A conducting rod A C of length 4 ℓis rotated about a point O in a uniform magnetic field directe

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

A conducting $\operatorname{rod} A C$ of length $4 \ell$ is rotated about a point $O$ in a uniform magnetic field directed into the paper. $A O=\ell$ and $O C=3 \ell$. Then (A) $V_{O}-V_{A}=\frac{B \omega l^{2}}{2}$ (B) $V_{O}-V_{A}=\frac{9}{2} B \omega l^{2}$ (C) $V_{A}-V_{C}=4 B \omega l^{2}$ (D) $V_{O}-V_{C}=\frac{9}{2} B \omega l^{2}$

A conducting $\operatorname{rod} A C$ of length $4 \ell$ is rotated about a point $O$ in a uniform magnetic field directed into the paper. $A O=\ell$ and $O C=3 \ell$. Then
(A) $V_{O}-V_{A}=\frac{B \omega l^{2}}{2}$
(B) $V_{O}-V_{A}=\frac{9}{2} B \omega l^{2}$
(C) $V_{A}-V_{C}=4 B \omega l^{2}$
(D) $V_{O}-V_{C}=\frac{9}{2} B \omega l^{2}$

Key Concepts

Motional Electromotive Force (EMF)

This concept refers to the EMF generated when a conductor moves in a magnetic field. In the context of a rotating rod, each element of the rod moving through the uniform magnetic field experiences an induced voltage due to its linear velocity, which is proportional to its distance from the rotation axis.

Rotating Conductor in a Uniform Magnetic Field

When a conductor rotates in a uniform magnetic field, different points along the rod have different tangential speeds. The induced emf in each segment depends on its radial distance from the pivot, and the net potential difference between two points is obtained by integrating these local contributions along the length of the conductor.

Integration of Differential EMF

The induced voltage over an infinitesimal segment of a rotating conductor can be expressed as dV = B * ? * r dr, where B is the magnetic field strength, ? is the angular velocity, and r is the distance from the rotation axis. Integrating this expression over a specified range of r yields the total EMF between two points, highlighting the quadratic dependence on the radial distance.

Voltage Distribution Along a Rotating Conductor

Due to the variation in tangential speed along the length of a rotating rod, the induced potential is not uniform. Calculating the voltage difference between any two points requires understanding that each segment contributes differently to the overall EMF, leading to expressions that depend on the squares of the distances from the axis.

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