Inductance

What is inductance?

In plain words: a coil resists changes in its current. Try to change the current and the coil produces an emf that pushes back, just as a heavy object resists changes in its speed. Inductance measures how hard it pushes back, and a nearby coil can feel the effect too.

Scope: L and M are treated as constants below, with fixed coil geometry and a linear, reciprocal magnetic medium. Beyond the NCERT text: real ferromagnetic cores can saturate (Wikipedia, “Inductance”) and show hysteresis (J. Bahrdt, CERN-2010-004, Fig. 5), so their inductance can depend on current and history (reasoned).

Mutual inductance

• Two ideal long air-core coaxial solenoids, with end effects neglected and the same length l (NCERT Figure 6.12, page 166): inner S₁ with radius r₁ and n₁ turns per unit length, outer S₂ with r₂ and n₂.
• A current I₂ in S₂ gives a flux linkage with S₁ of N₁Φ₁ = M₁₂I₂ (NCERT Eq. (6.7)). The field of S₂, μ₀n₂I₂ (see the solenoid), fills S₁, so (NCERT Eq. (6.9), page 166):

• The other way round, the field of S₁ is confined inside S₁, so the flux through S₂ also uses the area πr₁² and M₂₁ comes out the same (NCERT Eq. (6.11)). So M₁₂ = M₂₁ = M, a reciprocity result valid more generally for fixed coils in linear reciprocal media (NCERT Eq. (6.12), page 166). It is useful when one of the two is hard to calculate.
• With a uniform linear medium of constant relative permeability μ_r filling the long-solenoid magnetic region, M = μ_rμ₀n₁n₂πr₁²l. Mutual inductance also depends on the coils’ separation and relative orientation (NCERT page 167).
• A changing current in one coil induces an emf in the other (NCERT page 167):

• If geometry changes, use −d(MI₂)/dt, which also contains −I₂ dM/dt. For fixed coils this is Experiment 6.3 in numbers: pressing or releasing the key changes I₂, and the emf in C₁ is M times its rate of change (NCERT pages 167–168).
• NCERT slip: page 166 says “Using Eq. (6.9) and Eq. (6.10), we get M₁₂ = M₂₁”. The two expressions being compared are Eqs. (6.9) and (6.11); Eq. (6.10) only defines M₂₁.

Self-inductance

• A changing current in a coil also changes the flux through the coil itself, inducing an emf in it: self-induction. The flux linkage is proportional to the current (NCERT Eq. (6.13), page 168):

• The induced emf is (NCERT Eq. (6.14), page 168):

• Ideal long solenoid, neglecting end effects, of length l, cross-section A and n turns per unit length: B = μ₀nI, the flux linkage is (nl)(μ₀nI)A, so L = μ₀n²Al (NCERT Eq. (6.15)). With a uniform filling of constant relative permeability μ_r, approximating the unsaturated operating range of soft iron, L = μ_rμ₀n²Al (NCERT Eq. (6.16), page 168).
• The self-inductance depends on the coil’s geometry and on the permeability of the medium (NCERT page 168); strictly, it is independent of the current only while that permeability stays constant (reasoned; see the Scope note). Since L ∝ n², doubling the turns at the same length, area and permeability makes L four times as large (reasoned from NCERT Eq. (6.15)).

If L varies, Faraday’s law gives ε = −d(LI)/dt, including −I dL/dt; the simpler −L dI/dt assumes constant L.

Back emf

• The self-induced emf always opposes any change, increase or decrease, of the current in the coil (NCERT page 168). It is called the back emf (page 168).
• So self-inductance acts like inertia: it is the electromagnetic analogue of mass in mechanics, opposing the growth and the decay of current (NCERT pages 168–169).
• When two coils carry currents at once, the emf in coil 1 has both parts (NCERT page 169):

Energy stored in an inductor

• For an ideal linear inductor with fixed L, work must be done against the back emf to build up a current. At the rate dW/dt = |ε|I = LI dI/dt, the total work to raise the current from 0 to I is (NCERT Eq. (6.17), page 169):

• This is stored as magnetic energy, and it mirrors ½mv², with L in the place of m (page 169).
• For a solenoid, put I = B/(μ₀n) and L = μ₀n²Al into ½LI²: the energy is B²Al/(2μ₀). Dividing by the volume Al gives u_B = B²/(2μ₀) (NCERT Example 6.9 and Eq. (6.18), pages 169–170). It matches the electric energy density ½ε₀E²: both grow as the square of the field. Although derived for a solenoid and a parallel plate capacitor, they are general and valid for any region of space in which a magnetic or an electric field exists (NCERT page 170); give that in an exam. Strictly, these are the forms for fields in vacuum (or air): repeating Example 6.9 with L = μ_rμ₀n²Al and B = μ_rμ₀nI gives B²/(2μ_rμ₀) inside a linear magnetic material (reasoned from NCERT Eq. (6.16)).

