October 27 – The Maxwell-Faraday Equation
Meaning of Maxwell-Faraday Law Equation: “Changing number of magnetic field lines (magnetic flux) through a closed path produces an electric field (EMF) around that path”
Understanding how this equation works
The Maxwell-Faraday law is the general form of the flux rule of electromagnetism which says that the induced emf around a closed path is equal to the rate at which the magnetic flux through a conducting circuit is changing. Let us now understand how it works.
Imagine a set up as shown in the figure below.
There exists a uniform magnetic field B in space. We keep a simple loop of wire whose dimensions can be changed. So the loop has two parts: a fixed U-shaped part and a movable crossbar of length L. The crossbar can slide on the legs of the U shaped loop so the area of the circuit is variable. So far so good.
Now according to the flux rule, if we move the crossbar, there should an emf in the loop that is proportional to the rate of change of flux through the loop. The master equation that governs this law is the Lorentz force law which states that the total force on a charge is given by F = q(E + v x B), where v is the velocity of the charge and q is its magnitude. In the above scenario, as we move the crossbar which contains the charges, the charges experience a magnetic force v x B and thus the emf is given by E = LvB.
On the other hand, what happens when the circuit is held stationary and the value of the magnetic field is changed with time? If we do that, we decrease the number of field lines entering the stationary circuit. In other words, we change (decrease) the magnetic flux through the circuit. Experimentally, Faraday discovered that exactly the same phenomenon occurs in this case too. An emf is developed across the loop if we change the magnetic flux through the loop instead of moving the circuit itself.
Now the question is, from where does this force come from? It turns out that the Lorentz force formula has the answer to it. Now when the circuit is held stationary (v = 0), the magnetic forces vanish and what drives the electrons in the loop is the E term in that equation. Faraday’s discovery thus showed that the electric and magnetic fields are related by a new law: in a region where the magnetic field is changing with time, electric fields are generated.
It is this electric field responsible for the emf that develops around the wire when the circuit is held stationary and magnetic flux is changed. The equation written above is the differential form of the flux rule by Faraday. The differential form was developed by James Maxwell, hence the name Maxwell-Faraday equation. In the above equation, L.H.S is the curl of the electric field which by Stokes theorem changes to emf around the closed-loop. The mathematical derivation of this form can be found here.