Induced electric fields and induced magnetic fields confusion

[vish22]

So changing magnetic fields induce electric fields (Faraday's law when the magnetic field is changed by either moving the source or by changing the current in the source that's causing the magnetic field, ie. we're not moving the conductor where an emf is induced so there's no f=qvXB).
Also, changing electric fields induce magnetic fields (Maxwell-Ampere law, particularly with regards to displacement currents).

Now here's my query :
A changing magnetic field produces an electric field at a point in space. So now there's some electric field that's created hence there's dE/dt at that point in space. Wouldn't this give rise to another magnetic field (Maxwell-Ampere displacement current law)? And wouldn't this newly generated magnetic field (with an associated dB/dt) induce yet another electric field at that point in space and so on and so on infinitely? I'm not sure I understand how induced fields work.

 

[vanhees71]

This is a misconception, which unfortunately is sometimes even propagated by textbooks. Mathematically the only sources of the electromagnetic field are charge and current distributions. Correspondingly the em. fields are retarded solutions of the wave equations with charge and current distributions (and their derivatives) as sources.

This also makes sense from a relativistic point of view, because there is an electromagnetic field, described by an antisymmetric tensor field in 4D Minkowski space with electric and magnetic field components, but this split is dependent on the observer. Another observer, moving with respect to the former will split the electromagnetic field in a different way into electric and magnetic field components. The same holds true for the charge and current distributions which together form a Minkowski four-vector field.

Last but not least, you intuition gained from the somewhat questionable idea of time-dependent magnetic fields would be the cause of electric fields and vice versa, is not entirely wrong, because indeed there are electromagnetic waves in free space (i.e., in regions, where no charges or currents are present).

 

[vish22]

So if I'm looking at this correctly - the changing magnetic field is accompanied by an electric field such that Faraday's law is satisfied (ie. curl E = -dB/dt). The induced electric field isn't caused by the changing magnetic field, rather, the induced electric field is a consequence of the change in the magnetic field. Also, since an electric field is induced (as a consequence of the change in the magnetic field), there exists a dE/dT. This dE/dT fulfills the Maxwell-Ampere relation (ie. curl B = kdE/dT, k is some constant; assume there are no charge distributions or currents nearby). In other words, kdE/dt isn't creating any new B field, but rather, it equals the curl of B at any instant in which B is transforming.

The only real entity creating any field change here is the source of the changing magnetic field (ie. some charge/current distribution located far away - far enough for us to not include them in the domain we're considering here).

 

[vanhees71]

I find your first sentence describes it most accurately: "If the $\vec{B}$ field is time-dependent then there must be an electric field $\vec{E}$ due to Faraday's Law, $\vec{\nabla} \times \vec{E}=-\partial_t \vec{B}$. That equation doesn't express a cause-effect relation though. It's just a (local) property of the components of the electromagnetic field wrt. a (inertial) reference frame.

 

[vish22]

Thank you. I think it's clearer now.

 

[TSny]

Now here's my query :
A changing magnetic field produces an electric field at a point in space. So now there's some electric field that's created hence there's dE/dt at that point in space. Wouldn't this give rise to another magnetic field (Maxwell-Ampere displacement current law)? And wouldn't this newly generated magnetic field (with an associated dB/dt) induce yet another electric field at that point in space and so on and so on infinitely? I'm not sure I understand how induced fields work.

You might enjoy this discussion by Feynman regarding the interrelationship between changing E and B fields for a particular example. See section 23-2 in this lecture.

 

[vish22]

Beautiful - love how Feynman derived the J0 series using the exact same arguments I put forth. So it is indeed an infinite process - where a change in E is always accompanied by a change in B and vice versa. Let's just hope there's always a J0-like series we could use to get a resultant E(x,t) and B(x,t) field.

 

[TSny]

Beautiful - love how Feynman derived the J0 series using the exact same arguments I put forth.

Yes, this is one of my favorite chapters in the Feynman lecture series. (But there are many favorites!)

So it is indeed an infinite process - where a change in E is always accompanied by a change in B and vice versa. Let's just hope there's always a J0-like series we could use to get a resultant E(x,t) and B(x,t) field.

Well, the infinite process is one way we humans can sometimes calculate the resultant fields. I'm not sure Nature does it that way.

 

[alan123hk]

I saw this statement a long time ago in a physics book in the library, because the change of the magnetic field produces an electric field nearby, and the electric field produces a magnetic field nearby, and so on, causing the propagation of electromagnetic waves.

Now I think this statement seems to be very problematic. If you look at a simple planar electromagnetic wave, the electric and magnetic fields are in phase and traveling at 90 degrees to each other, they reach the highest point, the zero point, and the lowest point at the same time, there is no time for only the electric or magnetic field to happen.

Looking at the source of electromagnetic waves, it is to accelerate the charge and change the current. When they move or accelerate, the electric field and magnetic field also appear at the same time at the beginning, and they follow each other like shadows and never separate. So it looks like causality doesn't exist.

 

[vanhees71]

It's a common misinterpretation of Maxwell's equations. The correct analysis is that the homogeneous Maxwell equations are constraint equations, while the dynamical equations, i.e., the fields resulting from their sources are the inhomogeneous Maxwell equations, and the sources are solely charges and currents.

The most direct way to see this are the socalled Jefimenko equations, describing the field $(\vec{E},\vec{B})$ (or equivalently $F_{\mu \nu}$ in the relativistically covariant notation) as a retarded solution of the corresponding wave equations with the charge and current densities as sources, demonstrating the causal connection between the sources and the fields.