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Determination of Self- Induction by Raleigh’s method

The coil, whose self-inductance L is to be connected in the fourth arm of a Wheat stone’s bridge plug key K3 is connected across r so that it may be short-circuited. P, Q and R are non-inductive resistances.

(a)    Initially, K3 is kept closed. The ohmic resistance S of the inductance coil alone is included in the fourth arm. P is made equal to Q. Then R is adjusted for no deflection in the B.G., by first pressing battery key K1 and then galvanometer key K2. Under this condition, no current flows through the galvanometer.

(b)    If now the galvanometer key2 is closed first and then the battery key K1, then a throw θ1 is observed in the galvanometer. This throw arises due to an extra emf L di/dt induced in the coil while the current is growing. If G is galvanometer resistance, then current through it due to induced emf is

i’ = kL /G di /dt
Where k is constant which depends upon the relative resistance in the circuit. 

Hence the total charge passing through the galvanometer, as the current in the coil grows from zero to a steady maximum value i0 is given by

    q = ∫i0 I dt = kL /G ∫i0  di/ dt dt =  kL /G i0    ______________________________… (1)

If θ1 be the first throw of the galvanometer, then,
    Q = K θ1 (1 + λ /2)  _________________________________________________ … (2)
. :     kL/G i0 = K θ1 [1 + λ /2]  ____________________________________  (from 1 and 2)
    = T/2π C /nBA. θ1 [1 + λ /2]

(c)    To eliminate k and i0, the key K3 is opened and the resistance r is included in the arm CD. As r is small, it does not affect the current i0 in the arm CD appreciably. But it will introduce an additional emf ri0 in the arm CD: This causes a steady current (Kr/G)i0 through the galvanometer. K1 is closed first and then K2. The steady deflection Φ in the galvanometer is noted. Then,
(kr /G) i0 = C /nBA Φ.

Here, C /nAB is the current reduction factor of the galvanometer.

Dividing (3) by (4),
        L /r = T /2π θ1 / Φ [1 + λ/2]
Or         L = rT /2π θ1 / Φ [1 + λ/2]

self induction

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