Question

Explain the difference between self-inductance and mutual inductance. Define the coefficient of self-inductance and coefficient of mutual inductance. A plane coil of area 100 cm$^2$ is rotating in a magnetic field of 2 weber/m$^2$ with angular velocity 20 rad/s. What will be the maximum induced electromotive force in the coil?

Hint:

For a rotating coil in a magnetic field, the induced emf is directly proportional to the magnetic field strength, the area of the coil, and the angular velocity of the rotation.

Answer 1

Self-inductance (\(L\)) is a property of a coil (or any conductor) that opposes the change in current flowing through it. When the current in a coil changes, it induces an electromotive force (emf) in the coil itself, as given by the equation:
\[ \mathcal{E} = -L \frac{dI}{dt} \] Where:
- \(L\) is the self-inductance of the coil,
- \(\frac{dI}{dt}\) is the rate of change of current. Mutual inductance (\(M\)) is the property of two coils where a change in current in one coil induces an emf in the other coil. The induced emf is proportional to the rate of change of current in the first coil and the mutual inductance \(M\) between them. The equation is given by:
\[ \mathcal{E}_2 = -M \frac{dI_1}{dt} \] Where:
- \(\mathcal{E}_2\) is the induced emf in the second coil,
- \(M\) is the mutual inductance,
- \(\frac{dI_1}{dt}\) is the rate of change of current in the first coil. For the second part of the question, the induced electromotive force (\(\mathcal{E}\)) in a rotating coil in a magnetic field is given by:
\[ \mathcal{E} = B \cdot A \cdot \omega \] Where:
- \(B\) is the magnetic field strength,
- \(A\) is the area of the coil,
- \(\omega\) is the angular velocity.
Given:
- \(B = 2~\text{weber/m}^2\),
- \(A = 100~\text{cm}^2 = 100 \times 10^{-4}~\text{m}^2 = 10^{-2}~\text{m}^2\),
- \(\omega = 20~\text{rad/s}\).
Substituting the values into the formula:
\[ \mathcal{E} = 2 \times 10^{-2} \times 20 = 0.4~\text{V} \] Thus, the maximum induced electromotive force in the coil is \(0.4~\text{V}\).