Self-Induction

• If moving a coil in a magnetic field induces a current and that current induces a magnetic field, then the induced magnetic field will also have an effect on the coil’s own induced current.
• This self-induction will decrease the efficiency of the generation of a voltage.
• The self-induced emf produced is also known as “back-emf”. The effects of back-emf are more significant at higher currents so it needs to be considered as part of the design parameters.

Alternating Current (AC) Generators

∙ Recall Faraday’s law:

∙ From the formula we see that there are three ways to increase the induced emf of a rotating coil:
(1) Increase the number of turns \( N \) in the coil.
(2) Increase the flux change \( \Delta\Phi \).
(3) Decrease the time \( \Delta t \) over which the flux changes.

FYI

∙ Recall that \( \Phi = BA \cos \theta \). Given the uniform magnetic field and rotating coil, \( B \) and \( A \) are constant. Thus the flux change \( \Delta \Phi \) will be due only to the change in angle \( \Delta \theta \).

Alternating Current (AC) Generators – Flux and emf Variation

∙ Consider the rectangular loop of wire made to rotate in the fixed magnetic field:

• At this instant: \( \Phi = BA \cos 0^\circ = BA \)
• Later when \( \theta = 45^\circ \): \( \Phi = BA \cos 45^\circ = 0.7BA \)
• When \( \theta = 90^\circ \): \( \Phi = BA \cos 90^\circ = 0 \)
• As \( \theta \) continues, \( \Phi \) becomes negative → sinusoidal variation emerges

∙ Since \( \varepsilon = -\frac{\Delta \Phi}{\Delta t} \), the induced emf is the negative slope of the flux.
∙ Slope of cosine graph is a sine graph → \( \varepsilon \propto BA \sin \theta \)

This is the basis for how rotating coils in magnetic fields generate AC current.