The generalized equation for the operation of a motor is

 

 

When electricity is supplied to a coil and it is inserted within a permanent magnetic field, the two magnetic fields repel and attract each other causing the loop to rotate.  Recall the demonstration of the St. Louis motor. A DC motor is drawn as a circuit element in the following way:

 

 

The loop equation, ABCA, for this circuit would be

 

- ε_motor - Ir + ε = 0

 

ε _motor is called the back emf of the motor and r is the resistance of the coil. This back emf reduced the effective voltage of the battery and is also why capacitors are used to help jump start large circuits and there is a transient voltage to motors when they are initially turned off.

 

 

The generalized equation for a generator is

 

 

When a coil is rotated within a permanent external magnetic field, the changing flux lines generate a voltage within the coil.

 

Physlet: Faraday's Law of Induction
moving coil

 

Physlet: Generator
rotating coil

 

Since the coil has resistance, this induced emf will result in an induced electric current. The equation for the alternating voltage produced by the rotating coil is

 

v_{instantaneous} = V_osin(2πf)t

 

where V_o is the coil's maximum emf which equals NBA(2πf) and the frequency is in rev/sec [hz]. This equation can also be expressed as

 

 

where ω is the coil's angular velocity, ω = 2πf.

 

 

The Law of Transformers states

 

 

N_p and N_s are the number of loops in the primary and secondary coils. ε_p and ε_s are the emfs in the primary and secondary. A transformer only works when the number of flux lines through the iron core of the transformer keeps changing - therefore, the primary must be connected to an alternating source. Transformers will not operate in a DC circuit. Any change in flux in the primary is communicated to the secondary through the iron core.

 

The power utilized on both coils is the same since energy is conserved. Therefore, ε_pI_p = ε_sI_s, which explains how electricity is transferred by the power companies. High voltages will have smaller currents and less energy loss to Joule heating. Voltages are stepped down for use in buildings at transformers. When N_s < N_p, the transformer is a "step down" transformer since the induced AC voltage of the secondary will be less than the voltage across the primary.