Electric Generators

Electric generators take advantage of Inductance and Faraday’s Law by rotating a coil in a magnetic field, inducing an emf.

Basic Idea:

First we use Faraday’s Law: $\mathcal{E}=-N\frac{d{\Phi }_m}{dt}$ We can define the magnetic flux simply as: ${\Phi }_m=NBAcos\theta$ And after differentiating, we arrive at the following result: $\mathcal{E}=NBAsin\theta \frac{d\theta }{dt}$

General Formula:

If we consider this rectangular loop, we can derive a formula for all loops in this kind of generator. From Faraday’s Law, specifically our derivations on Motional EMF, we know that $\mathcal{E}=Blv$ The velocity in this case is not all perpendicular to the magnetic field. Thus, we find the perpendicular component. We also multiple by 2 for both sides of the loop. Thus: $\mathcal{E}=2Blvsin\theta$ To find the EMF as a function of time, we use $\theta =\omega t$: $\mathcal{E}=NBA\omega \sin (\omega t)$ We can see that the max EMF generated is $NBA\omega$ and follows a sinusoidal shape.