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?

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.