In a uniform magnetic field of induction B, a wire in the...

In a uniform magnetic field of induction $B$, a wire in the form of semicircle of radius $r$ rotates about the diameter of the circle with angular frequency $\omega$. If the total resistance of the circuit is $R$, the mean power generated per period of rotation is (A) $\frac{B \pi r^{2} \omega}{2 R}$ (B) $\frac{\left(B \pi r^{2} \omega\right)^{2}}{2 R}$ (C) $\frac{(B \pi r \omega)^{2}}{2 R}$ (D) $\frac{\left(B \pi r^{2} \omega\right)^{2}}{8 R}$

In a uniform magnetic field of induction $B$, a wire in the form of semicircle of radius $r$ rotates about the diameter of the circle with angular frequency $\omega$. If the total resistance of the circuit is $R$, the mean power generated per period of rotation is
(A) $\frac{B \pi r^{2} \omega}{2 R}$
(B) $\frac{\left(B \pi r^{2} \omega\right)^{2}}{2 R}$
(C) $\frac{(B \pi r \omega)^{2}}{2 R}$
(D) $\frac{\left(B \pi r^{2} \omega\right)^{2}}{8 R}$

Key Concepts

Faraday’s Law of Electromagnetic Induction

This law is fundamental to understanding how a time-varying magnetic flux through a circuit generates an electromotive force (emf). It is the principal concept behind the conversion of mechanical motion into electrical energy in generators and similar devices.

Rotation of Conductors in Magnetic Fields

When a conductor rotates in a uniform magnetic field, the area through which the field lines pass varies with time, resulting in a changing magnetic flux. This variation is crucial for inducing a time-dependent emf in the system.

Calculation of Induced Emf in Rotating Systems

The induced emf in a rotating conductor is determined by the rate of change of the magnetic flux, which often involves the use of trigonometric functions to describe the periodic changes in flux. This concept is central to analyzing systems where the geometry changes with rotation.

Joule Heating and Mean Power Dissipation

In circuits with a resistive element, the power converted from the induced emf is dissipated as heat via Joule's heating effect. Calculating the mean power generated over a cycle involves finding the average of the square of the induced emf divided by the resistance.

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