Definition

When the current in a coil changes, the magnetic flux linked with the coil also changes. According to Faraday’s law, this change induces an emf in the same coil. This phenomenon is called self-induction, and the emf is called self-induced emf.

Mathematical Expression

• Magnetic flux linkage through a coil of \(N\) turns:
  \[ \Phi_B = L I \]
  where \(L\) is called the self-inductance of the coil.
• According to Faraday’s law:
  \[ \varepsilon = – \frac{d\Phi_B}{dt} \]
• Substituting \(\Phi_B = L I\):
  \[ \varepsilon = – L \frac{dI}{dt} \]

Thus, self-induced emf is proportional to the rate of change of current.

Physical Meaning of L

• \(L\) depends on geometry of the coil (number of turns, area, length) and the medium inside (permeability).
• SI unit of \(L\) is henry (H).
• If current of 1 A in a coil produces flux linkage of 1 weber-turn, its inductance is 1 H.

Self-Inductance of a Solenoid

For a long solenoid of cross-sectional area \(A\), length \(l\), and \(n\) turns per unit length:

• Magnetic field inside: \( B = \mu_0 n I \)
• Total flux linkage: \( N \Phi_B = \mu_0 n^2 A l I \)
• Hence, self-inductance:
  \[ L = \mu_0 n^2 A l \]
  If the core has relative permeability \(\mu_r\), then
  \[ L = \mu_0 \mu_r n^2 A l \]

Energy Stored in an Inductor

Work is done against the back emf to establish current in a coil. This energy is stored in the magnetic field.

• Energy stored:
  \[ U = \frac{1}{2} L I^2 \]
• Energy density in magnetic field:
  \[ u = \frac{B^2}{2\mu_0} \]

Analogy with Mechanics

Self-inductance \(L\) is analogous to mass in mechanics:

• Just as mass opposes change in velocity, inductance opposes change in current.
• Self-induced emf acts like “electrical inertia.”

Applications

• Chokes in fluorescent tubes.
• Energy storage in inductors.
• Filters in AC circuits.
• Transformers (along with mutual induction).

Worked Example