Zachary W.

Animals in cold climates often depend on two layers of insulation: a layer of body fat (of thermal conductivity 0.20 W/m $\cdot$ K) surrounded by a layer of air trapped inside fur or down. We can model a black bear (Ursus americanus) as a sphere 1.5 m in diameter having a layer of fat 4.0 cm thick. (Actually, the thickness varies with the season, but we are interested in hibernation, when the fat layer is thickest.) In studies of bear hibernation, it was found that the outer surface layer of the fur is at 2.7$^\circ$C during hibernation, while the inner surface of the fat layer is at 31.0$^\circ$C. (a) What is the temperature at the fat-inner fur boundary so that the bear loses heat at a rate of 50.0 W? (b) How thick should the air layer (contained within the fur) be?

Consider the circuit shown in Fig. E29.31, but with the bar moving to the right with speed v. As in Exercise 29.31, the bar has length 0.360 m, $R$ = 45.0 $\Omega$, and $B =$ 0.650 T. (a) Is the induced current in the circuit clockwise or counterclockwise? (b) At an instant when the 45.0-$\Omega$ resistor is dissipating electrical energy at a rate of 0.840 J/s, what is the speed of the bar?

A long, thin solenoid has 900 turns per meter and radius 2.50 cm. The current in the solenoid is increasing at a uniform rate of 36.0 A/s. What is the magnitude of the induced electric field at a point near the center of the solenoid and (a) 0.500 cm from the axis of the solenoid; (b) 1.00 cm from the axis of the solenoid?

A very long, rectangular loop of wire can slide without friction on a horizontal surface. Initially the loop has part of its area in a region of uniform magnetic field that has magnitude $B =$ 2.90 T and is perpendicular to the plane of the loop. The loop has dimensions 4.00 cm by 60.0 cm, mass 24.0 g, and resistance $R =$ 5.00 $\times$ 10$^{-3} \Omega$. The loop is initially at rest; then a constant force $F_{ext}$ = 0.180 N is applied to the loop to pull it out of the field (Fig. P29.46). (a) What is the acceleration of the loop when $v =$ 3.00 cm/s? (b) What are the loop's terminal speed and acceleration when the loop is moving at that terminal speed? (c) What is the acceleration of the loop when it is completely out of the magnetic field?

In the circuit in Fig. P29.47, an emf of 90.0 V is added in series with the capacitor and the resistor, and the capacitor is initially uncharged. The emf is placed between the capacitor and switch $S$, with the positive terminal of the emf adjacent to the capacitor. Otherwise, the two circuits are the same as in Problem 29.47. The switch is closed at $t =$ 0. When the current in the large circuit is 5.00 A, what are the magnitude and direction of the induced current in the small circuit?

Suppose the loop in $\textbf{Fig. P29.50}$ is (a) rotated about the $y$-axis; (b) rotated about the x-axis; (c) rotated about an edge parallel to the $z$-axis. What is the maximum induced emf in each case if $A =$ 600 cm$^2$, $\omega$ = 35.0 rad/s, and $B =$ 0.320 T?