Question 263: Electromagnetic Induction

Question

A circular coil of resistance $R$, area $A$, number of turns ' $N$ ' is rotated about its vertical diameter with angular speed ' $\omega$ ' in a uniform magnetic field of magnitude ' $B$ '. The average power dissipated in a complete cycle is

Accepted Answer

The electromotive force (emf) induced in the coil as it rotates in a magnetic field is given by Faraday's law of electromagnetic induction. The expression for the emf in a rotating coil is:
$$ \epsilon(t) = NBA \omega \sin(\omega t) $$
where:
$\epsilon(t)$ is the induced emf as a function of time,
$N$ is the number of turns,
$B$ is the magnetic field,
$A$ is the area of the coil,
$\omega$ is the angular speed,
$t$ is time.
The instantaneous power dissipated in the coil due to its resistance $R$ is given by:
$$ P(t) = \frac{\epsilon(t)^2}{R} $$
Substituting the expression for $\epsilon(t)$:
$$ P(t) = \frac{(NBA \omega \sin(\omega t))^2}{R} = \frac{N^2 B^2 A^2 \omega^2 \sin^2(\omega t)}{R} $$
The average power dissipated over a complete cycle is calculated by integrating $P(t)$ over one period $T = \frac{2\pi}{\omega}$ of the sine function and then dividing by the period:
$$ \langle P \rangle = \frac{1}{T} \int_0^T P(t) \, dt = \frac{1}{T} \int_0^T \frac{N^2 B^2 A^2 \omega^2 \sin^2(\omega t)}{R} \, dt $$
The integral of $\sin^2(\omega t)$ over one period is $\frac{T}{2}$. Therefore:
$$ \langle P \rangle = \frac{1}{T} \cdot \frac{N^2 B^2 A^2 \omega^2}{R} \cdot \frac{T}{2} = \frac{N^2 B^2 A^2 \omega^2}{2R} $$
Thus, the average power dissipated in a complete cycle is:
Option A:
$$\frac{N^2 A^2 B^2 \omega^2}{2 R}$$

Answer

$\frac{\mathrm{N}^2 \mathrm{~A}^2 \mathrm{~B}^2 \omega^2}{2 \mathrm{R}}$

Answer

$\frac{\mathrm{BNA} \omega}{\mathrm{R}}$

Answer

$\frac{\mathrm{N}^2 \mathrm{AB}}{2 \mathrm{R} \omega^2}$

Answer

$\frac{B A \omega}{2 N R}$

Electromagnetic Induction

A circular coil of resistance R, area A, number of turns ' N ' is rotated about its vertical diameter with angular speed ' \omega ' in a uniform magnetic field of magnitude ' B '. The average power dissipated in a complete cycle is

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