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Faraday’s Law and Lenz’s Law of Induction

Faraday’s Law of Induction Lenz’s Law
Last semester in EE 381 Electric and Magnetic Fields, you saw that in
. Electrostatics: stationary charges produce `E (and D)
. Magnetostatics: steady currents (charges in constant
motion) produce `B (and H ). These are two distinct theories that were developed from two different experimentally derived laws: Coulomb’s law and Ampère’s force law. Now we are going to consider time-varying fields. While many of the concepts we’ve learned in statics will still apply, two new phenomenon we will observe are:
.Time varying `B produces `E, and
. Time varying `E produces `B!!
The complete electro-magnetic theory uses Coloumb’s and Ampère’s laws as a subset and requires one more experimentally derived law called Faraday’s law of induction. We’ve seen that a steady current in a wire produces a `B:

It may seem possible (by some type of “reciprocity”) that if we had a wire and a magnet, for example, that a current would be “induced” in the wire:

This doesn’t occur, however. If it did, there would be a clear violation of conservation of energy. What Faraday (ca. 1831) and Henry showed was that a timevarying magnetic field would produce (or “induce”) a current I in a closed loop!

Mathematically, Faraday’s law states that

In words, Faraday’s law states that the emf generated in a closed loop is equal to the negative time rate of change of the magnetic flux linking the loop.Substituting for the definitions of emf and ym yields an equivalent form of Faraday’s law of induction

Point Form of Faraday’s Law
By applying Stokes’ theorem to (2), as done in the text in Section 4.1.1, we can derive the point form of Faraday’s law. Specifically, applying Stokes’ theorem to the left-hand side of (2) gives

Substituting this result into (2) and combining terms gives

Since this result is valid for all s and c, then the integrand must vanish, leaving

This is called the point form of Faraday’s law.
Lenz’s Law
Why the minus sign in Faraday’s law? [For example, equations Because of Lenz’s law. states that the `B produced by an induced current (we’ll call this `Bind ) will be such that `Bind opposes the change in the ` that produced the induced current. If this weren’t the case, the `B field would grow indefinitely large even for the smallest induced current! Consider:

We can see here that the total `B (on the left-hand side) is increasing without bound if the induced `B enhances the original `B