The moving conductor of Figure 30.2 had an emf, but it couldn’t sustain a current because the charges had nowhere to go. It’s like a battery that is disconnected from a circuit. We can change this by including the moving conductor in a circuit.

FIGURE 30.5 shows a conducting wire sliding with speed $v$ along a U-shaped conducting rail. We’ll assume that the rail is attached to a table and cannot move. The wire and the rail together form a closed conducting loop—a circuit.

FIGURE 30.5 A current is induced in the circuit as the wire moves through a magnetic field.

Suppose a magnetic field $\vec{B}$ is perpendicular to the plane of the circuit. Charges in the moving wire will be pushed to the ends of the wire by the magnetic force, just as they were in Figure 30.2, but now the charges can continue to flow around the circuit. That is, the moving wire acts like a battery in a circuit.

The current in the circuit is an induced current. In this example, the induced current is counterclockwise (ccw). If the total resistance of the circuit is R, the induced current is given by Ohm’s law as

In this situation, the induced current is due to magnetic forces on moving charges.

We’ve assumed that the wire is moving along the rail at constant speed. It turns out that we must apply a continuous pulling force $\vec{F}{}_{\text{pull}}$ to make this happen. FIGURE 30.6 shows why. The moving wire, which now carries induced current I, is in a magnetic field. You learned in Chapter 29 that a magnetic field exerts a force on a current-carrying wire. According to the right-hand rule, the magnetic force $\vec{F}{}_{\text{mag}}$ on the moving wire points to the left. This “magnetic drag” will cause the wire to slow down and stop unless we exert an equal but opposite pulling force $\vec{F}{}_{\text{pull}}$ to keep the wire moving.

FIGURE 30.6 A pulling force is needed to move the wire to the right.

The magnitude of the magnetic force on a current-carrying wire was found in Chapter 29 to be $F_{\text{mag}}=IlB$. Using that result, along with Equation 30.4 for the induced current, we find that the force required to pull the wire with a constant speed $v$ is