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.
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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.
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
where
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
where
As
If
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
If
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.
S.I. Unit - Henry (H)
And
Its dimensional formula is
where K = coefficient of coupling.
If L=0 then M = 0
If K = 0 i.e case of No coupling then M = 0.
Consider two long co-axial solenoids of the same length
The turn density of these solenoids are n1 and n2 respectively are given as
Let i1 be the current flowing through solenoid 1, then the magnetic field produced inside it is given as
As the field lines of
So the magnetic flux linked with each turn of solenoid 2 due to solenoid 1 and is given by
The total flux linkage of solenoid 2 with total turns N2 is
And Using
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
Hence
So, In general, the mutual inductance between two long co-axial solenoids is given by
If a dielectric medium of permeability is present inside the solenoids, then
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
If Suppose a current
So the total flux passing through coil 2 will be given as
And using
we get
Where M=mutual inductance between two concentric coils.
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
1)
2)
3)
4)
Solution:
As
As given in the question
When a current of 3 A flows through coil 'P', a magnetic flux of
So
Now let the flux through 'P' is
So
from equation (1) and (2)
Hence, the answer is the option (1).
Example 2:
A time-varying current
1) 50.24
2) 12.56
3) 50.44
4) 12.56
Solution:
For Max emf
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)
2)
3)
4)
Solution:
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 , = 10)
1) 4
2) 5
3) 6
4) 7
Solution:
Given
we will take a larger coil as the primary
Mutual inductance
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
Hence, the answer is the option (4).
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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