Question

Define the coefficient of mutual inductance.
The coefficient of mutual inductance (\(M\)) is a measure of the ability of one coil to induce an electromotive force (EMF) in a nearby coil due to a changing current in the first coil. It is given by: \[ e = -M \frac{di}{dt} \]

If a current \( i = 10 \sin(100 \pi t) \, \mathrm{A} \) is flowing in a primary coil, and the maximum induced electromotive force in the secondary coil placed near it is \( 5 \pi \, \mathrm{volt} \), then the coefficient of mutual induction between these coils needs to be determined.

Hint:

The coefficient of mutual inductance can be calculated using \( \text{emf}_{\text{max}} = M \frac{di}{dt} \). Use the derivative of current to find \( \frac{di}{dt} \).

Answer 1

The maximum induced emf is related to mutual inductance \( M \) by: \[ \text{emf}_{\text{max}} = M \frac{di}{dt}. \] For \( i = 10 \sin(100 \pi t) \): \[ \frac{di}{dt} = 1000 \pi \cos(100 \pi t). \] The maximum value of \( \frac{di}{dt} \) is \( 1000 \pi \). Substituting values: \[ 5 \pi = M (1000 \pi) \quad \Rightarrow \quad M = 0.005 \, \mathrm{H}. \]