31.1.4 Faraday’s Law Revisited

The transformation of electric and magnetic fields gives us new insight into Faraday’s law. FIGURE 31.10a shows a reference frame A in which a conducting loop is moving with velocity $\vec{v}$ into a magnetic field. You learned in Chapter 30 that the magnetic field exerts a magnetic force $\vec{F}_{B} = q \vec{v} \times \vec{B} = (q v B, upward)$ on the charges in the leading edge of the wire, creating an emf $ϵ = v L B$ and an induced current in the loop. We called this a motional emf.

The figure labeled "a” illustrates the fields experienced when a conducting loop moves along a rectangular frame.

The figure labeled b illustrates a rectangular metallic frame, a positive charge, a magnetic field and an electric field.

Full Alternative Text

How do things appear to an experimenter who is in frame B that moves with the loop at velocity $\vec{v}_{BA} = \vec{v}$ and for whom the loop is at rest? We have learned the important lesson that experimenters in different inertial reference frames agree about the outcome of any experiment; hence an experimenter in frame B agrees that there is an induced current in the loop. But the charges are at rest in frame B so there cannot be any magnetic force on them. How is the emf established in frame B?

We can use the field transformations to determine that the fields in frame B are

\begin{array}{l} {\vec{E}}_{B} & = & {\vec{E}}_{A} + \vec{v} \times {\vec{B}}_{A} = \vec{v} \times \vec{B} \\ {\vec{B}}_{B} & = & {\vec{B}}_{A} - \frac{1}{c^{2}} \vec{v} \times {\vec{E}}_{A} = \vec{B} \end{array}

(31.13)

where we used the fact that $\vec{E}_{A} = \vec{0}$ in frame A.

An experimenter in the loop’s frame sees not only a magnetic field but also the electric field $\vec{E}_{B}$ shown in FIGURE 31.10b. The magnetic field exerts no force on the charges, because they’re at rest in this frame, but the electric field does. The force on charge q is $\vec{F}_{E} = q \vec{E}_{B} = q \vec{v} \times \vec{B} = (q v B, upward)$ . This is the same force as was measured in the laboratory frame, so it will cause the same emf and the same current. The outcome is identical, as we knew it had to be, but the experimenter in B attributes the emf to an electric field whereas the experimenter in A attributes it to a magnetic field.

Field $\vec{E}_{B}$ is, in fact, the induced electric field of Faraday’s law. Faraday’s law, fundamentally, is a statement that a changing magnetic field creates an electric field. But only in frame B, the frame of the loop, is the magnetic field changing. Thus the induced electric field is seen in the loop’s frame but not in the laboratory frame.