30.5.2 What Does Faraday’s Law Tell Us?

The induced current in the slide-wire circuit of Figure 30.26 can be understood as a motional emf due to magnetic forces on moving charges. We had not anticipated this kind of current in Chapter 29, but it takes no new laws of physics to understand it. The induced currents in Examples 30.8 and 30.9 are different. We cannot explain these induced currents on the basis of previous laws or principles. This is new physics.

Faraday recognized that all induced currents are associated with a changing magnetic flux. There are two fundamentally different ways to change the magnetic flux through a conducting loop:

The loop can expand, contract, or rotate, creating a motional emf.
The magnetic field can change.

We can see both of these if we write Faraday’s law as

ε = | \frac{d Φ_{m}}{d t} | = | \vec{B} \cdot \frac{d \vec{A}}{d t} + \vec{A} \cdot \frac{d \vec{B}}{d t} |

(30.19)

The first term on the right side represents a motional emf. The magnetic flux changes because the loop itself is changing. This term includes not only situations like the slide-wire circuit, where the area A changes, but also loops that rotate in a magnetic field. The physical area of a rotating loop does not change, but the area vector $\vec{A}$ does. The loop’s motion causes magnetic forces on the charge carriers in the loop.

The second term on the right side is the new physics in Faraday’s law. It says that an emf can also be created simply by changing a magnetic field, even if nothing is moving. This was the case in Examples 30.8 and 30.9. Faraday’s law tells us that the induced emf is simply the rate of change of the magnetic flux through the loop, regardless of what causes the flux to change.