A conducting rod P Q of length 4 ℓis rotated about a point O in a uniform magnetic field B →PO=ℓ

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A conducting rod P Q of length 4 ℓis rotated about a point O...

A conducting rod $P Q$ of length $4 \ell$ is rotated about a point $O$ in a uniform magnetic field $\mathrm{B} \rightarrow \mathrm{PO}=\ell$ Then (a) $\mathrm{V}_{\mathrm{Q}}-\mathrm{V}_{\mathrm{P}}=-\left[\left\{\mathrm{B} \omega \ell^{2}\right\} /\right.$ $\{2\}] \quad$ (b) $\mathrm{V}_{\mathrm{Q}}-\mathrm{V}_{\mathrm{O}}=(5 / 2) \mathrm{B} \operatorname{co} \ell^{2}$ (c) $\mathrm{V}_{Q}-\mathrm{V}_{\mathrm{O}}=(9 / 2) \mathrm{B} \omega \ell^{2}$ (d) $\mathrm{V}_{\mathrm{P}}-\mathrm{V}_{\mathrm{Q}}=4 \mathrm{~B} \omega \ell^{2}$

A conducting rod $P Q$ of length $4 \ell$ is rotated about a point $O$
in a uniform magnetic field $\mathrm{B} \rightarrow \mathrm{PO}=\ell$ Then
(a) $\mathrm{V}_{\mathrm{Q}}-\mathrm{V}_{\mathrm{P}}=-\left[\left\{\mathrm{B} \omega \ell^{2}\right\} /\right.$
$\{2\}] \quad$ (b) $\mathrm{V}_{\mathrm{Q}}-\mathrm{V}_{\mathrm{O}}=(5 / 2) \mathrm{B} \operatorname{co} \ell^{2}$
(c) $\mathrm{V}_{Q}-\mathrm{V}_{\mathrm{O}}=(9 / 2) \mathrm{B} \omega \ell^{2}$
(d) $\mathrm{V}_{\mathrm{P}}-\mathrm{V}_{\mathrm{Q}}=4 \mathrm{~B} \omega \ell^{2}$

Key Concepts

Integration of Electric Field Along a Conductor

To determine the total electromotive force across a non-uniform conductor, such as a rotating rod, it is necessary to integrate the local electric field (or the Lorentz force per unit charge) along the path of the conductor. This integration takes into account the variation of velocity along the length of the conductor, which is particularly important in rotational systems where the speed increases with distance from the center of rotation.

Rotational Motion in Magnetic Fields

When a conductor rotates in a magnetic field, different parts of the rod have different linear velocities due to the angular motion. This variation leads to a non-uniform distribution of the induced electric field and thus the potential difference across segments of the conductor. Analyzing such a system requires combining principles of rotational dynamics with electromagnetic induction to accurately compute the induced voltage differences.

Motional Electromotive Force (EMF)

Motional EMF is the voltage generated when a conductor moves through a magnetic field. The key idea is that free charges within the moving conductor experience the Lorentz force, which pushes them along the conductor to create a separation of charge, and therefore a potential difference. This concept is fundamental to understanding how electrical energy can be generated from mechanical motion in various devices such as generators.

Lorentz Force

The Lorentz force is the force acting on a charged particle moving through an electric and magnetic field. In the context of a moving conductor, the force (given by the cross product of the velocity vector and the magnetic field) causes charge carriers to deflect, establishing an electric field within the conductor. This phenomenon explains the origin of the motional EMF in systems where conductors traverse magnetic fields.

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