Question:
a conducting loop of radius r is placed concenteric with another loop of a much larger radius R so that both loop are coplanar find the mutual inductance of the system of the two loop take R>>r
Answer:
Let the current I pass through the outer coil of radius R. Due to this the magnetic field at the centre of the coil will be B = µ0 I / 2R
As it is given that R >> r, we can consider the magnetic field B to be constant over the entire cross-sectional area of the inner coil of radius r. Hence, flux associated with the small coil will be Ф1 = BA1 = B ∏ r2
Ф1 = (µ0 I / 2R) * ∏ r2
By the definition of mutual inductance Ф1 = M12 I2 = M12 I
Hence, M21 = M = M12 = Ф1 / I
= (µ0 I / 2R) * ∏ r2 * (1/I)
= µ0 ∏ r2 / 2R
The value of M shows that it is independent of the value of current and depends on radius of the two coils only.
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