Question 256: Electromagnetic Induction

Question

Two coils have a mutual inductance $5 	imes 10^{-3} \mathrm{H}$. The current changes in the first coil according to the equation $I_1=I_0 \sin \omega t$ where $I_0=10 \mathrm{~A}$ and $\omega=100 \pi \mathrm{rad} / \mathrm{s}$. What is the value of the maximum e.m.f. in the coil?

Accepted Answer

The electromotive force (e.m.f.) induced in the second coil due to a changing current in the first coil is given by Faraday's law of electromagnetic induction and the concept of mutual inductance. The e.m.f. $\mathcal{E}$ in the second coil is related to the rate of change of current in the first coil as follows:
$$ \mathcal{E} = -M \frac{dI_1}{dt} $$
where $M$ is the mutual inductance and $I_1$ is the current in the first coil.
Given:
$M = 5 	imes 10^{-3} \, \mathrm{H}$
$I_1 = I_0 \sin \omega t$
$I_0 = 10 \, \mathrm{A}$
$\omega = 100 \pi \, \mathrm{rad/s}$
To find the maximum e.m.f., we first differentiate the current $I_1$ with respect to time $t$:
$$ \frac{dI_1}{dt} = \frac{d}{dt}(I_0 \sin \omega t) = I_0 \omega \cos \omega t $$
Substituting the values, we have:
$$ \frac{dI_1}{dt} = 10 	imes 100\pi \cos(100\pi t) = 1000\pi \cos(100\pi t) $$
The maximum value of $\cos(100\pi t)$ is 1, therefore the maximum rate of change of current is:
$$ \left(\frac{dI_1}{dt}ight)_{	ext{max}} = 1000\pi $$
Now, substituting this into the equation for the induced e.m.f., we calculate the maximum e.m.f.:
$$ \mathcal{E}_{	ext{max}} = -M 	imes 1000\pi = -(5 	imes 10^{-3}) 	imes 1000\pi $$
Simplifying, we find:
$$ \mathcal{E}_{	ext{max}} = -5 	imes \pi = -5\pi \, \mathrm{V} $$
Since e.m.f. is generally discussed in magnitude, the answer is:
Option D
: $5\pi \, \mathrm{V}$

Answer

$2 \pi \mathrm{~V}$

Answer

$3 \pi \mathrm{~V}$

Answer

$4 \pi \mathrm{~V}$

Answer

$5 \pi \mathrm{~V}$

Electromagnetic Induction

Two coils have a mutual inductance 5 \times 10^{-3} \mathrm{H}. The current changes in the first coil according to the equation I_1=I_0 \sin \omega t where I_0=10 \mathrm{~A} and \omega=100 \pi \mathrm{rad} / \mathrm{s}. What is the value of the maximum e.m.f. in the coil?

Choose the correct answer: