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INDUCTANCE, An electric current can be induced in a coil by flux, change produced by another coil in its vicinity or, flux change produced by the same coil. These, two situations are described separately. In both, the cases, the flux through a coil is proportional to, the current., ΦB α I, For a closely wound coil of N turns, NΦB ∝ I, The constant of proportionality, in this relation, is called, inductance., The two types of inductance are, 1) Mutual inductance 2) Self inductance
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Mutual induction, When current through a coil changes an e.m.f. is induced, in the neighbouring coil. This is called mutual induction., For the inner solenoid S 1, Radius – r1., Number of turns per unit length- n1., Total number of turns – N1, For the outer solenoid S 2, Radius – r 2, Number of turns per unit length- n 2, Total number of turns-N 2
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Relation connecting induced emf and mutual inductance, , We know that N1 ϕ1 = MI2, For currents varying with time, From Faraday’s law, , or, , The magnitude of the induced emf depends, upon the rate of change of current and mutual, inductance of the two coils.
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Self-inductance, When the current through a coil changes, an, e.m.f. is induced in it. This is called self induction., In this case, flux linkage through a coil of N turns is proportional to the current through the coil, and is expressed as NΦB ∝ I or N ΦB = L I, where constant of proportionality L is called self-inductance of the coil. It is also called the, coefficient of self-induction of the coil., When the current is varied, the flux linked with the coil also changes and an emf is induced, in the coil. Using Equation, the induced emf is given by, Thus, the self-induced emf always opposes any, change (increase or decrease) of current in the coil.
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Equation of self inductance, The self-inductance of a long solenoid of cross-sectional area A and length ℓ, having n turns per, unit length. The magnetic field due to a current I flowing in the solenoid is, B = μ0nI., The total flux linked with the solenoid is, , NΦ B = ( nℓ )( μ0nI )( A )= μ0n2 Aℓ I----------(1), We know that NΦB = L I ------------(2), Comparing these equation L = μ0n2 Aℓ, If we fill the inside of the solenoid with a material of relative permeability μr, , L = μr μ0 n2 Aℓ, The SI unit of Self inductance is henry and is denoted by H, The self-inductance of the coil depends on its geometry and on the permeability of the medium., The self-induced emf is also called the back emf as it opposes any change in the current in a, circuit. Physically, the self-inductance plays the role of inertia. It is the electromagnetic analogue, of mass in mechanics.
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Energy stored in an inductor, The work to be done against the back emf in an inductor, is stored as magnetic potential energy. For the current I, at an instant in a circuit, the rate of work done is, Total amount of work done in establishing the, current I is, , This expression reminds us of mv2 /2 for the (mechanical) kinetic energy of a particle of, mass m, and shows that L is analogous to m (i.e., L is electrical inertia and opposes growth, and decay of current in the circuit)
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Combined effect of self inductance and, mutual inductance of coil, When currents are flowing simultaneously, through two coils we get, , Where L1 – self inductance of the first coil, M12 – mutual inductance, of the first coil with respect to the second coil.
