On magnetic fields 'inducing' electric fields

Question

Griffith's Introduction to Electrodynamics, 4th Ed. introduces the concept of Faraday's law by providing us with three experiments: one in which a coil is pulled out of a region of a constant magnetic field (the field is perpendicular to the area of the loop, not perpendicular to the area vector), the second where the constant magnetic field is moved (in the opposite direction as in experiment 1) with the coil held stationary, and the third where both are held stationary and the intensity of the field is varied with time. All three experiments have current induced in the circuit.

While the observation of current in the first experiment is explained by 'motional EMFs', the second is because 'a changing magnetic field induces an electric field'. To this, Griffith's adds a lengthy footnote, which I have paraphrased:

"Induced is a slippery verb, for it inevitably reminds us of causation. There is an ongoing debate of as to whether a changing magnetic field should be regarded as an independent "source" of electric fields; after all, the magnetic field is itself due to electric currents. Ultimately, $\rho$ and $\overline{J}$ produce all electromagnetic fields, and changing the magnetic field merely delivers electromagnetic news from currents elsewhere."

It is the statement in bold that I would like to consider. Which 'currents from elsewhere' contribute to the field?

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Here's something I tried, but seem to have gone horribly wrong somewhere:

A constant magnetic field is famously produced by an infinite sheet of current. So, consider a sheet of steady surface current $\overline{K} = \sigma\overline{v}$ along the $\hat{x}$ direction, confined to the $xy$ plane. Then, the field is along the $\hat{y}$ direction; it is $\overline{B} = \frac{\mu_{o}}{2}\overline{K}\hat{y}$ (on one side; the other is along $-\hat{y}$). Consider now a square loop of conducting material parallel to the $xz$ plane, completely on the side of the current carrying plane with field along $\hat{y}$. We have achieved the set up of experiment 2.

My question is: can I consider the induced emf in the loop as because the current in my sheet changes direction because of the plane's motion? This might result in a change in the direction of the net field, which may now have a component along the length of the coil, thus inducing an emf.

If the plane moves as in experiment 2, then we need a motion that is perpendicular to the direction of the field. If we move our current carrying plane away from the loop at constant velocity $\overline{u}$ along $\hat{z}$, then we are done. The net velocity of the charge on the plane is now $\overline{w} = v\hat{x} + u\hat{z}$. We can consider this current to be contained in a plane that is inclined with respect to our original $xy$ plane, but shares the y axis of the originial plane. The field is still along $\hat{y}$, and so still doesn't have a component along the length of the loop (which is parallel to the original $xz$ plane).

However, if we move our current carrying plane towards the loop (or, in the direction of the $\overline{B}$), then our surface current has velocity along $\overline{w} = v\hat{x} + u\hat{y}$. The resulting $\overline{B}$ is now still contained in the $xy$ plane, but no longer along $\hat{y}$ alone. So, there is now a component along the length of the loop, thus inducing an emf.

This seems quite contrary to the experiment, for the first translation is supposed to produce an emf while the second isn't. Where am I going wrong? Also, which 'currents from elsewhere' contribute to the generated emf in this experiment?

Answer 1

Ultimately, 𝜌 and 𝐽 produce all electromagnetic fields, and changing the magnetic field merely delivers electromagnetic news from currents elsewhere.

Here Griffith is referring to Jefimenko’s equations. Griffith correctly points out the fact that Maxwell’s equations do not have the form of a causal relationship. So when we talk about a changing magnetic field inducing an electric field and vice versa (Faraday’s law and Ampere’s law) we are often tempted to think causally: that the changing magnetic field causes the electric field and vice versa. If we could think of induction as not a causal relationship then we would be OK, but in practice that is hard.

The reason that it is not causal is because it has the wrong form. A cause is something that happens earlier and an effect happens later. So a causal relationship has the form: $f(t)=g(t_r)$ where $t_r<t$ is called the retarded time. Then $g$ is the cause of $f$ and $f$ is the effect of $g$. Causes happen before effects, not at the same time.

The laws of EM can be written in terms of causal effects. That is Jefimenko’s equations and the closely related retarded potentials: $$\phi(\vec r, t) = \frac{ 1 }{ 4\pi \epsilon_0 } \int \frac{ \rho(\vec r’,t_r)}{ |\vec r - \vec r’|} d^3 \vec r’$$ $$\vec A (\vec r, t) = \frac{ \mu_0 }{ 4\pi } \int \frac{ \vec J(\vec r’,t_r)}{ |\vec r - \vec r’|} d^3 \vec r’$$

The retarded time, $t_r = t - \frac{|\vec r - \vec r’|}{c}$ changes from location to location but is always less than $t$. So in the retarded potentials expression $\rho$ causes $\phi$ and $\vec J$ causes $\vec A$. The causes happen before the effects, and at different locations.

So the phrase “currents from elsewhere” simply refers to the term $\vec J(\vec r’,t_r)$. The magnetic field is caused (in the correct sense of cause and effect) by the current at the retarded time in a distant location.

Answer 2

Any data communication relies on propagating electromagnetic waves. They are generated by a current in one place and converted into data, images, sound and the news elsewhere. This includes light, radio waves, internet etc. This is what is meant by the bold statement.

By the way, the statement 'a changing magnetic field induces an electric field' is misleading. A better statement would be 'a changing magnetic field is the same thing as an electric field'.