Q: A 10-cm-long wire is pulled along a U-shaped conducting rail in a perpendicular magnetic field. The total resistance is 0.20 Ω. If a force of 1.0 N causes 4.0 W of power to be dissipated, what is the speed of the wire?

$$v = \frac{P}{F} = 4.0\ \text{m/s}$$

Q: In the same scenario as above, what is the strength of the magnetic field if the wire is moving at 4.0 m/s?

$$B = \sqrt{\frac{R F^2$}{P v^2$ \$$ell^2$$}} = 2.2\ \text{T}$$

Q: A 15-cm-long metal rod is pulled along two frictionless, conducting rails at a constant speed of 3.5 m/s in a 1.4 T magnetic field. The rod has a resistance of 0.65 Ω. What is the current induced in the rod?

$$I = \frac{B \ell v}{R} = \frac{1.4 \times 0.15 \times 3.5}{0.65} \approx 1.1\ \text{A}$$

Q: What force is required to keep the rod moving at a constant speed in the previous scenario?

$$F = B I \ell = 1.4 \times 1.13 \times 0.15 \approx 0.24\ \text{N}$$

Q: A 50 g horizontal metal bar, 12 cm long, slides between two vertical rods in a 0.060 T magnetic field. A 1.0 Ω resistor is connected across the rods. What is the terminal speed at which the bar falls?

$$v = \frac{mg R}{(\ell B)^2$} = \frac{0.050 \times 9.8 \times 1.0}{(0.12 \times 0.060)^2$} \approx 9500\ \text{m/s}$$

Q: A microphone coil with diameter 1.2 cm and 100 turns moves in a 0.010 T magnetic field. If the emf at some instant is 10 mV, what is the speed of the coil?

$$v = \frac{\Delta V}{2 \pi r N B} = \frac{0.010}{2 \pi \times 0.006 \times 100 \times 0.010} \approx 0.27\ \text{m/s}$$

Q: A rectangular loop (15 cm × 30 cm) is placed in a region where the left half has a magnetic field of 2.5 T and the right half has 1.5 T, both perpendicular to the loop. What is the total magnetic flux through the loop?

$$\Phi = (0.15 \times 0.15 \times 2.5) + (0.15 \times 0.15 \times 1.5) = 0.056\ \text{Wb}$$

Q: A 2.0-cm-diameter solenoid passes through the center of a 6.0-cm-diameter loop. The magnetic field inside the solenoid is 0.20 T. What is the magnetic flux through the loop when it is perpendicular to the solenoid?

$$\Phi = \pi r^2$ B = \pi \times (0.010)^2$ \times 0.20 = 6.3 \times 10$$^{-5}$$\ \text{Wb}$$

Question 1

A 12.0-cm-long wire moves through a 0.27 T uniform magnetic field at a speed of 5.0 m/s, developing a potential difference of 53 mV across its ends. The magnetic field is perpendicular to the axis of the wire. What is the angle between the wire’s velocity and the magnetic field?

Accepted Answer

$$\sin \alpha = \frac{\Delta V}{v B \ell}$$, \alpha = 19^\circ$

Question 2

A 10-cm-long wire is pulled along a U-shaped conducting rail in a perpendicular magnetic field. The total resistance is 0.20 Ω. If a force of 1.0 N causes 4.0 W of power to be dissipated, what is the speed of the wire?

Accepted Answer

$$v = \frac{P}{F} = 4.0\ 	ext{m/s}$$

Question 3

In the same scenario as above, what is the strength of the magnetic field if the wire is moving at 4.0 m/s?

Accepted Answer

$$B = \sqrt{\frac{R F^2$}{P v^2$ \$$ell^2$$}} = 2.2\ 	ext{T}$$

Question 4

A 15-cm-long metal rod is pulled along two frictionless, conducting rails at a constant speed of 3.5 m/s in a 1.4 T magnetic field. The rod has a resistance of 0.65 Ω. What is the current induced in the rod?

Accepted Answer

$$I = \frac{B \ell v}{R} = \frac{1.4 	imes 0.15 	imes 3.5}{0.65} \approx 1.1\ 	ext{A}$$

Question 5

What force is required to keep the rod moving at a constant speed in the previous scenario?

Accepted Answer

$$F = B I \ell = 1.4 	imes 1.13 	imes 0.15 \approx 0.24\ 	ext{N}$$

Question 6

A 50 g horizontal metal bar, 12 cm long, slides between two vertical rods in a 0.060 T magnetic field. A 1.0 Ω resistor is connected across the rods. What is the terminal speed at which the bar falls?

