Relationship Between Self Inductance And Mutual Inductance

Inductance is the property of an electric conductor that causes an electromotive force to be generated due to a change in the current flow. A better understanding of electromagnetism and the electric and magnetic fields can be achieved by studying inductance (self-inductance and mutual inductance).

Inductance and its SI Unit

Inductance is the phenomenon of the electric conductor opposing any change in electric current that flows through it by developing an electromotive force. The SI unit of inductance is Henry (H). 1 Henry is the inductance needed to produce an electromotive force of 1 V in the conductor. The rate of flow of current through that conductor is one Ampere per second. 

The current that flows through the conductor creates a magnetic field around that conductor. The developed magnetic field’s strength is entirely based on the current’s magnitude flowing through the conductor. 

Faraday’s law of induction states that the interaction of electric and magnetic fields creates an electromotive force that opposes any change in the current flowing. Accordingly, induction can also be expressed as the voltage ratio, which is used to determine the rate of current change that causes it. 

Self-inductance and Mutual Inductance

Self Inductance

The induction of electromotive force due to changes in the flowing current or the coil or the conductor’s magnetic flux is defined as self-inductance. The magnetic flux at any point inside the circuit is directly proportional to the current flowing through it. Self-inductance is denoted by L (self-inductance coefficient). The self-inductance of a conductor depends on the cross-section area, the total number of turns in the coil, and the coil material’s permeability. The following is the relation:

∅=LI

The rate of change of the magnetic flux can be stated as:

e = – (d∅)/dt

e = -L

Therefore:

-L = dI/dt

Here, ∅ stands for magnetic flux.

L is self-induced.

I stand for the current in amperes.

Mutual inductance

When the arrangement of coils is such that the current flowing through one coil changes, an emf is induced in the other coil. These coils are said to have a mutual inductance between them. Mutual inductance is denoted by M and is also measured in Henry. 

The setup is created such that there are two coils, primary and secondary. A key and a battery are connected to the primary coil, and a galvanometer is attached to the secondary. When the magnetic flux of the two coils changes, an electromotive force, opposing in nature, is produced all across the coils. This is called mutual inductance. The following is the relation:

∅=MI

Here, M is the mutual inductance coefficient, and I is the change in current.

e = -d∅/dt

e = -M

Therefore: 

-M = dI/dt

M stands for the mutual inductance coefficient.

I denote current in Ampere.

The formula for mutual inductance is used in various calculations and derivations of several quantities related to it. The formula is:

M = μ₀μ_rN₁N₂A / L 

μ₀ is the free space’s permeability.

μ_r is the relative permeability of the used iron core.

A stands for the cross-section area in m².

L is the coil’s length in m, and N stands for the total number of turns in the coil.

Relationship between self inductance and mutual inductance :

L₁= ∅₁/ l

M = ∅2/ I

L₂ = (∅₂)/ I

∅₁ = B₁A

∅₁ = (μ₀N₁I / L) x N₁A

∅₁  = (μ₀(N₁ )²IA / L)

So, L₁ = μ₀(N₁)² A / l

Similarly, L₂ = μ₀(N₂)²A/ l

∅₂ = B₁ . A

∅₂ = μ₀N₁IN₂A/L

Considering the values of L₁ and L₂

√L₁L₂ = μ0 N₂ N₁ A/ l

Therefore, √L₁L₂ = M.

Conclusion

Self-inductance and mutual inductance are both measured in Henry. They have a relation of √L₁L₂ = M. The relationship between these two can be used to understand the phenomenon’s behaviour and how the various quantities involved play their respective roles. This is highly helpful across multiple induction cookers, motors, and more. 

The characteristics of self-inductance and mutual inductance are not similar, but play essential roles in their way. There is one significant difference between the characteristics of self-inductance and mutual inductance, and that is the involvement of a single coil in the former and two coils in the latter.