FIGURE 23-10
-A conducting rod whose length is 25 cm is placed on a U-shaped metal wire that has a resistance of 8.0 Ω as shown in Figure 23-10. The wire and the rod are placed in the plane of the paper. A constant magnetic field of strength 0.40 T is applied out of the paper. An applied force moves the rod to the left with a constant speed of 6.0 m/s. What is the magnitude and direction of the induced current in the wire?

Question

FIGURE 23-10 
-A conducting rod whose length is 25 cm is placed on a U-shaped metal wire that has a resistance of 8.0 Ω as shown in Figure 23-10. The wire and the rod are placed in the plane of the paper. A constant magnetic field of strength 0.40 T is applied out of the paper. An applied force moves the rod to the left with a constant speed of 6.0 m/s. What is the magnitude and direction of the induced current in the wire?

Quizplus · Accepted Answer

The induced emf in the wire can be found using Faraday's law:
$\mathcal{E}=-\frac{\Delta\Phi_B}{\Delta t}$,
where $\Phi_B$ is the magnetic flux through the wire. Since the magnetic field is out of the paper, the magnetic flux is given by:
$\Phi_B=B\cdot A $,
where $A$ is the area enclosed by the wire. Since the wire is a U-shaped loop, $A$ is equal to the length of the wire times the distance between the wire and the rod:
$A=2L\cdot d$,
where $L$ is the length of the wire and $d$ is the distance between the wire and the rod. The induced emf is therefore given by:
$\mathcal{E}=-B\cdot 2L\cdot d\cdot \frac{\Delta v}{\Delta t}$,
where $\Delta v/\Delta t$ is the velocity of the rod. Since the rod is moving at a constant speed, the induced emf is also constant. 
The induced current in the wire is given by Ohm's law:
$I=\frac{\mathcal{E}}{R}$,
where $R$ is the resistance of the wire. Substituting numerical values, we obtain:
$I=\frac{-0.40	ext{ T}\cdot(2	imes 0.25	ext{ m})\cdot(8.0	ext{ cm})\cdot(6.0	ext{ m/s})}{8.0	ext{ }\Omega}= -0.45	ext{ A}$.
The negative sign indicates that the induced current is counterclockwise, as shown in the diagram. The magnitude of the current is $|I|=0.45$ A = 450 mA, which is closest to choice B, 75 mA counterclockwise.

FIGURE 23-10 -A Conducting Rod Whose Length Is 25 Cm Is Placed