Accepted Answer

$$v = \frac{mg R}{(\ell B)^2$} = \frac{0.050 	imes 9.8 	imes 1.0}{(0.12 	imes 0.060)^2$} \approx 9500\ 	ext{m/s}$$

Question 7

A microphone coil with diameter 1.2 cm and 100 turns moves in a 0.010 T magnetic field. If the emf at some instant is 10 mV, what is the speed of the coil?

Accepted Answer

$$v = \frac{\Delta V}{2 \pi r N B} = \frac{0.010}{2 \pi 	imes 0.006 	imes 100 	imes 0.010} \approx 0.27\ 	ext{m/s}$$

Question 8

Rank the magnetic fluxes $$\$$Phi_1$$ through $$\$$Phi_4$$ for the four loops shown, given that loops 3 and 4 have twice the radius of loops 1 and 2, and the magnetic field is the same for each. Which is correct?

Accepted Answer

$$\$$Phi_4$$ \u003e \$$Phi_3$$ \u003e \$$Phi_1$$ \u003e \$$Phi_2$$

Question 9

A rectangular loop (15 cm × 30 cm) is placed in a region where the left half has a magnetic field of 2.5 T and the right half has 1.5 T, both perpendicular to the loop. What is the total magnetic flux through the loop?

Accepted Answer

$$\Phi = (0.15 	imes 0.15 	imes 2.5) + (0.15 	imes 0.15 	imes 1.5) = 0.056\ 	ext{Wb}$$

Question 10

A 2.0-cm-diameter solenoid passes through the center of a 6.0-cm-diameter loop. The magnetic field inside the solenoid is 0.20 T. What is the magnetic flux through the loop when it is perpendicular to the solenoid?

Accepted Answer

$$\Phi = \pi r^2$ B = \pi 	imes (0.010)^2$ 	imes 0.20 = 6.3 	imes 10$$^{-5}$$\ 	ext{Wb}$$

Question 11

A metal equilateral triangle, 20 cm on each side, is halfway into a 0.10 T magnetic field. What is the magnetic flux through the triangle?

Accepted Answer

$$\Phi = \frac{1}{2} 	imes 	ext{area} 	imes B = \frac{1}{2} 	imes \frac{1}{2} 	imes 0.20 	imes 0.1732 	imes 0.1 = 8.7 	imes 10$$^{-4}$$\ 	ext{Wb}$$

Question 12

If the magnetic field strength decreases in the previous triangle scenario, what is the direction of the induced current in the triangle?

Accepted Answer

Clockwise

Question 13

For the loop shown in Figure Q25.3(a), where the magnetic field is out of the page and the loop area is increasing, what is the direction of the induced current?

Accepted Answer

Clockwise

Question 14

For Figure Q25.3(b), where the magnetic field is into the page and the loop area is increasing, what is the direction of the induced current?

Accepted Answer

Counterclockwise

Question 15

For Figure Q25.3(c), where the magnetic field is parallel to the plane of the page, what is the direction of the induced current?

Accepted Answer

Zero

Question 16

A loop of wire is horizontal and a bar magnet is pushed toward the loop from below, along the axis. What is the direction of the induced current in the loop?

Accepted Answer

Clockwise (viewed from above)

Question 17

A 100-turn coil of radius 12 cm in a 0.15 T magnetic field is rotated 90° in 0.30 s, ending up parallel to the field. What is the average emf induced in the coil?

Accepted Answer

$$\mathcal{E}_{	ext{avg}} = \frac{N \pi r^2$ B (\cos 90^\circ - \cos 0^\circ)}{\Delta t} = \frac{100 	imes \pi 	imes (0.12)^2$ 	imes 0.15 	imes (-1)}{0.30} \approx 2.3\ 	ext{V}$$

Question 18

A 10-cm-diameter loop with resistance 0.10 Ω is placed in a changing magnetic field. If the field is increasing out of the page at 0.50 T/s, what is the induced current and its direction?

Accepted Answer

$$I = \frac{\mathcal{E}}{R} = \frac{\pi r^2$ \frac{dB}{dt}}{R} = \frac{\pi 	imes (0.05)^2$ 	imes 0.50}{0.10} \approx 39\ 	ext{mA}$$, clockwise

Question 19

A solenoid passes through the center of a wire loop. The solenoid has 1200 turns, diameter 2.0 cm, length 7.5 cm, and the loop resistance is 0.032 Ω. If the current in the solenoid increases by 1.3 A in 50 ms, what is the induced current in the loop?

Accepted Answer

$$I = \frac{\mu_0$ N A \Delta I}{R L \Delta t} = \frac{4\pi 	imes 10$$^{-7}$$ 	imes 1200 	imes \pi 	imes (0.01)^2$ 	imes 1.3}{0.032 	imes 0.075 	imes 0.050} \approx 5.1\ 	ext{mA}$$

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