30. Induction and Inductance

A 100-turn, 2.0-cm-diameter coil is at rest with its axis vertical. A uniform magnetic field 60° away from vertical increases from 0.50 T to 1.50 T in 0.60 s. What is the induced emf in the coil?

CALC A 10 cm×10 cm square loop of wire lies in the xy-plane. The magnetic field in this region of space is $\vec{B}=(0.30t\hat{i}+0.50t^2\hat{k})\text{ T}$, where t is in s. What is the emf induced in the loop at (a) t = 0.5 s and (b) t = 1.0 s?

FIGURE P30.48 shows two 20-turn coils tightly wrapped on the same 2.0-cm-diameter cylinder with 1.0-mm-diameter wire. The current through coil 1 is shown in the graph. Determine the current in coil 2 at (a) t = 0.05 s and (b) t = 0.25 s. A positive current is into the figure at the top of a loop. Assume that the magnetic field of coil 1 passes entirely through coil 2.

A small, 2.0-mm-diameter circular loop with R = 0.020 Ω is at the center of a large 100-mm-diameter circular loop. Both loops lie in the same plane. The current in the outer loop changes from +1.0 A to −1.0 A in 0.10 s. What is the induced current in the inner loop?

A rectangular metal loop with 0.050 Ω resistance is placed next to one wire of the RC circuit shown in FIGURE P30.53. The capacitor is charged to 20 V with the polarity shown, then the switch is closed at t = 0 s. What is the current in the loop at t = 5.0 μs? Assume that only the circuit wire next to the loop is close enough to produce a significant magnetic field.

CALC Your camping buddy has an idea for a light to go inside your tent. He happens to have a powerful (and heavy!) horseshoe magnet that he bought at a surplus store. This magnet creates a 0.20 T field between two pole tips 10 cm apart. His idea is to build the hand-cranked generator shown in FIGURE P30.57. He thinks you can make enough current to fully light a 1.0 Ω lightbulb rated at 4.0 W. That's not super bright, but it should be plenty of light for routine activities in the tent. With what frequency will you have to turn the crank for the maximum current to fully light the bulb? Is this feasible?

CALC Your camping buddy has an idea for a light to go inside your tent. He happens to have a powerful (and heavy!) horseshoe magnet that he bought at a surplus store. This magnet creates a 0.20 T field between two pole tips 10 cm apart. His idea is to build the hand-cranked generator shown in FIGURE P30.57. He thinks you can make enough current to fully light a 1.0 Ω lightbulb rated at 4.0 W. That's not super bright, but it should be plenty of light for routine activities in the tent. Find an expression for the induced current as a function of time if you turn the crank at frequency f. Assume that the semicircle is at its highest point at t = 0 s.

INT You've decided to make the magnetic projectile launcher shown in FIGURE P30.58 for your science project. An aluminum bar slides along metal rails through a magnetic field B. The switch closes at t = 0 s, while the bar is at rest, and a battery of emf ε_bat starts a current flowing around the loop. The battery has internal resistance r. The resistances of the rails, which are separated by distance l, and the bar are effectively zero. Show that the bar reaches a terminal speed v_term, and find an expression for v_term.

INT You've decided to make the magnetic projectile launcher shown in FIGURE P30.58 for your science project. An aluminum bar slides along metal rails through a magnetic field B. The switch closes at t = 0 s, while the bar is at rest, and a battery of emf ε_bat starts a current flowing around the loop. The battery has internal resistance r. The resistances of the rails, which are separated by distance l, and the bar are effectively zero. Evaluate v_term for ε_bat = 1.0 V, r = 0.10 Ω, l = 6.0 cm, and B = 0.50 T.

INT FIGURE P30.59 shows a U-shaped conducting rail that is oriented vertically in a horizontal magnetic field. The rail has no electric resistance and does not move. A slide wire with mass m and resistance R can slide up and down without friction while maintaining electrical contact with the rail. The slide wire is released from rest. Determine the value of v_term if l = 20 cm,m = 10 g, R = 0.10 Ω, and B = 0.50 T.

CALC Let's look at the details of eddy-current braking. A square loop, length l on each side, is shot with velocity v₀ into a uniform magnetic field B. The field is perpendicular to the plane of the loop. The loop has mass m and resistance R, and it enters the field at t = 0 s. Assume that the loop is moving to the right along the x-axis and that the field begins at x = 0 m. Find an expression for the loop's velocity as a function of time as it enters the magnetic field. You can ignore gravity, and you can assume that the back edge of the loop has not entered the field.

CALC Let's look at the details of eddy-current braking. A square loop, length l on each side, is shot with velocity v₀ into a uniform magnetic field B. The field is perpendicular to the plane of the loop. The loop has mass m and resistance R, and it enters the field at t=0 s. Assume that the loop is moving to the right along the x-axis and that the field begins at x = 0 m. Calculate and draw a graph of v over the interval 0 s ≤ t ≤ 0.04 s for the case that v₀=10 m/s, l = 10 cm, m = 1.0 g, R = 0.0010 Ω, and B=0.10 T. The back edge of the loop does not reach the field during this time interval.

CALC The rectangular loop in FIGURE CP30.81 has 0.020 Ω resistance. What is the induced current in the loop at this instant?

One way to measure the strength of a magnetic field is with a flip coil. Suppose a 200-turn, 4.0-cm-diameter coil with a resistance of 2.0 Ω is connected to a ballistic galvanometer, a device that measures the total charge passing through. The coil is held perpendicular to the field, then quickly flipped 180° so that the opposite side is facing the magnetic field. Afterward, the galvanometer reads 7.5 μC. What is the field strength? Hint: Use I = dq/dt to relate the net change of flux to the amount of charge that flows through the galvanometer.

A rocket zooms past the earth at v=2.0×10⁶ m/s. Scientists on the rocket have created the electric and magnetic fields shown in FIGURE EX31.4. What are the fields measured by an earthbound scientist?