Faraday's Law

Faraday found that if the magnetic field varied with time, it would create an electric field around it (principle of electromagnetic induction).

Magnetic Flux

Consider an open surface . Then the differential magnetic flux is defined by:

where is the angle between and .

So the total flux through a surface will be:

Equation

(see integral)

Warning

Not to be confused with Maxwell's second equation, which is for a closed surface, not open.

Intuition

This means the B-field is actually magnetic flux density (see Interpretation of Integrals)

Faraday's Law

Law

Consider a closed loop, and the open surface made by this loop.

Faraday's law states that if the magnetic flux enclosed within the loop changes with time, then an E-field is created which loops around the flux.
Furthermore, the strength of this E-field is proportional to the rate of change of the enclosed magnetic flux.

The negative sign is due to Lenz's Law.

(see integral, dot product, derivative)

Note the connection to Ampere's law, which states that . If we walk around loop and integrate , we get the total current enclosed by .

Faraday's law has the same form as Ampere's law, except with a time derivative. A time-varying magnetic flux creates an electric field which curls around the magnetic field.

Faraday's Law is Non-Conservative

For electrostatic fields, we have:

If we start from a point, and walk on a closed loop and come back to the same point, we do not create a potential difference.

However, if the magnetic field is time-varying and we walk on the loop and then return to the same point, we create a potential difference across the same point.

Important

Faraday's law is non-conservative. That is, path matters in Faraday's law.