Electromotive force (emf)

• Consider the magnetic field (B-field) provided by a horseshoe magnet ( a curved bar magnet).
• If we place a stationary charge q within the B-field, it will feel NO MAGNETIC FORCE.
• Yet if we project the charge q through the B-field with a velocity v it will feel a force:

Induced electromotive force (emf)

∙It should thus come as no surprise that moving a piece of wire (a conductor) through a magnetic field produces a magnetic force on the charges in the moving wire:
∙Don’t forget the right-hand-rule!

• Consider this new experiment: If the north pole of a magnetic is suddenly thrust through a looped conductor, a current is created.
• T or F: Current travels through the circuit only while the magnet is moving through the loop.
• T or F: Current direction depends on which direction the magnet is being moved through the loop.
• 
• Since moving a conductor through a magnetic field produces a current, this very action must therefore induce an electromotive force (emf) in the conductor.
• Since moving a magnetic field trough a conductor produces a current, this very action must therefore induce an emf in the conductor.
  ∙We have shown, therefore, that all we need is relative motion between a conductor and a magnetic field in order to induce an emf.

FYI

• This is the principle behind electricity generation using turbines and generators. Motion of a conductor through a B-field produces an emf which can drive a current.

EXAMPLE:

Show that the emf \(\varepsilon\) induced in a straight conductor of length \(\ell\) moving at velocity \(v\) through a magnetic field of strength \(B\) is

SOLUTION:

Note that since \(\mathbf{v} \perp \mathbf{B}\), then \(\phi = 90^\circ\):
 \(F = qvB \sin \phi = qvB \sin 90^\circ = qvB\).
 Recall that \(E = \frac{V}{x} = \frac{V}{\ell}\) and that \(F = qE\).
 Since \(F = qE = \frac{qV}{\ell}\) and \(F = qvB\), we have
\[
\frac{qV}{\ell} = qvB
\]
\[
V = Bv\ell = \varepsilon.
\]

Magnetic flux

• Consider the experiment where the plane of the loop is in the same plane as the moving magnetic field:
• T or F: Because of the orientation of the loop most of the magnetic field lines to NOT pass through the area of the loop.
• T or F: There is no current generated in this experiment.
• Because of the importance in orientation between the area A of the loop and the magnetic field B, a new quantity called magnetic flux Φ is here defined.
•  
• Obviously we have to somehow define the direction of an area. Quite simply, the direction of an area is perpendicular (or normal) to the plane of that area.
• 

Magnetic flux density

Magnetic flux is measured in \( \text{T m}^2 \), which are also known as webers (Wb). Thus \( 1 \, \text{T m}^2 = 1 \, \text{Wb} \).
We define the magnetic flux density as the magnetic flux per unit area. Thus
\[
\text{magnetic flux density} = \frac{\Phi}{A \cos \theta}
\]
 But from the definition of magnetic flux, we see that
\[
\Phi = BA \cos \theta
\]
\[
\text{magnetic flux density} = \frac{BA \cos \theta}{A \cos \theta}
\]
\[
\text{magnetic flux density} = B.
\]

FYI

 Be aware that the magnetic flux density and magnetic field strength are the same thing—namely the B-field.

Magnetic flux linkage

• If instead of a single loop we make a coil of N loops, the flux Φ through each loop is “linked” to each of the other loops in what is termed flux linkage.
• 
• Each loop produces its own emf, and the emfs from each loop add to the total emf.
• Note that an emf is only produced while the flux is changing.
• 
• As all of our demonstrations have shown, only while the flux is changing does an emf get produced.
  ∙Since flux Φ = BA cos θ, there are three ways to change the flux:
  (1) Change the B-field.
  (2) Change the area A.
  (3) Change the relative orientation θ of A and B

FYI

Recall that (3) is the way a generator produces electricity at a power plant. A coil in a generator is rotated by a turbine, thus changing θ.

Faraday’s law of induction and Lenz’s law

• Faraday’s law states that the emf induced in a coil is equal to the rate of change in the flux linkage in the coil.
• Lenz’s law states that an induced current will have a direction such that it will oppose the change in flux that produced it. Hence, the (-) sign in Faraday’s law.
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