A rectangular loop of dimensions l and h moves with a...

A rectangular loop of dimensions $l$ and $h$ moves with a constant velocity $u$ away from a long wire that carries a steady current $I_1$ in the plane of the loop. The total resistance of the loop is $R$. Derive an expression for the current $I_2$ in the loop at the instant the closer side of the loop is a distance $r$ from the wire. We have done a similar problem before but use the flux rule this time. Notice that the resistor heats up. What is providing this energy or rather, doing work on the system? The magnetic force seems to be doing work! Resolve this apparent paradox. (GRAPH CAN'T COPY)

Key Concepts

Energy Conservation in Electromagnetic Systems

This concept emphasizes that even though magnetic forces appear to do work in the system, they do not actually contribute energy since magnetic forces cannot change the kinetic energy of moving charges directly. Instead, the energy to overcome induced electromagnetic forces comes from the mechanical work done by an external agent moving the loop, ensuring overall energy conservation.

Joule Heating

Joule heating refers to the process by which electrical energy is converted into thermal energy (heat) in a resistor. When an induced current flows through a resistive loop, part of the electrical energy is dissipated as heat, illustrating an irreversible conversion of energy resulting from resistance.

Non-Work-Doing Nature of Magnetic Forces

Magnetic forces are always perpendicular to the velocity of a charge, meaning that they do no work on the moving charges directly. In the context of electromagnetic induction, even though magnetic forces are responsible for redistributing energy within the system, the actual work required to drive the current (and consequently to convert energy to heat via resistance) must be supplied by an external force acting against the induced electromagnetic reaction.

Magnetic Flux

Magnetic flux is a measure of the total magnetic field passing through a given area. It is defined as the integral of the magnetic field over the surface area perpendicular to the field lines. In scenarios where a loop moves through a magnetic field, any change in the area exposed to the magnetic field alters the flux, thereby inducing an emf.

Lenz's Law

Lenz's Law is the rule that determines the direction of the induced current. It states that the induced current will flow in such a way that its magnetic field opposes the change in the original magnetic flux. This opposition is a manifestation of the conservation of energy in electromagnetic systems.

Faraday's Law

Faraday's Law of electromagnetic induction states that a changing magnetic flux through a circuit induces an electromotive force (emf) in that circuit. This principle underlies the operation of generators and explains how movement relative to a magnetic field can produce an electrical current in a conductor, as seen in the moving loop in the problem.

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