Does a time-varying electric field, which is produced by a time-varying magnetic field, itself induce an other magnetic field?

Based on the Maxwell equations we know that

A time-varying magnetic field induces an electric field

A time-varying electric field induces a magnetic field

Suppose that an electric field, which is induced by a time-varying magnetic field, is itself time-varying (not stationary). Then the induced time-varying electric field itself induce an other time-varying magnetic field?

Time-varying magnetic field -> Time-varying electric field -> Time-varying magnetic field -> Time-varying electric field -> ...

If we measure the net magnetic (or electric) field in a point, we actually measure the resultant of all individually induced magnetic (or electric) fields in this chain?

• Well, that's why electromagnetic waves propagate in vacuum... Aug 11, 2022 at 15:06
• I believe the accepted answer is correct. Nevertheless, this question reminded me of a chapter in Feynmann's lecture where he indeed starts with a varying electric field and goes on to make corrections similar to the argument you make in the question. You might find it interesting! Aug 12, 2022 at 16:11

You are thinking about it in the wrong way. What Maxwell's equations tell you is that when you have a time-varying magnetic field, there must also be an electric field present that satisfies the Maxwell-Faraday equation. The two fields co-exist (and indeed are different aspects of THE electromagnetic field). Similarly, there must be the right value of current density present such that both sides of the Ampere-Maxwell equation agree.

In each case, the equality sign does not imply a causal relationship, it merely says that the left and right hand sides must be equal.

• One could argue that if the rate of change of something is sourced by something else it defines a causal relationship between the two? Aug 15, 2022 at 7:40
• @chris Unclear what you mean by sourced. You can produce a time-varying magnetic field with a changing current. That simultaneously means there is an electric field commensurate with Maxwell's equations. Ditto for if you were to move a permanent magnet - the time-varing B-field would coexist with the curling E-field. Aug 15, 2022 at 10:05
• I mean d/dt of something = source. I understand your point that E,B are components of Fmunu so they are closely linked but nonetheless IMHO Maxwell's equation clearly define a causal relationship where changes in B are sourced by gradients in E and vice versa (well curls and div to be more precise). Aug 15, 2022 at 10:12

Maxwell was being unnecessarily pedantic. The only way an electromagnetic field can vary is with time. Any field can only vary over time. So he was really only describing a simple variation of the field.

A variation of the electromagnetic field necessarily implies a variation in both its electric and its magnetic properties. Maxwell unified electricity and magnetism, into a single theory. The inescapable consequence is that both the electric and magnetic components of the field must vary simultaneously, and in proportion to each other, since there is only a single field present, such that the effects we had previously perceived as separate effects must therefore be related: must in fact be inter-dependent.

We might therefore view a shift in the electric field as being caused by the shift in the magnetic field, or vice versa; but there is no logical consequence -- such as that as suggested by the question -- that the electromagnetic field causes some effect outside that field.

Maxwell's description of the unified field describes only how the field itself changes, and does not imply that such changes have an external effect separate from the field.

If the field is moving, that still does not imply that it is causing changes in a separate field. The changes, logically, are still occuring in the initial field, albeit that the field has motion.

• What do you exactly mean by "moving" and "motion"? Do you mean propagation of weave in space? Aug 14, 2022 at 4:32
• An electromagnetic wave propagates through spacetime, so by definition it is in motion. This differs it from a static field, such as that created by an electromagnet. But the only significant difference is that the former is moving. Sep 1, 2022 at 5:30