Mutual Inductance of Coaxial Solenoids - JEE Important Topic

It is the property by which the magnetic field of one coil leads to the voltage generation in a coil placed nearby. Sometimes, even an already existing voltage or current might be changed through the introduction of a new coil through mutual inductance. While mutual inductance can be observed between two circular coils, two concentric coplanar loops, and two solenoids. In this section, we will discuss the mutual inductance of coaxial solenoids.

Coaxial Solenoids

A solenoid is used to transform electrical energy into mechanical energy. Solenoids consist of coils with many windings and hence produce a magnetic flux when an electric current passes through them. Two coaxial solenoids are the ones having a common axis.

The following are the most important characteristics of a solenoid:

• They are made up of many windings of a tightly wound copper wire. 

• The solenoids’ coil is strongly protected by an outer housing made up of steel or iron.

• The plunger is attached to the stop through the concentration of the magnetic field providing the mechanical force to do the work.

Derivation of Formula of Mutual Inductance Between Two Coaxial Solenoids

In order to derive the mutual inductance of two coil formulas, let us consider two solenoids S₁ and S₂ of equal length. Let r₁ and r₂ be the radius of the solenoids S₁ and S₂, respectively. And let N₁ and N₂ be the number of turns on the solenoids S₁ and S₂, respectively.

The magnetic induction caused by the current I₂ in the solenoid S₂ is given by:

${B_2}={\mu_0}{n_2}{I_2}={\mu_0} \dfrac {N_2}{l} {I_2}$

The corresponding flux in solenoid S₁  is given by:

${N_1}{\phi_{12}}={N_1}{B_2}{A_1}$

${N_1}{\phi_{12}}={N_1}{\left[{\mu_0} \dfrac {N_2}{l} {I_2}\right]}{\pi~{r_1}^2}$.......(1)

Also,

${N_1}{\phi_{12}}={M_{12}}{I_2}$.....(2)

Now, comparing (1) and (2), we get,

${M_{12}}=\dfrac {{\mu_0}{N_1}{N_2}{\pi}{r_1}^2}{l}$

If the same method is followed to find M₂₁ considering the magnetic induction is produced in the solenoid S₂ due to current I₁, we can see that

M₂₁ = M₁₂ = M

This formula is known as the mutual inductance formula and used for finding the mutual inductance of one coil due to current passing through another coil in coaxial solenoids. This formula is applicable for the case of mutual inductance of two long coaxial solenoids or mutual inductance of two coaxial solenoids The unit of mutual inductance is $kg~{m^2}{s^{-2}}{A^{-2}}$. 

Steps to Find the Mutual Inductance

• Assume that current flows through one of the coils.

• Derive the equation or expression for the magnetic field induced in the neighbouring coil due to the current flowing in the assumed coil.

• Obtain the flux linkage. 

• Like the expression derivation, compare the flux linkages and find the mutual inductance M.

Applications of Mutual Inductance

The most common electronic applications of mutual inductance are transformers, generators, and motors. Let us discuss in detail how exactly mutual inductance is used in a few of these applications.

• Transformers

A transformer consists of two coils that are separate with different numbers of turns wound around a laminated iron core (transformer core). The primary winding has a connection to an energy source which allows primary current into the setup. The secondary winding is connected to the incoming load through mutual inductance to produce the desired secondary current. The secondary current and secondary voltage are controlled by the number of windings and a few other factors.

The transformer can be a step-up or step-down transformer. Accordingly, the primary circuit can be of high or low voltage. With the help of mutual inductance, transformers are employed to transform the alternating current produced at the source into a high voltage system for high efficiency during the distribution of electricity.

• Generators

The generators are another important application of mutual inductance property. The emf in generators are generated and their direction is governed by Lenz law. According to Lenz law, the magnetic field must be directed to oppose the flux change due to the approaching magnetic field. The induced current will or must be directed in such a way that the above condition gets satisfied.

Coupling Coefficient

Theoretically, magnetic flux linkage between two coils is taken as 100%. But in practical scenarios, some or other sort of leakage will reduce this efficiency. The coupling coefficient is the fraction of loss in flux linkage to the total achievable flux linkage between two coils. It is denoted by the letter k.

The fractional value is expressed as numbers rather than percentages and lies between 0 and 1. Here, 0 and 1 indicate zero inductive coupling and maximum inductive coupling, respectively.

• If k > 0.5, the coils are tightly coupled. 

• If k < 0.5, the coils are loosely coupled. 

The expression for the coupling coefficient k is given as

$k=\dfrac {M}{\sqrt {{L_1}{L_2}}}$

where

• L₁ - The self-inductance of the coil 1

• L₂ - The self-inductance of the coil 2

Summary

When two coils are placed near each other and current passes through one coil, in addition to self-inductance, there is mutual inductance also, i.e., the current passing through one coil generates a magnetic flux in the second coil. While this happens, the efficiency might not be 100% and the fraction of actual flux linkage and maximum attainable flux linkage is called the coupling coefficient. Mutual inductance plays a vital role in various electric applications such as transformers, motors, and generators.