Faraday's Law: UNIZOR.COM - Physics4Teens - Electromagnetism - The Laws of Induction
Channel: Zor Shekhtman on UNIZOR Education
Date: 2020-06-17

UNIZOR.COM - Physics through a Looking Glass of Mathematician 

Notes to a video lecture on http://www.unizor.com

Faraday's Law

Let's quantify the voltage (or electromotive power - EMF) generated by a straight line wire moving in a uniform magnetic field parallel to itself in a plane perpendicular to magnetic field lines.

In a setup presented in the video the straight line wire PQ is moving parallel to itself and parallel to Y-axis along the rails AP and OQ stretched parallel to X-axis and connected by a resistor R, making a closed circuit with a wire PQ being a part of it.

The uniform magnetic field of intensity B is directed along the Z-axis perpendicularly to XY-plane in a direction of viewing the picture.

"Free" electrons, carried by a wire's motion along X-axis, experience the Lorentz force directed perpendicularly to their movement inside a wire, that is they will move to one end of a wire. 
As a result, the difference in electric potential (or EMF) is generated and an electric current I will start running in the closed circuit OAPQ.


So, Lorentz force on electrons moving with a wire along the X-axis causes the electrons to move to one end of a wire, that causes the electric current in the wire. The latter causes the force exerted by the magnetic field onto a wire in the direction opposite to its movement.
Since we are moving the wire ourselves with constant speed V towards the positive direction of X-axis, we have to exert work against the force of magnetic field onto a wire directed in the negative direction.

This force of magnetic field onto a wire directed in the negative direction equals, as we mentioned above, to
F = I·L·B
where L is the length of a wire PQ.
If during time t the wire moves by a distance S=V·t, the work we have to do against the force F equals to
W = F·S = I·L·B·V·t
To do this work during a time interval t, we have to exert a power
P = W/t = F·S = I·L·B·V

Energy does not disappear without a trace. In this case the energy we spend generated voltage (EMF) U at wire's ends and an electric current I in a closed circuit. It will be transformed into heat in the resistor R with a rate I·U.
Therefore, we can equate two powers - the one we exerted to move a wire and the one consumed by a resistor:
I·L·B·V = I·U
from which we obtain the amount of EMF generated by moving the wire
U = L·V·B

Let's examine the expression on the right.
The value L·V represents the rate of increase of the area of a circuit OAPQ per unit of time.
The product of the area and the intensity of the magnetic field with magnetic field lines going through this area perpendicularly to it is called magnetic flux Φ.
Since our magnetic field is uniform, an expression L·V·B represents the rate of increase of magnetic flux, which we can safely denote using the language of calculus as dΦ/dt.

In this language we can state that the electromotive force (EMF) generated by a moving wire - a part of a closed circuit - equals to a rate of increase of magnetic flux going through this closed circuit.
U = dΦ/dt.
This is the Faraday's Law of electromagnetic induction.