# What is Mutual Induction ?

Mutual Induction : Two coils C1 and C2 are placed very close to each other. A source of emf is connected in the coil C1. When current flows through the coil C1 magnetic field is produced . The coil C2 lies within the magnetic field of C1 and hence magnetic flux is associated with it. So long current changes in the coil C1 its magnetic field changes and hence flux associated with C2 also changes , which induces an emf in the coil . Induction of an emf due to variation of current in the neighbouring coil is known as Mutual Induction .

Flux associated with C2 depends on current in C1 and number of turns in C2.

φ(C2) = M IC1 , where M is the mutual inductance of the given pair of coils.

$\displaystyle \xi = -\frac{d\phi}{dt}$

$\displaystyle \xi_{C_2} = -M \frac{d I_{C_1}}{dt}$

SI unit of mutual inductance is henry.

Example : What is the self inductance of a system of co-axial cables carrying current in opposite directions as shown. Their radii are ‘ a ‘ and ‘ b ‘ respectively.

Solution : The ‘ B ‘ between the space of the cables is

$\displaystyle B = \frac{\mu_0 I}{2 \pi r }$ The Ampere’s law tells that ‘ B ‘ outside the cables is zero, as the net current through the amperian loop would be zero.  Taking an element of length l and thickness ‘ dr ‘ ;  dφ through it is

$\displaystyle d\phi = \frac{\mu_0 I}{2 \pi r} l dr$

$\displaystyle \phi = \frac{\mu_0 I l}{2 \pi } \int_{a}^{b} \frac{1}{r} dr$

$\displaystyle \phi = \frac{\mu_0 I l}{2 \pi } ln\frac{b}{a}$