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Ch. 29 - Electromagnetic Induction and Faraday's Law
Giancoli Douglas - Physics for Scientists and Engineers 5th edition
Giancoli Douglas5th editionPhysics for Scientists and EngineersISBN: 9780137488179Not the one you use?Change textbook
All textbooksGiancoli Douglas 5th editionCh. 29 - Electromagnetic Induction and Faraday's LawProblem 30
Chapter 28, Problem 30

(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 B→\(\overrightarrow{B}\) (⊥ 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?

Verified step by step guidance
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Step 1: Analyze the forces acting on the conducting rod. The rod experiences a magnetic force due to the current flowing through it, given by the Lorentz force formula: F = IℓB, where I is the current, ℓ is the length of the rod, and B is the magnetic field strength. This force will accelerate the rod along the rails.
Step 2: Write the equation of motion for the rod. The net force acting on the rod is the magnetic force minus the resistive force due to the rod's resistance. Using Newton's second law, the equation becomes: m(dv/dt) = IℓB - F_resistive, where m is the mass of the rod and dv/dt is its acceleration.
Step 3: For case (a), where the source provides a constant current I, substitute the constant current into the force equation. The resistive force arises due to the induced emf in the rod as it moves through the magnetic field. The induced emf is given by Faraday's law: ε_induced = Bℓv, where v is the velocity of the rod. The resistive force is then F_resistive = ε_induced / R = (Bℓv) / R.
Step 4: For case (b), where the source provides a constant emf ε₀, the current in the rod is determined by Ohm's law: I = ε₀ / R. Substitute this current into the force equation and follow a similar process as in case (a) to account for the resistive force due to the induced emf.
Step 5: For part (c), determine if the rod reaches a terminal speed. A terminal speed occurs when the net force on the rod becomes zero, meaning the magnetic force equals the resistive force. Solve for the terminal speed by setting IℓB = (Bℓv_terminal) / R in case (a) and ε₀ℓB = (Bℓv_terminal) / R in case (b).

Key Concepts

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

Electromagnetic Induction

Electromagnetic induction is the process by which a changing magnetic field within a closed loop induces an electromotive force (emf) in the wire. This principle, described by Faraday's law, is crucial for understanding how the conducting rod experiences a force when placed in a magnetic field, leading to its motion. The induced emf can be influenced by the speed of the rod and the strength of the magnetic field.
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Introduction to Induction

Lorentz Force

The Lorentz force is the force experienced by a charged particle moving through a magnetic field. In this scenario, the current flowing through the rod interacts with the magnetic field, resulting in a force that propels the rod along the rails. The direction of this force is given by the right-hand rule, which helps determine the motion of the rod in response to the magnetic field.
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Lorentz Transformations of Velocity

Terminal Velocity

Terminal velocity occurs when the net force acting on an object is zero, resulting in a constant speed. In the context of the conducting rod, this can happen when the magnetic force opposing the motion equals the force due to the current. Understanding whether the rod reaches terminal velocity in the given scenarios involves analyzing the balance of forces and the effects of resistance and induced emf over time.
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Related Practice
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Textbook Question

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