At the beginning of the notes, we said that induction and Faraday's law was the start of tying magnetic fields back to electric fields, but so far we haven't used the electric field yet. These notes will use the relationship between electric potential and electric field to make this connection more obvious.

We have already talked about how Faraday's Law says that when the magnetic flux is changing, it can produce an induced current or an induced voltage. If there is an induced current, this means that there are charges moving in a wire that is not (necessarily) connected to a battery. So what is actually making those charges move?

If we go back to our equation for Faraday's Law, we have a way to relate the electric potential to the changing magnetic flux: $$V_{ind} = - \frac{d \Phi_B}{dt}$$

We can then use the relationship between electric potential ($V_{ind}$) and the electric field to rewrite the right hand side of the equation. Namely that: $$\Delta V=-\int_{r_i}^{r_f} \vec{E}\bullet d\vec{r}$$

Find the negative problem….

So we see that: $$ \int \vec{E} \bullet d\vec{l} = - \frac{d \Phi_B}{dt}$$

This tells that the changing magnetic flux actually creates an electric field. It is this electric field that is pushing the charges, creating the induced current and the induced electric potential. This is a very different situation than when we talked about current in circuits. Before in circuits, we said that the battery created a surface charge gradient, which in turn created an electric field in the wire and created current in the wire. In this case, we do not have battery or surface charges. Instead, it is the changing magnetic flux that creates the electric field, which in turn creates the current through the wire.

Faraday's law is conceptually very important because it tells us how electric and magnetic fields are related (finally!). From a more practical stand point, Faraday's law provides a means of creating an electric current when there previously was not any. This is actually how electric generators work to create the electricity that comes out of the wall outlets. There is generally some sort of coil placed in a large magnetic field. The coil is then moved by some mechanical means (i.e., by wind in a turbine or by steam from burning coal or nuclear material in power plant). When the coil rotates in the magnetic field, the flux through the coil changes and creates a current that can then be used.