# Mutual Inductance

### What is Mutual Inductance?

We know that electric currents can be induced in closed coils when subjected to varying magnetic fields. This phenomenon of inducing current or emf in a coil by changing magnetic fields is called the electromagnetic induction or EMI.

We also know that if a current flows through any coil, whether the current is increasing, or decreasing, the coil opposes the change in the current’s strength passing through it. This means supplying varying current is necessary.

So, if we use two coils in place of one, what type of phenomenon will occur here?

Well, mutual inductance takes place between these two coils.

### Mutual Inductance

To understand this concept, let us take two coils P and S (Distinct coils) and keep them side-by-side.

We connect one coil to the switch, and the other to a galvanometer.

As soon as a varying current is generated in coil P, automatically current induces in the coil S.

P coil is known as the primary coil, and the S coil in which we see the deflection is the secondary coil.

So What happens Next?

Well, the varying current in the P coil generates varying magnetic field lines that pass through both the coils.

This means increasing the current; the magnetic field lines increase because of which the flux at the secondary coil increases.

When this flux increases, an induced EMF is generated in the coil because of which an induced current starts flowing in it.

Therefore, the galvanometer shows a deflection.

For finding the direction of the magnetic field lines, we curl our fingers of our right hand around the wire, the direction in which the thumb points, is the direction of the magnetic field.

We can see that the magnetic field lines are in the direction of the current.

If these lines change (because of the changing current), the flux in the secondary coil changes because of which an induced emf and the induced current generates in it.

### Mutual Inductance Derivation

We know that on increasing the current in the primary coil, the flux in the secondary coil increases.

I.e.,   (ф2)T  α I

We are not sure of the number of turns in the S coil. So, to calculate the total flux,  we have taken the subscript T in (ф2)T.

On removing the sign of proportionality constant, we get,

(ф2)T  = MI

Where M is the constant of proportionality and is called the coefficient of mutual induction or mutual inductance of two coils.

The unit of mutual inductance is:

M =  (ф2)T  /I = $\frac{Weber}{Ampere}$ = $\frac{Volt-sec}{Ampere}$ = Henry

∴ The unit of M is Henry.

If I = 1,  (ф2)T  = M x 1  M = (ф2)T

Thus the coefficient of mutual inductance of two coils is equal to the amount of flux that generates in one coil because of the current flow in the primary coil.

M doesn’t depend on (ф2)T, and I because it is a constant term.

However, it depends upon the following factors:

1. Geometry (shape) of the coils,

2. Their separation (or the radius of the coils),

3. The orientation (coils kept parallel or inclined at some angle), and

4. The medium in which we keep these coils.

We know that an EMF is induced in the secondary coil. Now we will apply Faraday’s law here:

e2  = - d(ф2)T/dt = - d(MI1)/dt

(Its because the flux of S coil, i.e., (ф2)T depends on the current (I1) in the P coil)

e2 = - M dI1/dt

If dI1/dt = 1, then M = -  e2

EMF in the secondary coil generates only when there is a change in the current I1.

∴The coefficient of mutual inductance of two coils is equal to the induced emf in the S coil when the rate of change of current in the P coil is unity.

### Important Formulas in Mutual Induction

1. Coefficient of Coupling (K)

The coefficient of coupling of two coils is a measure of the coupling between the two cells. It is given by K = $\frac{M}{\sqrt{L1L2}}$

Where L1 and L2  are coefficients of the self-inductance of the two coils.

The value of K is always < 1.

If two coils are arranged in series, then their K = 1, then we can show that

L = L1 + L2 + 2M (When current in two coils is in the same direction), and

L = L1 + L2 - 2M (When current in two coils in the opposite directions).

1. Mutual Inductance of Two Long Coaxial  Solenoids (S1 and S2)

M =  $\frac{\mu_{0}N1N2A}{l}$

Where μ0 = Magnetic constant,

N1 and N2 = Total number of turns in a solenoid S1, and S2, respectively,

l = Length of the longer solenoid, and

A = πr2= Cross-sectional area of the inner solenoid.

### Application of Mutual Inductance

Mutual inductance is the basic operating principle for the following:

1. Transformers

2. Motors

3. Generators

Q1: A Current of 10 A Flowing in the Primary of a Circuit Reduces to Zero at a Rate of 10-2s.If M is 5 H, What is the Induced EMF in the Secondary?

Ans: Here, I = 10 A, I = 0, dt = 10-2s, M = 5 H

Putting these value in this equation:

e = - M dI/dt

= - 5 * ( 0 - 10)/10-2

= 50 x 100 = 5000 or 5 x 103V is the emf in the secondary.

Q2: The Magnetic Flux through a Cell Perpendicular to its Plane and Directed into Paper is Varying According to the Relation, ф = (4t2+ 9t + 5)miWb. Calculate the EMF Produced in the Loop at t = 4s.

Ans: We know that e = |- dф/dt| = dф/dt

We are given, ф = (4t2+ 9t + 5)  x 10-3(1 miWb = 10-3Wb)

Now, differentiating (4t2+ 9t + 5)w.r.t. time,

= d (4t2+ 9t + 5)/dt = (8t + 9)  x 10-3

On putting t = 4 sec, we get,

((8 x 4) + 9)10-3  = - 0.041 Volt

Q3: Is Mutual Inductance Always Positive?

Ans: No. It can either be positive or negative depending on the polarity of the mutual voltage about the direction of the inducing current.

Q4: Write the Dimensional Formula for Mutual Inductance (M).

Ans: The dimensional formula of the mutual inductance is [M1L2T-2A-2]