For this post I will be introducing possibly my favorite of Maxwell’s four equations, the Maxwell-Faraday Equation that relates the electric field to the magnetic field in the following way:
This equation was quite revolutionary at the time it was first discovered as it revealed that electricity and magnetism are much more closely related than we thought. In fact, originally electricity and magnetism were thought of as completely separate entities but thanks to the works of Faraday and Maxwell (actually, Einstein as well later on) we have a much richer understanding of the true nature of the two fields now. In this post I will be exploring the origin of this equation as well its mathematical and physical significance. If you have not read my previous posts regarding the first two Maxwell equations you can read them here and here.
Origin of the Maxwell-Faraday Equation
Our story begins with Michael Faraday and his famous law known as Faraday’s law of induction that states that if the magnetic flux through a closed loop of wire changes, an electromotive force (and hence a current) will form. Mathematically it can be represented in the following equation:
Where the fancy E is the electromotive force, and Φ is the magnetic flux (think of it as the surface integral of the magnetic field through the loop). Originally, this law was found through experiments conducted by Faraday himself as he discovered that moving a loop of wire through a magnetic field causes a current to form. But why does this happen?
Well, suppose we have some region with a constant magnetic field and a loop of wire that is partially in the region of magnetic field. Currently, no current flows through the wire. Now, if we pull the loop to the right (away from the region of magnetic field) we will notice a current flow in the wire. This situation is illustrated in the diagram below:
The reason why current flowed when the wire moved and why it didn’t when the wire was stationary lies in the Lorentz force law that explains how charged objects experience forces due to surrounding electric and magnetic fields:
Focusing on the magnetic field part of the equation, we can see that if there is a velocity in the charged objects (represented by v) it will experience a force in the direction of the cross product between v and B. Well, if we look back at our situation, we can notice that if the wire as a whole is moving, the free charges in the wire must be moving as well and hence experience a force. If we assume the magnetic field is pointing into the page, we can use the Fleming’s hand rule to show that free charges on the left edge of the wire will flow upward and hence product a current (actually, it is a bit more complicated than that as the upward movement of the charges will change the direction of the velocity vector and hence the force it experiences, but it does work out nicely in the end).
After doing a bit of math, the equation for Faraday’s law of induction I showed earlier can in fact be derived using this principle. However, the story doesn’t end here.
Faraday performed yet again another similar experiment but this time instead of moving the loop of wire, he moved the magnetic field itself to the opposite direction:
Unsurprisingly, the same current we saw in the previous case was produced in the wire. Why is this unsurprising you ask? Well, if we compare this situation to the previous case, the magnetic flux through the loop must be decreasing just like it did before since the relative motion of the objects aren’t changing. If we then use Faraday’s law, the same electromotive force must be produced due to the changing magnetic flux.
Although you may conclude this as trivial and move on, stop and think about it for a second. In the previous example our entire explanation for the induced current hinged on the movement of the free charges in the wire, but in this situation they actually aren’t moving at all. How can a magnetic field that only affects moving charges cause a force on stationary charges? The only thing that can possibly cause a force on stationary charges is the electric field which is seemingly not present in this situation. It was at this moment that Faraday had perhaps one of the most important and revolutionary ideas in the history of classical electrodynamics as well as physics in general: “What if a changing magnetic field produces an electric field?” If this is the case, then the electromotive force must be caused directly by the electric field (and only indirectly due to the magnetic field).
If we recall that the electromotive force is a measure of the work done per unit charge, we can express this in terms of a closed line integral of the electric field around the loop (since F=qE). Now, equating this with the magnetic flux we get the following:
Actually, we are already half way there as the equality between the second and forth expression above is already the integral form of the Maxwell-Faraday law. We can now convert it to differential form. If we apply Stokes’ theorem on the electric field line integral, we can turn it to a surface integral of ∇×E. Since this surface is the same surface we are integrating the magnetic field over, we can drop the integral to get the following:
Which is precisely the Maxwell-Faraday equation. While there was a lot of mathematical nuances and derivations I didn’t mention, I think seeing how this equation shows up from just considering two experiments is quite interesting and helps in intuitively understand the equation in terms of its relation to the physical world.
Interpreting the Maxwell-Faraday equation
This section will be a bit shorter since we have already touched upon the most important interpretation of the equation that a changing magnetic field produces an electric field. However, it does not just produce any type of electric field. Looking at the Maxwell-Faraday equation, we can see that a changing magnetic field will actually produce some curl in the electric field. Since the curl is a measure of how much the electric field rotates about a point, this means that the electric field produced by a magnetic field will intrinsically have some rotating property to it. As you may notice, this is very different from what Gauss’s law stated as in that case the electric field due to a stationary charge had no rotation but instead pointed radially outwards/inwards. If we think back on the second experiment we looked at, this actually makes sense as an electric field with a curl will naturally cause a rotation (or circulation) of the free charges around the loop which is precisely what causes the current!
While for many years we believed that electricity and magnetism were completely separate, it actually turns out that they were really just two sides of the same coin. In fact, in the next article where I will look at the final Maxwell equation, we will see that a changing electric field can cause a magnetic field as well, showing how closely related these two fields really are. Thank you for reading.
References
GRIFFITHS, D. J. (2023). Introduction to electrodynamics. CAMBRIDGE UNIV PRESS.
