Question

What is meant by mutual inductance? Define coefficient of mutual inductance and write its unit.

Hint:

Relate the flux in the second coil to the current in the first, \( \phi_2 = M I_1 \); the induced emf gives \( M = -\varepsilon_2/(dI_1/dt) \), and the SI unit is the henry.

Answer 1

Step 1: Mutual inductance is the phenomenon in which a changing current in one coil (the primary) induces an emf in a neighbouring coil (the secondary), because the magnetic flux linked with the secondary keeps changing.
Step 2: Let a current \( I_1 \) in the primary produce a magnetic flux \( \phi_2 \) linked with the secondary. This flux is proportional to the current: \( \phi_2 = M I_1 \), where \( M \) is the coefficient of mutual inductance.
Step 3: Hence \( M = \dfrac{\phi_2}{I_1} \). It equals the flux linked with the secondary per unit current in the primary.
Step 4: By Faraday's law the emf induced in the secondary is \( \varepsilon_2 = -\dfrac{d\phi_2}{dt} = -M\dfrac{dI_1}{dt} \). So \( M = -\dfrac{\varepsilon_2}{dI_1/dt} \), i.e. \( M \) is the emf induced in the secondary per unit rate of change of current in the primary.
Step 5: Unit: from \( M = \dfrac{\phi_2}{I_1} \) the unit is weber per ampere, which is named the henry (H). Also \( 1\ \text{H} = 1\ \text{V·s/A} \).
\[\boxed{\text{Unit of } M = \text{henry (H)}}\]