Ch. 30 - Inductance, Electromagnetic Oscillations, and AC Circuits

Giancoli Douglas - Physics for Scientists and Engineers 5th edition

Giancoli Douglas5th editionPhysics for Scientists and EngineersISBN: 9780137488179Not the one you use?Change textbook

Giancoli Douglas 5th edition

Ch. 30 - Inductance, Electromagnetic Oscillations, and AC Circuits

Problem 28

Chapter 29, Problem 28

(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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Identify the forces acting on the rod: The rod experiences a downward gravitational force $ F_g = m g $, where $ m $ is the mass of the rod and $ g $ is the acceleration due to gravity. Additionally, as the rod moves through the magnetic field, it induces an electromotive force (EMF) and a current, which results in an upward magnetic force $ F_B $. At terminal velocity, these forces balance: $ F_g = F_B $.

Determine the induced EMF: The motion of the rod through the magnetic field induces an EMF given by $ \mathcal{E} = B \ell v $, where $ B $ is the magnetic field strength, $ \ell $ is the length of the rod, and $ v $ is the velocity of the rod.

Relate the EMF to the current: Using Ohm's law, the current in the circuit is $ I = \frac{\mathcal{E}}{R} = \frac{B \ell v}{R} $, where $ R $ is the resistance of the rod.

Calculate the magnetic force: The magnetic force on the rod is given by $ F_B = I \ell B = \left( \frac{B \ell v}{R} \right) \ell B = \frac{B^2 \ell^2 v}{R} $.

Set up the force balance equation: At terminal velocity, the downward gravitational force equals the upward magnetic force: $ m g = \frac{B^2 \ell^2 v}{R} $. Solve for the terminal velocity $ v_t $: $ v_t = \frac{m g R}{B^2 \ell^2} $. Substitute the given values for $ m $, $ g $, $ R $, $ B $, and $ \ell $ to find the terminal velocity.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Terminal Velocity

Terminal velocity is the constant speed that a freely falling object eventually reaches when the resistance of the medium through which it is falling prevents further acceleration. In this scenario, the gravitational force acting on the rod is balanced by the electromagnetic force generated due to its motion in the magnetic field, leading to a steady speed.

Electromagnetic Induction

Electromagnetic induction is the process by which a changing magnetic field within a closed loop induces an electromotive force (EMF) in the conductor. In this case, as the rod falls through the magnetic field, it cuts through the magnetic lines of force, generating an induced current that interacts with the magnetic field, producing a force opposing the motion.

Ohm's Law

Ohm's Law states that the current flowing through a conductor between two points is directly proportional to the voltage across the two points and inversely proportional to the resistance. In this problem, the induced EMF from the falling rod creates a current that can be calculated using Ohm's Law, which is essential for determining the forces acting on the rod and its terminal speed.

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Related Practice

Textbook Question

Two tightly wound solenoids have the same length and circular cross-sectional area. But solenoid 1 uses wire that is 1.5 times as thick as solenoid 2. What is the ratio of their inductive time constants? (Assume the only resistance in the circuits is that of the wire itself.)

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Textbook Question

Two tightly wound solenoids have the same length and circular cross-sectional area. But solenoid 1 uses wire that is 1.5 times as thick as solenoid 2.

(a) What is the ratio of their inductances?

(b) What is the ratio of their inductive time constants? (Assume the only resistance in the circuits is that of the wire itself.)

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Textbook Question

A 10.0-k Ω resistor is in series with a 34.0-mH inductor and an ac source. Calculate the impedance of the circuit if the source frequency is (a) 55.0 Hz; (b) 55.0 kHz.

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Textbook Question

(II) A 25-mH coil whose resistance is 0.80 Ω is connected to a capacitor C and a 420-Hz source voltage. If the current and voltage are to be in phase, what value must C have?

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Textbook Question

(II) (a) In Fig. 30–28, assume that the switch has been in position A for sufficient time so that a steady current I₀ = V₀/R flows through the resistor R. At time t = 0, the switch is quickly switched to position B and the current decays through resistor R' (which is much greater than R) according to $I = I_{0} e^{- t / τ^{'}}$ $I = I_{0} e^{- t / τ^{'}}$ . Show that the maximum emf ε_max induced in the inductor during this time period is (R'/R)V_o. (b) If R' = 45R and V_o = 145 V, determine ε_max. [When a mechanical switch is opened, a high-resistance air gap is created, which is modeled as R' here. This Problem illustrates why high-voltage sparking can occur if a current-carrying inductor is suddenly cut off from its power source. The very high voltage can produce an electric field great enough to ionize atoms of air, which emit light when electrons recombine with the ions.]

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Textbook Question

(II) A capacitor is placed in parallel with some device, B, as in Fig. 30–18b, to filter out stray high-frequency signals, but to allow ordinary 60.0-Hz ac to pass through with little loss. Suppose that circuit B in Fig. 30–18b is a resistance R = 530 Ω connected to ground, and that C = 0.35 μF. Calculate the ratio of the capacitor’s current amplitude to the incoming current’s amplitude if the incoming current has a frequency of (a) 60.0 Hz; (b) 60.0 kHz.