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Mutual Inductance

Last updated date: 03rd Dec 2023
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An Introduction

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.


About the Concept of 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.


To find 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


(Image  will be uploaded Soon)


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,


2)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 = \[\frac{M}{\sqrt{L_{1}L_{2}}}\]

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).


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

M =  \[\frac{{\mu_{0}N_{1}N_{2}A}}{l}\]

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

What is Inductance?

In the field of electronics and electromagnetic, inductance is a key notion that describes a conductor's tendency to oppose current flow. A magnetic field is created across the conductor by the current flow. The field strength is determined by the size of the current and changes as the current changes.

Types of Inductance

There are two main types of inductance:

  1. Self Inductance

  2. Mutual Inductance

Here we will study about Mutual Induction

Mutual Inductance

When two coils are brought near together, the magnetic field in one of the coils directs to connect with the other coil, according to the definition of mutual induction. This leads to voltage development in the second coil. A mutual inductance property describes the situation in which one coil influences or alters the voltage and current values in the other coil.

Leakage and stray inductance are two negative consequences of mutual inductance. Through the process of electromagnetic induction, when they are released from one coil, they alter the functionality of another element.

It has a fairly simple theory that may be grasped by employing two or more coils. In the 18th century, an American scientist named Joseph Henry characterized it. One of the qualities of the coil or conductor utilized in the circuit is this. If the current in one coil fluctuates with time, the EMF will induce in another coil, according to the property inductance. The transformer, for example, is a fundamental example of mutual inductance.

The fundamental disadvantage of mutual inductance is that leakage of one coil's inductance might cause the operation of another coil using electromagnetic induction to be disrupted. Electrical screening is essential to reduce leakage.

Mutual Inductance Formula

The formula of two coils is given as

M =  \[\frac{{\mu_{0}N_{1}N_{2}A}}{l}\]

Where μ0= permeability of free space = 4π10-2

μ = permeability of the soft iron core

N1= turns of coil 1

N2= turns of coil 2

A= cross-sectional area in m2

L = length of the coil in meters

The unit of mutual inductance is kg. m2.s-2.A-2

The amount of inductance produces the voltage of one volt due to the rate of change of current of 1 Ampere/second.

Joseph Henry, a scientist from the United States, coined the term to describe the phenomenon of two coils.

FAQs on Mutual Inductance

1.What do you understand by mutual inductance between coils?

The mutual inductance between the two coils can be considerably increased by placing them on a common soft iron core or increasing the number of turns in each coil, as in a transformer. If the two coils are tightly wound one on top of the other over a common soft iron core, unity coupling is said to exist since any flux leakage losses will be minimal. 

2.Give a few examples of mutual  inductance applications?

The following are a few examples of Mutual inductance applications:

  • Electric motors and generators use it.

  • Implemented in the digital signal processing functionality

  • Used in transformers - A transformer is the principal component that operates on the mutual inductance principle and is designed to convert alternating current from one voltage level to another voltage level.

3.Differentiate between mutual induction and self induction

Difference between mutual induction and self induction are:

Mutual induction

  1. The characteristic of a pair of conductors is mutual inductance.

  2. When the main current in the coil decreases, the induced current advanced in the surrounding coil faces the decay of the current in the coil.

  3. When the primary current in the coil grows, the induced current produced in the adjoining coil faces the growth of current in the coil.

Self induction

  1. The conductors themselves have a property called self inductance.

  2. When the main current in the coil decreases, the induced current faces the degradation of current in the coil.

  3. When the primary current in the coil grows, the induced current opposes the development of current in the coil.

4.Mention the factors that affect mutual inductance.

Factors affecting mutual inductance are as follows:-

a.The number of turns per unit length in each coil

b. the coils' area

c. the coils' length

d. the permeability of the medium inside the coils

e. the coils' separation, and 

f. the coils' relative orientation.

NOTE : For More Information Logon To Vedantu App.

5. What are the principles of mutual induction?

The principle of mutual induction states that current flowing through a conductor generates a magnetic field, and a changing magnetic field induces current in another conductor. The magnetic field strength and orientation relative to the second conductor determine the size of the current response in the second conductor. When the second conductor is perpendicular to the field and experiences as much of the field as feasible, this is maximised.The basic functioning principle of a transformer, motors, generators, and any electrical component that interacts with another magnetic field is said to have mutual inductance.

6. 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?

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.

7. 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.

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

8. Is Mutual inductance always positive?

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

9. Write the Dimensional Formula for Mutual Inductance (M).

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