A short rod of length a moves with its velocity 𝐯 parallel to an infinite wire carrying a curren

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A short rod of length a moves with its velocity 𝐯 parallel...

A short rod of length $a$ moves with its velocity $\overrightarrow{\mathbf{v}}$ parallel to an infinite wire carrying a current $I$ (see below). If the end of the rod nearer the wire is a distance $b$ from the wire, what is the emf induced in the rod?

Key Concepts

Motional EMF

Motional EMF refers to the electromotive force generated when a conductor moves through a magnetic field. The induced voltage arises due to the Lorentz force acting on the charge carriers in the conductor, pushing them along the length of the conductor. This concept is fundamental in electromagnetic induction and in devices such as generators.

Magnetic Field of a Current-Carrying Conductor

The magnetic field around a long straight conductor carrying current is given by the Biot–Savart law, which simplifies to B = (?? I) / (2?r), where r is the distance from the wire. This relation is crucial in understanding how the magnetic field strength varies with distance from the current, affecting the induced EMF in nearby moving conductors.

Faraday's Law of Electromagnetic Induction

Faraday's Law states that a changing magnetic flux through a circuit induces an electromotive force in that circuit. In the context of a moving conductor in a nonuniform magnetic field (due to a current-carrying wire), the movement results in a time-varying magnetic flux that generates an EMF. This law underpins the process of electromagnetic induction in dynamic systems.

Integration Over a Distributed Magnetic Field

When a conductor moves through a spatially varying magnetic field, the induced EMF must be calculated by integrating the local contributions along the length of the conductor. This involves summing the product of the local magnetic field and the differential element of the conductor, taking into account the varying distance from the source of the magnetic field.

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