Week 10

Basic Facts

Electro-Motive Force: Electro-motive force is a generalisation of the idea of electric potential to non-conservative electric fields. The electromotive force in a circuit Γ is defined to be
. The EMF is measured in volts and it plays the same role in Ohm's Law that the electrostatic potential did. Thus an EMF V will cause a current
to flow in a resistance R.

Faraday's Law: When the magnetic flux through a closed circuit Γ changes, an EMF is introduced in the circuit according to the equation
where S is any surface bounded by Γ.

Lenz's Law: The induced current acts to oppose the change that caused it. That is, the current flows in such a direction as to create a magnetic field that treis to keep the total flux constant.

Faraday's Law in differential form: application of Stoke's Theorem to Faraday's Law gives a differential form of the law
.

A magnetic field B stores energy. The energy density (energy per unit volume) stored in a field of intensity B is
.

Useful Facts

Self-Inductance: Faraday's Law causes a coil that carries a changing current I to create an induced EMF that opposes changes in the current. The induced EMF is given by
where L is a constant that describes the geometry of the coil. It is called the Self Inductance of the coil and is measured in units of Henries.

For a given coil geometry the inductance L can be calculated by computing the total magnetic flux, Φ_M, that is created in the coil by a current I and using
. For example, a long solenoid of length l (that is a letter el), radius R, wound with N turns of wire has an inductance
.

If two coils are placed so that flux from one coil passes through the other coil then a changing current I₁ in the first coil will induce an EMF V₂ in the second coil where
. The constant M is called the Mutual Inductance and it depends only on the geometry of the two coils.