Question

Define self-inductance and mutual inductance. Find an expression for mutual inductance of two coaxial solenoids.

Hint:

Self-inductance opposes current change in a coil, while mutual inductance measures the induced emf due to a current in another coil. For coaxial solenoids: \[ M = \frac{\mu_0 N_1 N_2 A}{l} \]

Answer 1

Self-Inductance:
Self-inductance (\(L\)) is the property of a coil or solenoid that causes it to oppose any change in the current passing through it. When the current through a coil changes, a time-varying magnetic field is created, which induces an emf (electromotive force) in the coil, opposing the change in current. The self-inductance is defined as the ratio of the induced emf to the rate of change of current: \[ L = \frac{N \cdot \Phi_B}{I} \] Where: - \(L\) is the self-inductance,
- \(N\) is the number of turns of the coil,
- \(\Phi_B\) is the magnetic flux linked with the coil,
- \(I\) is the current passing through the coil.
Mutual Inductance:
Mutual inductance (\(M\)) is the property of two coils such that the change in current in one coil induces an emf in the other coil due to the time-varying magnetic field produced by the first coil. The mutual inductance is defined as the ratio of the induced emf in one coil to the rate of change of current in the other coil: \[ M = \frac{N_2 \cdot \Phi_{21}}{I_1} \] Where: - \(M\) is the mutual inductance,
- \(N_2\) is the number of turns in the second coil,
- \(\Phi_{21}\) is the magnetic flux through the second coil due to the current in the first coil,
- \(I_1\) is the current in the first coil.
Expression for Mutual Inductance of Two Coaxial Solenoids:
Consider two solenoids with lengths \(l_1\) and \(l_2\), radii \(r_1\) and \(r_2\), and number of turns \(N_1\) and \(N_2\), placed coaxially. The mutual inductance \(M\) between the two coils can be expressed as: \[ M = \frac{\mu_0 N_1 N_2 A}{l} \] Where: - \(\mu_0\) is the permeability of free space,
- \(N_1\) and \(N_2\) are the number of turns in the first and second solenoids,
- \(A\) is the cross-sectional area of the solenoids,
- \(l\) is the length of the solenoids.
This expression holds when the solenoids are placed coaxially with their magnetic fields aligned.