Question
Define mutual inductance between a pair of coils. Derive an expression for the mutual inductance of two long coaxial solenoids of the same length wound one over the other.
✓
Answer
$i$. Mutual inductance is numerically equal to the induced emf in the secondary coil when the current in the primary coil changes by unity.
$ii$. in the question mutual inductance of two long coaxial solenoids of same length $l$ wound one over the other is given by: $-$
Let a current $l_2$ flow in the secondary coil
$\therefore B_2=\frac{\mu_0 N_2 i_2}{l}$
$\therefore$ Flux linked with the primary coil $=\frac{\mu_0 N_2 N_1 A_1 i_2}{l}$
$= M _{12} i _2$
Hence $, M _{12}=\frac{\mu_0 N_2 N_1 A_1}{l}$
$\mu_0 n_2 n_1 A_1 l\left(n_1=\frac{N_1}{l} ; n_2=\frac{N_2}{l}\right)$
$ii$. in the question mutual inductance of two long coaxial solenoids of same length $l$ wound one over the other is given by: $-$
Let a current $l_2$ flow in the secondary coil
$\therefore B_2=\frac{\mu_0 N_2 i_2}{l}$
$\therefore$ Flux linked with the primary coil $=\frac{\mu_0 N_2 N_1 A_1 i_2}{l}$
$= M _{12} i _2$
Hence $, M _{12}=\frac{\mu_0 N_2 N_1 A_1}{l}$
$\mu_0 n_2 n_1 A_1 l\left(n_1=\frac{N_1}{l} ; n_2=\frac{N_2}{l}\right)$
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