30. Induction and Inductance

(III) Suppose a conducting rod (mass m, resistance R) rests on two frictionless and resistanceless parallel rails a distance ℓ apart in a uniform magnetic field $\overrightarrow{B}$ (⊥ to the rails and to the rod) as in Fig. 29–53. At t = 0, the rod is at rest and a source of emf is connected to the points a and b. Determine the speed of the rod as a function of time if (a) the source puts out a constant current I, (b) the source puts out a constant emf ε₀. (c) Does the rod reach a terminal speed in either case? If so, what is it?

(III) A simple generator has a square armature 6.0 cm on a side. The armature has 85 turns of 0.65-mm-diameter copper wire and rotates in a 0.65-T magnetic field. The generator is used to power a lightbulb rated at 12.0 V and 25.0 W. At what rate should the generator rotate to provide 12.0 V to the bulb? Consider the resistance of the wire on the armature.

If 75 MW of power at 45 kV (rms) arrives at a town from a generator via transmission lines of total resistance 3.0 Ω, calculate (a) the emf at the generator end of the lines, and (b) the fraction of the power generated that is wasted in the lines.

The moving rod in Fig. 29–12b is 12.0 cm long and is pulled at a speed of 15.0 cm/s. If the magnetic field is 0.650 T, calculate the emf developed.

A conducting rod of length $l$ moves with constant velocity $v$ perpendicular to a uniform magnetic field of magnitude $B$. What is the magnitude $e$ of the motional emf induced in the rod?

A long horizontal wire carries 24.0 A of dc current due north. What is the net magnetic field 20.0 cm due west of the wire if the Earth’s field there points downward, 38° below the horizontal, and has magnitude 5.0 x 10^-5 T?

A 22.0-cm-diameter coil consists of 36 turns of circular copper wire 2.6 mm in diameter. A uniform magnetic field, perpendicular to the plane of the coil, changes at a rate of 8.65 x 10^-3 T/s. Determine (a) the current in the loop, and (b) the rate at which thermal energy is produced.

INT A 20-cm-long, zero-resistance slide wire moves outward, on zero-resistance rails, at a steady speed of 10 m/s in a 0.10 T magnetic field. (See Figure 30.26.) On the opposite side, a 1.0 Ω carbon resistor completes the circuit by connecting the two rails. The mass of the resistor is 50 mg. If the wire is pulled for 10 s, what is the temperature increase of the carbon? The specific heat of carbon is 710 J/kg K.

In a certain region of space near Earth’s surface, a uniform horizontal magnetic field of magnitude B exists above a level defined to be y = 0. Below y = 0, the field abruptly becomes zero (Fig. 29–63). A vertical square wire loop has resistivity ρ, mass density ρ_m, diameter d, and side length ℓ. It is initially at rest with its lower horizontal side at y = 0 and is then allowed to fall under gravity, with its plane perpendicular to the direction of the magnetic field. (a) While the loop is still partially immersed in the magnetic field (as it falls into the zero-field region), determine the magnetic “drag” force that acts on it at the moment when its speed is υ. (b) Assume that the loop achieves a constant terminal velocity V_T before its upper horizontal side exits the field. Determine a formula for V_T. (c) If the loop is made of copper and B = 0.80 T, find V_T.

A thin rod moves in a perpendicular, unknown magnetic field. If the length of the rod is 10 cm and the induced EMF is 1 V when it moves at 5 m/s, what is the magnitude of the magnetic field?

(II) Suppose that the U-shaped conductor and connecting rod in Fig. 29–12a are oriented vertically (but still in contact) so that the rod is falling due to the gravitational force. Find the terminal speed of the rod if it has mass m = 3.6 grams, length 𝓁 = 18 cm, and resistance R = 0.0013 Ω. It is falling in a uniform horizontal field B = 0.080 T. Neglect the resistance of the U-shaped conductor, and friction.

A simple generator is designed to generate a peak output voltage of 24.0 V. The square armature consists of windings that are 5.15 cm on a side and rotates in a field of 0.380 T at a rate of 60.0 rev/s. How many loops of wire should be wound on the square armature?

A potential difference of 0.050 V is developed across the 10-cm-long wire of FIGURE EX30.3 as it moves through a magnetic field perpendicular to the figure. What are the strength and direction (in or out) of the magnetic field?

INT A 20-cm-long, zero-resistance slide wire moves outward, on zero-resistance rails, at a steady speed of 10 m/s in a 0.10 T magnetic field. (See Figure 30.26.) On the opposite side, a 1.0 Ω carbon resistor completes the circuit by connecting the two rails. The mass of the resistor is 50 mg. How much force is needed to pull the wire at this speed?

(II) A conducting rod rests on two long frictionless parallel rails in a magnetic field $\overrightarrow{B}$ (⊥ to the rails and rod) as in Fig. 29–53. (a) If the rails are horizontal and the rod is given an initial push, will the rod travel at constant speed even though a magnetic field is present? (b) Suppose at t = 0, when the rod has speed v = v₀, the two rails are connected electrically by a wire from point a to point b. Assuming the rod has resistance R and the rails have negligible resistance, determine the speed of the rod as a function of time. Discuss your answer.