Solved examples

Common mistakes to avoid

Quick revision summary

• In the fixed-geometry linear model, inductance is flux linkage per unit current, NΦ_B/I; a scalar measured in henry, depending on geometry and material.
• Mutual inductance: N₁Φ₁ = MI₂ and ε₁ = −M dI₂/dt; M₁₂ = M₂₁; for coaxial solenoids M = μ₀n₁n₂πr₁²l.
• Self-inductance: NΦ_B = LI and ε = −L dI/dt; for a long solenoid L = μ_rμ₀n²Al.
• The self-induced (back) emf opposes any change in current: L acts as electrical inertia.
• Energy stored: W = ½LI²; vacuum magnetic energy density B²/(2μ₀).

Questions students ask

What is the difference between self-inductance and mutual inductance, and how are they related?

Self-inductance L belongs to one coil and sets the back emf −L dI/dt of its own changing current, while mutual inductance M belongs to a pair of coils and sets the emf −M dI₂/dt that a changing current in one induces in the other; both are in henry (NCERT pages 165–168). Beyond the NCERT text, |M| = k√(L₁L₂), where k from 0 to 1 measures coupling strength. The sign of M depends on the chosen winding/current directions (US Navy, NEETS Module 2, 1998, page 2-16). For NCERT’s coaxial solenoids, Eqs. (6.9) and (6.15) give k = r₁/r₂, so M = √(L₁L₂) only if the inner coil fills the outer.

What is the difference between self-induction and self-inductance?

Self-induction is the phenomenon, an emf induced in a coil by a change in its own current, while self-inductance L is the quantity that measures it, the flux linkage per unit current, NΦ_B = LI, in henry (NCERT page 168). NCERT also calls L the coefficient of self-induction, and in the same way mutual induction is the effect of one coil on another, while mutual inductance M, the coefficient of mutual induction, is its measure (NCERT pages 166 and 168). So ‘a coil’s self-induction is 3 H’ is loose wording: the phenomenon has no unit, and 3 H is the coil’s self-inductance.

What is one henry of inductance?

One henry is the self-inductance of a coil in which a current changing at 1 A s^⁻¹ induces an emf of 1 V, so 1 H = 1 V s A^⁻¹; equivalently, the coil’s flux linkage is 1 Wb for each ampere, 1 H = 1 Wb A^⁻¹ (reasoned from NCERT Eqs. (6.13)–(6.14), page 168; US Department of Energy, DOE Fundamentals Handbook: Electrical Science, 1992, Module 3). Two coils have a mutual inductance of 1 H when a current changing at 1 A s^⁻¹ in one induces 1 V in the other. For example, a 40 mH (0.040 H) coil whose current changes at 45 A s^⁻¹ has a back emf of 0.040 × 45 = 1.8 V.

What is the difference between inductance and resistance?

Resistance opposes the current itself and turns electrical energy into heat at the rate I^²R, while inductance acts only while the current changes, through the back emf L dI/dt, and stores energy ½LI^² that it returns as the current falls (NCERT pages 168–169). So a steady direct current through a coil meets only the resistance of its wire: with dI/dt = 0 there is no back emf, and the inductance only slows the rise and fall of the current at switching (beyond the NCERT text: US Navy, NEETS Module 2, 1998, Chapter 2). With alternating current the coil also has an inductive reactance.

How do inductances combine in series and in parallel?

Coils too far apart to share flux combine like resistors: in series L = L₁ + L₂, since the same changing current gives back emfs that add, and in parallel 1/L = 1/L₁ + 1/L₂, since the same emf acts on both and their rates of change of current add (beyond the NCERT text: US Navy, NEETS Module 2, 1998, pages 2-17 to 2-19; reasoned from NCERT Eq. (6.14)). So 30 mH and 60 mH give 90 mH in series and 20 mH in parallel. Coils in series that share flux through a mutual inductance M give L₁ + L₂ ± 2M, plus when their fields aid and minus when they oppose (reasoned from NCERT page 169).

Is the energy stored in an inductor deleted from NCERT, and is inductance in the JEE and NEET syllabus?

No, the energy stored in an inductor is not deleted: the 2026–27 NCERT Class 12 book derives W = ½LI^² in Section 6.7 (Eq. (6.17), page 169) and the magnetic energy density B^²/(2μ₀) in Example 6.9 (pages 169–170). Inductance is in both exams: Unit 14 of the JEE Main syllabus lists “self and mutual inductance”, and Unit 14 of the NEET (UG) 2026 syllabus lists “Self and mutual inductance.” Neither syllabus names the stored energy separately, but it follows directly from self-inductance, in the same NCERT section.

Practice questions

Try each question first, then open the answer to check your working.

1. Q1. What are the SI unit and the dimensions of inductance?

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   The henry (H); dimensions [M L² T⁻² A⁻²], those of flux divided by current (NCERT page 165).

2. Q2. The current in a coil of inductance 0.20 H rises at 50 A/s. Find the size of the self-induced emf.

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   |ε| = L dI/dt = 0.20 × 50 = 10 V (NCERT Eq. (6.14), page 168).

3. Q3. How much energy is stored in a 0.50 H inductor carrying 4.0 A?

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   W = ½LI² = ½ × 0.50 × 4.0² = 4.0 J (NCERT Eq. (6.17), page 169).