A metal rod with resistance R can slide without friction on...

A metal rod with resistance $R$ can slide without friction on two zero-resistance rails, as shown in Figure 23.26. The rod and the rails are within a region of constant magnetic field pointing out of the page. Describe the motion of the rod when the switch is closed and be sure to include the effects of motional emf.

Key Concepts

Energy Conversion in Electromagnetic Systems

The system converts mechanical energy into electrical energy through the process of electromagnetic induction. As the rod accelerates and moves through the magnetic field, kinetic energy is transformed into electrical energy via the induced current. This conversion highlights the interplay between mechanical motion and electromagnetic effects, and it can result in self-limiting acceleration due to induced opposing forces.

Magnetic Force on a Current-Carrying Conductor

When current flows through the rod in the presence of a magnetic field, the rod experiences a magnetic force according to the Lorentz force law. This force is perpendicular to both the direction of the current and the magnetic field, and it influences the motion of the rod by accelerating it or opposing its movement, depending on the current’s direction dictated by Lenz’s Law.

Motional EMF

Motional electromotive force (emf) is generated when a conductor moves through a magnetic field. In the context of the rod sliding on rails, the movement of the rod through the constant magnetic field creates an emf that drives current in the circuit. This emf is proportional to the velocity of the rod, the magnetic field strength, and the effective length of the rod within the field.

Faraday’s Law of Electromagnetic Induction

Faraday’s Law states that an emf is induced in a circuit when there is a change in magnetic flux through the circuit. As the rod moves, it changes the area of the circuit enclosed by the rails and the rod, leading to a changing magnetic flux, and thus inducing an emf according to this law.

Lenz’s Law

Lenz’s Law describes the direction of the induced current, specifying that the induced emf will generate a current whose magnetic field opposes the change in the original magnetic flux. In this situation, the current induced by the motional emf produces a magnetic field that opposes the rod’s motion, effectively acting as a braking force.

Transcript

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