Inductance

Inductance is a fundamental property of any conductor or coil that quantifies its ability to store energy in a magnetic field when current flows through it. It forms the backbone of transformer design, filter circuits, and energy storage systems. Understanding inductance is essential for GATE aspirants tackling electromagnetics and circuit theory problems.

Core Concept Explanation

When current flows through a conductor, it creates a magnetic field around it. If the conductor is wound into a coil, the field lines thread through each turn, producing a net magnetic flux linkage denoted by the symbol λ (lambda) or N·Φ. Inductance is defined as the ratio of this flux linkage to the current producing it.

Self inductance L describes the flux linkage a coil creates with itself. It is purely a geometric property: it depends on the number of turns, cross-sectional area, length of the coil, and the permeability of the core material. The defining equation is L = N·Φ / I, where N is the number of turns, Φ is the magnetic flux through one turn, and I is the current in the coil. The SI unit of inductance is the henry (H), equal to one weber per ampere.

Mutual inductance M describes the coupling between two separate coils. When current I1 flows in coil 1, it produces a flux Φ12 that links coil 2 with N2 turns. Then M = N2·Φ12 / I1. The reciprocity theorem in magnetics proves that M12 = M21, meaning the coupling is symmetric. A useful dimensionless number called the coupling coefficient k relates M to the self inductances: M = k·√(L1·L2), where 0 ≤ k ≤ 1.

The voltage across an inductor is determined by how fast the current changes. For self inductance: v = L·(dI/dt). For mutual inductance, the voltage induced in coil 2 due to changing current in coil 1 is v2 = M·(dI1/dt). The negative sign in the full form comes from Lenz law, which is discussed in a separate article.

Mathematical Expression

The formal definition of inductance uses the concept of flux linkage. For a coil with N turns each carrying flux Φ, the total flux linkage λ is λ = N·Φ. Self inductance is then L = λ / I. Using Faraday's law, the induced emf is e = -dλ/dt = -L·(dI/dt). For a linear medium (constant permeability), L is constant and depends only on geometry.

For a medium with permeability μ = μ0·μr, the flux Φ = B·A = μ·H·A. Since H = N·I / l (for a solenoid of length l), we get Φ = μ·N·I·A / l. Substituting into L = N·Φ / I gives L = μ·N²·A / l. This is the standard solenoid inductance formula. Notice that L scales as N², meaning doubling the turns quadruples the inductance.

Practical Understanding

Inductors oppose changes in current rather than current itself. This is why a large inductor in a DC circuit acts as a short circuit at steady state but impedes AC current. In power electronics, inductors smooth out current ripple in switching converters. In RF circuits, inductors form resonant tanks with capacitors. In transformers, mutual inductance transfers energy between primary and secondary windings without a direct electrical connection.

The energy stored in an inductor is W = (1/2)·L·I². This energy resides in the magnetic field and is returned to the circuit when current decreases. This is unlike a resistor, which permanently dissipates energy as heat. The inductance therefore acts as a lossless energy buffer in ideal conditions.

Given:
Coil: N = 500 turns, cross-section A = 4 cm² = 4×10⁻⁴ m², length l = 20 cm = 0.2 m
Core: air (μ = μ0 = 4π×10⁻⁷ H/m)
Current I = 2 A

Why this formula applies:
The coil is a solenoid so L = μ₀·N²·A / l applies directly.

Formula:
L = μ₀·N²·A / l

Substitution:
L = (4π×10⁻⁷) × (500)² × (4×10⁻⁴) / 0.2

Calculation:
L = (4π×10⁻⁷) × 250000 × (4×10⁻⁴) / 0.2
L = (4π×10⁻⁷) × 0.5
L = 2π×10⁻⁷ × ... let us compute step by step:
Numerator = (4π×10⁻⁷) × (2.5×10⁵) × (4×10⁻⁴)
         = (4π×10⁻⁷) × 0.1
         = 4π×10⁻⁸
L = 4π×10⁻⁸ / 0.2 = 20π×10⁻⁸ ≈ 6.28×10⁻⁷ H

Final Answer: L ≈ 628 nH (nano-henry) for the air-core solenoid.

Exam Tip: GATE often asks for inductance scaling. Remember L is proportional to N² and to μr. If μr of the core doubles, L doubles. If N doubles, L becomes four times. Do not confuse flux linkage (N·Φ) with flux (Φ) in the formula.

Mechanism of Inductance in Physical Terms

Key Points on Mechanism

• When current I flows through a coil of N turns, it creates a magnetic field B = μ·n·I inside the coil, where n = N/l is the turn density.
• The flux through each turn is Φ = B·A, and the total flux linkage is λ = N·Φ. Inductance is the proportionality constant: L = λ/I.
• Any change in current causes a change in flux linkage, which by Faraday's law induces a back-emf: e = -L·dI/dt. This back-emf always opposes the change causing it.
• For mutual inductance, flux from coil 1 partially links coil 2. The fraction that links is determined by the coupling coefficient k. Ideal transformer has k = 1.
• Inductance increases with higher permeability core material. Ferromagnetic cores (μr up to thousands) allow compact high-L inductors used in power supplies.

Quick Revision

• Self inductance: L = N·Φ / I = μ·N²·A / l. Unit: henry (H).
• Mutual inductance: M = N2·Φ12 / I1 = k·√(L1·L2). Reciprocal: M12 = M21.
• Voltage law: v = L·dI/dt for self; v2 = M·dI1/dt for mutual.
• L scales as N². Doubling turns gives 4× inductance.
• Energy stored: W = (1/2)·L·I². This is magnetic field energy.
• Trap: L = N·Φ/I uses flux through ONE turn, not total linkage. Total linkage λ = N·Φ = L·I.
• Coupling coefficient k ranges from 0 (no coupling) to 1 (perfect coupling, as in ideal transformer).