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Mutual Inductance

Mutual Inductance

Edited By Vishal kumar | Updated on Sep 25, 2024 02:53 PM IST

Mutual inductance is a fundamental concept in electromagnetism, crucial to the operation of transformers, inductors, and many types of electrical circuits. It occurs when the magnetic field created by the current flowing through one coil induces a voltage in a nearby coil. This phenomenon not only forms the basis of many electrical devices but also has practical applications in everyday life. For example, the wireless charging of smartphones and electric toothbrushes relies on mutual inductance to transfer energy from a charging pad to the device without direct electrical contact. In this article, we will understand mutual inductance is essential for both designing efficient electrical systems and appreciating the invisible forces at work in our technologically driven world.

What is Mutual Induction?

Mutual induction is the process by which a change in the electric current in one coil induces an electromotive force (EMF) in a nearby coil through electromagnetic induction. This occurs due to the magnetic field created by the current in the first coil (the primary coil) affecting the second coil (the secondary coil). Whenever the current passing through a coil or circuit changes, the magnetic flux linked with a neighbouring coil or circuit will also change. Hence an emf will be induced in the neighboring coil or circuit. This phenomenon is called ‘mutual induction’. or The phenomenon of producing an induced emf in a coil due to the change in current in the other coil is known as mutual induction.

Coefficient of Mutual Induction

The coefficient of mutual induction (also known as mutual inductance) is a measure of how effectively a change in current in one coil induces a voltage in another coil. It is denoted by the symbol MMM and is defined as the ratio of the induced electromotive force (EMF) in the secondary coil to the rate of change of current in the primary coil. If two coils (P-primary coil or coil 1, S-secondary coil or coil 2) are arranged as shown in the figure below.

If we change the current through coil P then flux passing through Coil S will change.

i.e N2ϕ2αi1N2ϕ2=M21i1=Mi1

where

M21= mutual induction of Coil 2 w.r. t Coil 1
N1= Number of turns in the primary coil
N2= Number of turns in the secondary coil
i1= current through the primary coil or coil 1

Similarly, if we exchange the position of Coil 1 and Coil 2

then

If we change the current through coil S then flux passing through Coil P will change.

i.e N1ϕ1αi2N1ϕ1=M12i2=Mi2

where

M12= mutual induction of Coil 1 w.r. t Coil 2N1= Number of turns in the primary coil N2= Number of turns in the secondary coil i2= current through the Coil 2 or Coil S

As N2ϕ2=Mi1

If i1=1amp,N2=1 then, M=ϕ2

I.e coefficient of mutual induction of two coils is numerically equal to the magnetic flux linked with one coil when unit current flows through the neighbouring coil.

Using Faraday's Second Law of Induction emf we get

ε2=N2dϕ2dt=Mdi1dt

If di1dt=1ampsec and N2=1 then |ε2|=M

I.e The coefficient of mutual induction of two coils is numerically equal to the emf induced in one coil when the rate of change of current through the other coil is unity.

Units and dimensional formula of ‘M’

S.I. Unit - Henry (H)

And 1H=1VsecAmp

Its dimensional formula is ML2T2A2

Dependence of Mutual Inductance

  • Number of turns (N1, N2) of both coils
  • The coefficient of self inductances (L1, L2) of both the coilsand the relation between them is given asM=KL1L2
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where K = coefficient of coupling.

If L=0 then M = 0

If K = 0 i.e case of No coupling then M = 0.

  • Distance(d) between two coils (i.e As d increases then M decreases)
  • The magnetic permeability of the medium between the coils (μr)

Consider two long co-axial solenoids of the same length l. Let A1 and A2 be the area of the cross-section of the solenoids with A1 being greater than A2 as shown in the figure below.

The turn density of these solenoids are n1 and n2 respectively are given as n1=N1l and n2=N2l

Let i1 be the current flowing through solenoid 1, then the magnetic field produced inside it is given as

B1=μon1i1

As the field lines of B1 are passing through the area A2

So the magnetic flux linked with each turn of solenoid 2 due to solenoid 1 and is given by

Φ21=A2B¯1dA=B1A2=(μ0n1i1)A2

The total flux linkage of solenoid 2 with total turns N2 is

(ϕ21)total =N2Φ21=(n2l)(μ0n1i1)A2(ϕ21)total =N2Φ21=(μ0n1n2A2l)i1

And Using (ϕ21)total =N2Φ21=M21i1 we get

M21=μ0n1n2A2l

Where M21 is the mutual inductance of the solenoid 2 with respect to solenoid 1.

