} {\bf \hat{\mathfrak r}} - \frac{{\bf {\dot J}} ({\bf r}',t_r)}{c^2 {\mathfrak r}}] d \tau',$$ $${\bf B}({\bf r},t) = \frac{\mu_0}{4 \pi} \int [\frac{{\bf {J}} ({\bf r}',t_r)}{{\mathfrak r}^2} + \frac{{\bf {\dot J}} ({\bf r}',t_r)}{c {\mathfrak r}} ] \times {\bf \hat{\mathfrak r}} d \tau'.$$

These equations show that to create a magnetic field you require either a steady current or/as well a changing current. If the current density is steady (so that $${\bf {\dot J}} \equiv 0$$) then you can see that you can arrange for no electric field by having the charge density $$\rho$$ vanish everywhere. Another way to create a magnetic field is to have a time varying current density, which necessarily creates an electric field.