Definition

When the current in one coil changes, the magnetic flux linked with a nearby coil also changes, inducing an emf in it. This is called mutual induction. The coefficient of mutual induction is called mutual inductance \(M\).

Mathematical Expression

• If current in coil 2 is \(I_2\), then flux linkage with coil 1:
  \[ N_1 \Phi_1 = M I_2 \]
• Induced emf in coil 1:
  \[ \varepsilon_1 = -M \frac{dI_2}{dt} \]
• Similarly, if current in coil 1 is \(I_1\), emf in coil 2 is:
  \[ \varepsilon_2 = -M \frac{dI_1}{dt} \]

Thus, emf in one coil is proportional to the rate of change of current in the other coil.

Derivation for Two Co-axial Solenoids

Consider two long co-axial solenoids, length \(l\), inner radius \(r_1\), outer radius \(r_2\).

• Magnetic field due to current \(I_2\) in solenoid 2:
  \[ B = \mu_0 n_2 I_2 \]
  where \(n_2\) = turns per unit length of solenoid 2.
• Flux through solenoid 1:
  \[ \Phi_1 = B \cdot A_1 = \mu_0 n_2 I_2 \cdot \pi r_1^2 \]
• Total flux linkage:
  \[ N_1 \Phi_1 = \mu_0 n_1 l \cdot n_2 I_2 \cdot \pi r_1^2 \]
• Comparing with \( N_1 \Phi_1 = M I_2 \), we get:
  \[ M = \mu_0 n_1 n_2 \pi r_1^2 l \]

If the medium has relative permeability \(\mu_r\), then
\[ M = \mu_0 \mu_r n_1 n_2 \pi r_1^2 l \]

Key Points

• \(M\) depends only on geometry of the coils and the medium, not on the current.
• SI unit of \(M\) is henry (H).
• Mutual inductance is symmetric: \( M_{12} = M_{21} \).

Applications

• Transformers: step-up and step-down voltages.
• Wireless charging: induced emf in nearby coil.
• Induction cooktops and guitar pickups.

Worked Example