30. Induction and Inductance

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) 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.

(II) 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?

(I) 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?

(II) A generator rotates at 85 Hz in a magnetic field of 0.030 T. It has 950 turns and produces an rms voltage of 150 V and an rms current of 60.0 A. (a) What is the peak current produced? (b) What is the area of each turn of the coil?

(II) 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.

(I) 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.

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(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.

Two parallel resistanceless rails are 32 cm apart on a 6.0° ramp. They are joined at the bottom by a 0.60-Ω resistor. At the top a copper bar of mass 0.040 kg (ignore its resistance) is laid across the rails. Assuming a vertical 0.65-T magnetic field, what is the terminal (steady) velocity of the bar as it slides frictionlessly down the rails?

A flashlight can be made that is powered by the induced current from a magnet moving through a coil of wire. The coil and magnet are inside a plastic tube that can be shaken by your hand causing the magnet to move back and forth through the coil. Assume the magnet has a maximum field strength of 0.05 T. Make reasonable assumptions and specify the size of the coil and the number of turns necessary to light a standard 1-watt, 3-V flashlight bulb.

"(II) Part of a single rectangular loop of wire with dimensions shown in Fig. 29–49 is situated inside a region of uniform magnetic field of 0.650 T. The total resistance of the loop is 0.250 Ω. Calculate the force required to pull the loop from the field (to the right) at a constant velocity of 3.40 m/s. Neglect gravity.

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(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.

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(II) A conducting rod rests on two long frictionless parallel rails in a magnetic field →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 , 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.

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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 Pₘ, 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 VT before its upper horizontal side exits the field. Determine a formula for VT. (c) If the loop is made of copper and B = 0.80 T, find VT.

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In an experiment, a coil was mounted on a low-friction cart that moved through the magnetic field B of a permanent magnet. The speed of the cart υ and the induced voltage V were simultaneously measured, as the cart moved through the magnetic field, using a computer-interfaced motion sensor and a voltmeter. The Table below shows the collected data:

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(b) Find the % error between the slope of the experimental graph and the theoretical value for the slope.