Question

(a) Show that the energy stored in an inductor i.e., the energy required to build current in the circuit from zero to I is $\frac{1}{2} {LI}^{2},$ where L is the self-inductance of the circuit.
(b) Extend this result to a pair of coils of self-inductances L₁ and L₂ and mutual inductance M. Hence obtain the inequality M² < L₁ L₂.

Solution

(a)
Energy spent by the source to increase current from i to i + di in time dt in an inductor is,

= L \frac{di}{dt} \times i \times dt = Li di

Energy required to increase current from 0 to I

E = \int_{0}^{I} Li di = L {[\frac{i^{2}}{2}]}_{0}^{I} E = L [\frac{I^{2}}{2} - 0] = \frac{1}{2} {LI}^{2}

which, is the energy stored in a conductor.

(b) The energy stored in the two inductors is independent of the manner of building up current in the coils.

Let I₂ = 0 initially and also suppose that the current be built up from 0 to I₁ in coil 1.
The required energy is

E = \frac{1}{2} L_{1} I_{1}^{2} .

...(I)

Let us now build up current in coil 2. Let the current increase from i₂ to i₂ + di₂ in time dt.

Work done for coil 2 , W₂

= i_{2} L_{2} \frac{{di}_{2}}{dt} dt

But this change in i₂ causes change in flux in 1 and induces emf in coil 1 given by,

M \frac{{di}_{2}}{dt}

Work done in time dt to maintain current I_1,W₁

= I_{1} M \frac{{di}_{2}}{dt} dt

Total work done in increasing current from 0 to I₂ in coil 2 and for maintaining current I₁in coil 1 is,

L_{2} \int_{0}^{I_{2}} i_{2} {di}_{2} + {MI}_{1} \int_{0}^{I_{2}} {di}_{2}

Therefore,
Energy,

E_{2} = \frac{1}{2} L_{2} I_{2}^{2} + {MI}_{1} I_{2}

...(II)

Total energy stored in a pair of coupled coils

E = \frac{1}{2} L_{1} I_{1}^{2} + \frac{1}{2} L_{2} {I_{2}}^{2} + {MI}_{1} I_{2}

from (I, II)

= \frac{1}{2} L_{1} (I_{1}^{2} + \frac{2 M}{L_{1}} I_{1} L_{2} + \frac{M^{2}}{L_{1}^{2}} + I_{2}^{2}) + \frac{1}{2} L_{2} I_{2}^{2} - \frac{1}{2} \frac{M^{2}}{L_{1}} I_{1}^{2} = \frac{1}{2} L_{1} {(I_{1} + \frac{M}{L_{1}} I_{2})}^{2} + \frac{1}{2} (L_{2} - \frac{M^{2}}{L_{1}}) I_{2}^{2}

In order to that, the energy be non-negative for all values of I₁ and I₂ , a necessary and sufficient condition is that

L_{2} > \frac{M^{2}}{L_{1}}

i.e,

M^{2} < L_{1} L_{2} .

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