Maxwell's 3rd Equation

northWe now turn to Maxwell’s 3rd equation that is based on Faraday’s law of induction, which was discovered independently by Michael Faraday and Joseph Henry in 1831, although it was Faraday who first published his results. In words, this law might be expressed as follows:

The induced electromotive force (VE) in a closed loop equals the negative of the rate of change of magnetic flux through the loop .

Basically, moving the magnet shown in the diagram back and forth through the wire loop causes a current, i.e. a flow of charge particles, in the wire [L]. In physics, the energy supplied to drive a unit charge around the circuit is defined by the following unit of measurement:

[1]      1

In this case, the difference in charge potential between two points in the circuit is being created by the change in the magnetic field, i.e. the change in magnetic flux [φB]. However, this difference in charge effectively gives rise to an electrical potential energy, which is known as the electromotive force, which is being denoted as [VE] on the basis that it is primarily associated with the electric field [E]. As such, we may translate the definition above to its mathematical counterpart:

[2]      2

If we look at the units in [1], we might wonder if the term ‘electromotive force’ is misleading as the process seems to involve energy, not force, with respect to charge. In fact, [VE] corresponds to the work done, i.e. energy, associated with moving one unit of charge around the circuit shown. We might also remember that the electric field [E] is equal to the force per unit charge, as outlined in the earlier introduction of electric fields . As such, it is the closed line integral of the electric field [E] that corresponds to the work done in moving a charge [q] around the circuit [L]:

[3]      3

However, before proceeding, it might yet again be useful to do a sanity check in terms of the units on both sides of [2]:

[4]      4

As such, we might feel reassured that the quantities on either side of [2] are comparable. Also, in the previous discussion of Maxwell’s 2nd law, it was shown that the magnetic flux [φB] across any arbitrary surface is given by the integral form:

[5]      5

However, unlike Maxwell’s 2nd equation, the conceptual Gaussian surface being considered does not enclose the magnetic field [B], it only circles it in the form of the wire loop [L], i.e. we are talking about a line integral not a surface integral. In fact, it might be seen that we can combine [2], [3] and [5] into the form:

[6]      6

As such, [6] can be rationalised to the integral form of Maxwell’s 3rd equation in which the units of each parameter has been expanded to check for equality:

[7]      7

As such, we can see that the units in [7] remain consistent with [4], but the integration with respect to [L] means that [7] is accounting for all the charge movement in the loop. To obtain the differential form, Stokes’ theorem has to be applied to the left-hand side, which may not really add any clarity to the physics, but a few steps are presented for general reference:

[8]      8

However, this law may be better understood in terms of moving a permanent magnet through a loop of wire, i.e. moving the magnet produces a changing magnetic field in time (dB/dt) when measured relative to the loop, which then induces an electric field in the wire loop surrounding the moving magnet.

lens law

The significance of the negative sign (-) in both [6] and [8] can best be explained in terms of Lenz's Law, which states that the electric field produced by the changing magnetic field causes a current to circulate in the wire loop, which in-turn produces another magnetic field. This secondary magnetic field always opposes the original changing magnetic field and is linked to the conservation of energy.  Initially, in position-1, shown in the diagram above, the moving magnet produces no measurable flow of current in the distant wire loop. However, as the magnet passes through the loop a small current is induced in the wire loop, which in-turn generates a secondary magnetic field. The direction of the secondary magnetic field opposes that of the primary magnetic field and so conserves the net energy of the system.