Similarly, M12 =mutual inductance of solenoid 1 with respect to solenoid 2 is given as

M12=μ0n1n2A2l

Hence M21=M12=M

So, In general, the mutual inductance between two long co-axial solenoids is given by

M=μ0n1n2A2l

If a dielectric medium of permeability \mu is present inside the solenoids, then

M=μn1n2 A2l or M=μ0μrn1n2 A2l

Consider two circular coils one of radius 'r1' and the other of radius' r2'placed coaxially with their centres coinciding as shown in the below figure.

Since r1≫≫>r2 so we can assume coil 2 is at the center of coil 1.

If Suppose a current i1 flows through the outer circular coil. Then Magnetic field at the center of coil 1 is given as

B1=μ0N1i12r1

So the total flux passing through coil 2 will be given as

(ϕ2)total =N2B1A2=μ0N1N2i1A22r1

And using (ϕ2)total =Mi1

we get M=μ0N1N2A22r1=μ0N1N2(πr22)2r1

Where M=mutual inductance between two concentric coils.

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Solved Examples Based on Mutual Inductance

Example 1: Two coils 'P' and 'Q' are separated by some distance. When a current of 3 A flows through coil 'P', a magnetic flux of 103 Wb passes through 'Q'. No current is passed through 'Q'.When no current passes through 'P' and a current of 2A passes through 'Q', the flux through 'P' is :

1) 6.67×104 Wb
2) 3.67×103 Wb
3) 6.67×103 Wb
4) 3.67×104 Wb

Solution:

As

ϕ=Mi

As given in the question

When a current of 3 A flows through coil 'P', a magnetic flux of 103 Wb passes through 'Q'

So

103=M(3)(1)

Now let the flux through 'P' is ϕ when a current of 2A passes through 'Q

So

ϕ=M(2)(2)

from equation (1) and (2)

103ϕ=32ϕ=23×103=6.67×104 Wb

Hence, the answer is the option (1).

Example 2:

A time-varying current I(t)=2coswt is flowing in the primary coil of 200 turns with a frequency of 40 Hz. The coefficient of mutual induction is 10 m H. Find the emf induced (max) (in mV) is a secondary coil of 400 turns

1) 50.24

2) 12.56

3) 50.44

4) 12.56

Solution:

ε2=N2dϕ2dt=MdI1dt Induced emf. ε=μdI1dtε=μddt(2coswt)ε=μ2wsinwt

For Max emf |sinwt|=1
εmax=2Mwεmax=2M×2πfεmax=2×10×103×2×3.14×40εmax=50.24mV

Hence, the answer is the option (1).

Example 3: Find the mutual inductance in the arrangement, when a small circular loop of wire of radius 'R ′ is placed inside a large square loop of wire of side (L >> R). The loops are coplanar and their centres coincide :

1) M=2μoR2L
2) M=22μoRL2
3) M=2μoRL2
4) M=22μoR2L

Solution:

ϕ=MIϕ2=MI1 B1 A2=MI1M=B1 A2I1
B1 magnetic field due to square frame A2 Area of circle I1 current in square frame.
B1B1=4BAB=4[μ0I12π12[sin45+sin45]]B1=2μ0I1πL(12+12)=22μ0I1πLM=B1A2ILM=(22μ0I1πL)×πR2I1=22μ0R2 L

Hence, the answer is the option (4).

Example 4: Two concentric circular coils with radii 1 cm and 1000 cm, and number of turns 10 and 200 respectively are placed coaxially with centers coinciding. The mutual inductance of this arrangement will be _______ × 10-8 H. (Take , \pi^{2} = 10)

1) 4

2) 5

3) 6

4) 7

Solution:

Given
a=1000 cm b=1 cm or ba
we will take a larger coil as the primary
B=μ0ipN2a flux ϕs=BA=μ0ipN2a×πb2×n Mutual inductance M=ϕsip

Mutual inductance M
M=μ0Nnπb22×a or M=4π×107×200×10×π×1×1042×1000×102=4π2×109 or M=4×108 (usin π2=10 )

Hence, the answer is the option (1).

Example 5: The mutual inductance of a pair of coils is 2 H. If the current in one of the coils changes from 10 A to zero in 0.1 s, the emf induced in the other coil is

1) 2 V

2) 20 V

3) 0.2 V

4) 200 V

Solution:

The induced emf in the other coil (coil 2 ) is
e2=Mdi1dt=MΔi1Δt=M(i2i1)Δt=2(010)0.1=200 V

Hence, the answer is the option (4).

Summary

Mutual inductance is the process where a changing current in one coil induces an electromotive force (EMF) in a nearby coil, foundational to devices like transformers and wireless chargers. The coefficient of mutual induction, denoted as M, quantifies this effect, influenced by factors such as coil turns, distance, and magnetic permeability. Various practical examples and problems illustrate the principles and calculations of mutual inductance in different configurations.

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