A conducting rod of length ℓmoves with velocity 𝐯 parallel...

A conducting rod of length $\ell$ moves with velocity $\mathbf{v}$ parallel to a long wire carrying a steady current $I$. The axis of the rod is maintained perpendicular to the wire with the near end a distance $r$ away, as shown in Figure P31.66. Show that the magnitude of the emf induced in the rod is $$|\boldsymbol{\varepsilon}|=\frac{\mu_{0} I}{2 \pi} v \ln \left(1+\frac{\ell}{r}\right).$$

Key Concepts

Electromagnetic Induction

This concept refers to the creation of an electromotive force (emf) in a conductor due to its exposure to a changing magnetic field or its motion through a magnetic field. It is the underlying principle behind many electrical devices and is governed by Faraday’s law of induction.

Magnetic Field of a Long Straight Conductor

This key concept involves understanding that a long, straight wire carrying a steady current produces a magnetic field that circulates around the wire. The strength of this field decreases with distance from the wire according to the expression B = (??I)/(2?r), a result derived from Ampère's law.

Motional Electromotive Force (EMF)

Motional emf arises when a conductor moves in a magnetic field, leading to the separation of charges due to the Lorentz force acting on the charge carriers. The induced emf is determined by integrating the contributions along the length of the conductor, which is especially important when the magnetic field is non-uniform.

Integration in Non-uniform Magnetic Fields

In cases where the magnetic field varies along the path of a moving conductor, calculating the total induced emf requires integrating the infinitesimal contributions from each segment of the conductor. This approach accounts for the spatial variation of the magnetic field, such as that produced by a long straight current-carrying wire.

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