Coupling of inductors - Practice problems, FAQs

Resistors are electrical components that are not capable of storing electrical energy when current is passed through them. Joule observed that the heat dissipated in a resistor varies as the square of the current which passes through it. On the other hand, inductors store energy in the form of magnetic fields. When a time-varying current is passed through the inductor, then a magnetic flux is produced in and around it, which causes an emf to be developed between its ends. Now imagine a situation where a change in flux in one coil can cause emf to be induced in the other. Such a phenomenon is called mutual induction–commonly used in transformers. In this article, we will explore such coupling of inductors in detail.

Can you imagine the situation where the electrical energy is stored in a conducting coil? Or there is a transmission of energy without any medium ? Just the mutual inductance which does the magic!

Table of contents

Coupling of inductors
Different cases of coupling
Practice problems
FAQs

Coupling of Inductors

When two inductor coils are kept in such a way that the current flowing in the first coil causes flux change in the other coil, such an arrangement is called coupling of inductors. Coupling is due to a phenomenon called mutual induction.

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$ϕ_{21} \propto i_{1};$ where i₁ is the current flowing in the first coil(also known as primary coil) and $ϕ_{21}$ denotes flux change in the second coil (also known as secondary coil) due to i₁.Then,

$ϕ_{21} = M i_{1} .$

Similarly, the flux linked with the first coil $ϕ_{21}$ due to current i₁ flowing in the second coil would be,

Here M is mutual inductance.

Self inductance is the property by which a coil opposes growth of current in it. Its unit is Henry(H). Its dimensional formula is

Let L₁ indicate the self inductance of the first coil, then

The self inductance of the second coil,

Where and are the number of turns in the coils and is the area of the coil. is the permeability of free space.

The field B₁ due to current in coil 1 can be written as,

Magnetic field due to current in coil can be written as,

Substituting the value of in equation we get,

But indicates the flux linked with coil 1.

Comparing equations (iii) and (iv), we get

Now ,

indicates the relation between self induction and mutual induction.
is called the coefficient of coupling or simply the coupling coefficient.

For iron or ferrite materials, the value of is and for air cored solenoids, the value ranges between to

When it means that the coils are perfectly coupled.
On the other hand, if K < 0.5 means that the coils are loosely coupled.

Different cases of coupling

(i) When the coils are placed as shown in arrangement (a), there is maximum flux linkage. Hence, the induced emf would be maximum in this case.

(ii) When the coils are placed as shown in arrangement (b), there is zero flux linkage since they are perpendicular. Hence, the induced emf would be zero(minimum) in this case.

(iii) When the coils are placed as shown in arrangement (c), zero flux is linked between the two coils, since they are perpendicular. Hence, zero emf is induced in the other coil.

Practice problems

Q.If two coils having inductances L₁= 2 mH and L₂= 8 mH are placed coaxially over one another. The coefficient of mutual induction between the coils would be

(a) 2 mH
(b) 4 mH
(c) 6 mH
(d) 8 mH

A.b
Given,

L₁= 2mH ; L₂= 8mH

So, the coefficient of mutual inductance, $M = \sqrt{2 \times 8} = \sqrt{16} = 4 m H$ .

Q. Two solenoids are placed coaxially with each other. The time varying current flowing in one of the coils is 0.1 A/s. If the coefficient of mutual induction between the coils is equal to 10 H, the emf induced in the second coil would be

(a) 1 V
(b) 2 V
(c) 2.5 V
(d) 1.5 V

A.a
M = 10 H.

Given, $\frac{d i}{d t} = 0.1 A / s$ .

emf induced in the second coil, $ε_{2} = M \frac{d i}{d t} = 10 \times 0.1 = 1 V .$

Q. Two coils have 100 and 200 turns respectively. The coefficient of mutual induction between the coils, if they have both an area of and a length of 1 m would be
(a)8 mH (b)2 mH (c) 4 mH (d)1 mH

A.a

Given,

Q. The coefficient of mutual induction between two coils is 8 mH. When an iron core having relative permeability μ_{_r}= 2000 is inserted between the axis of the coils, the mutual inductance now becomes
(a)1 H (b)4 H (c)16 H (d)2 H

A.c
Given, μ_{_r}= 2000; M = 8 mH.

Mutual inductance, ∝

FAQs

Q. On what factors does mutual inductance depend?
A. It depends upon: the number of turns, the area of the coils, the length of the coils and relative permeability of the core used.

Q. Can mutual inductance be negative?
A. The mutual inductance can be negative or positive depending upon the polarity of the induced voltage between the terminals of the coil.

Q. Where is mutual induction used?
A. It is used in transformers, generators and motors.

Q. How many coils are used in mutual induction?
A. Two coils are used in mutual induction. It happens when the flux of one is also linked to the other.