To summarize what we just learned, we found that a changing magnetic field will create a curly electric field. This means that we now have two source of electric fields - one being the static charges (that we talked about at the beginning of the semester) and the second being changing magnetic fields.

Mathematically, we represent this relationship by thinking about how curly the electric field is around a loop and relate that to the changing magnetic flux through that loop. Namely: $$-\int \vec{E}_{nc} \bullet d\vec{l}= \frac{d \Phi_B}{dt}$$ We can also rewrite this to be in terms of the induced voltage or induced current in the loop if that better suits what we would like to find. $$-V_{ind}=-I_{ind}R=\frac{d \Phi_B}{dt}$$

We also have a new right hand rule that let's us figure out what will be the direction of the induced current (or the direction of the curly electric field) in the wire. For this right hand rule, we first point our thumb in direction of the change of the magnetic flux. Then flip the direction 180 degrees because the induced current should oppose the change in flux that created it. Then the direction that your fingers curl in will be the direction of the induced current around the loop.

Faraday's law is conceptually extremely 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 - as long as you can provide the energy to change the flux. 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.