Sections 22.1-22.3 in Matter and Interactions (4th edition)
The Curly Electric Field and Induced Current
Now that we have talked about the changing magnetic flux part of Faraday's law, we should go back to the right hand side and talk about the curly electric field (the
Induced Voltage and the Electric Field
We know from Faraday's Law that a changing magnetic field will create a curly electric field. If we put a loop of wire nearby, we can add up the little bits of length along that loop with the curly electric field, which tells us how curly that electric field is. (This is very similar to what we did with Ampere's Law.) The units of this integral (
So instead we will define this quantity as the induced voltage.














Perhaps more intuitively, we can also define the








































































Why do we need the negative sign?
When we're talking about Faraday's Law, the negative sign plays an important role. It must be there to satisfy momentum and energy conservation, which we'll talk about in the next few paragraphs. If we think about the third equation above, the negative sign says that the induced current generates a magnetic field that will be in the direction that opposes the change in magnetic flux. The curly electric field that the changing magnetic field generates points in the same direction as this current. This should make some intuitive sense because the conventional current always points in the same direction as the electric field that is driving it. However, we generally talk about the direction of the induced current (rather than the electric field or induced voltage) since it make more intuitive sense and is easy to measure.
As an example of how to figure out which direction the induced current flows, let's say we have a bar that is sliding down a pair of connected conductive rails (so current is free to flow through the loop created by the bar and rails), which is sitting in a magnetic field that points into the page (shown in the top figure to the right). Initially there would be some magnetic flux through the loop (directed into the page following the magnetic field). At a later time (shown in the second picture to the right), after the bar has moved down the rails, there would be a larger magnetic flux through the loop (still directed into the page following the B-field) because the area of the loop will have increased. Since the magnetic flux increased, we know that there should be an induced current in the loop - but what direction should it flow around the loop? We can use another right hand rule to figure it out.
For this right hand rule, you want to first point your right thumb in the direction of the change in magnetic flux. For our example, the change in magnetic flux would point into the page. (Because
As we said before, the fact that the induced current will always generate a magnetic field to oppose the change in flux is an important result and ties back to energy and momentum conservation (it even is sometimes referred to as it's own law: Lenz's law). We can see this by thinking about what actually happens to the bar as it is moving through the magnetic field. Now that there is a induced current flowing through the bar, it would also feel a magnetic force from the magnetic field. Using our right hand rule again, this time with the