Self and Mutual Induction

When an electric current passes through a coil, it induces flux change either in itself or in another coil in its vicinity. Self-induction occurs if the flux change is produced within the same coil due to the current, whereas mutual inductance occurs if the flux changes are produced in another coil. This process is called inductance. There are two types of inductances. 

1.  Self-inductance

2.  Mutual inductance

Self-inductance is represented as L, whereas mutual inductance is represented as M.

Inductance depends solely on the geometric orientation of the coil and intrinsic material properties of the coil. This is similar to capacitance in which a parallel plate capacitor depends on the plate area and plate separation with the dielectric constant K of the intervening medium.

Scalar quantity and its SI unit is Henry, denoted by H. It is named after a physicist Joseph Henry who discovered electromagnetic induction.

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Self induction

Self-inductance may be referred to as the resistance of the current-carrying coil against the change of current running through it. This resistance or the opposition is generated in the coil as electromotive force or EMF. In other words, the work of self-induced EMF or electromotive force is to maintain a constant flow of current through the coil. It resists the rise of the current and the fall of the current. The SI unit of self-inductance is Henry. This self inductance property of the coil is always found for the change of the current, which is the alternating current. The resistant property does not exist in direct current. “L” represents inductance. 

Some factors responsible for self-inductance are cross-sectional area, the number of turns of the wire the coil consists of, and the permeability of the material used in the core. The formula for self-inductance is 

e = -L (di/dt)

Here L is the inductance of the coil, i is the current passing in the coil and dt is the small time period.

Mutual Inductance

Inductance is represented as the proportionality constant in the equation representing the relation between magnetic flux produced and the current flowing through the coil.

If a coil has N number of turns and a current is passed through it, the flux change produced is contributed by each coil turn. 

NΦ = LI

Mutual inductance occurs when two coils are present close to each other and the magnetic field produced in one of the coils gets linked with another. This magnetic field may occur due to the current flowing through the coil. The property of a coil that affects or changes the current and voltage or the magnetic flux of the coil in the environment is called mutual inductance. 

Mutual inductance is a Major Principle commonly seen in transformers, generators, and motors. This principle of mutual inductance is applicable wherever there is an interaction between two coils or parts that are caused by altering the magnetic field. 

Mutual Inductance of Coaxially Placed Solenoids

In order to find mutual inductance of two long coaxials, we placed solenoids with each of length l. The methodology is as follows.

Consider two solenoids, solenoid 1 and solenoid 2

 Solenoid 1 (S1), inner solenoid

 Length of solenoid S1 = l

 The radius of solenoid S1 = r1

 The number of turns per unit length of solenoid S1 = n1

 The number of turns of solenoid S1 = N1 

 The mutual inductance produced in solenoid S1 due to solenoid S2 = M12

Current flowing through solenoid S1 = I1

Solenoid 2 (S2), outer solenoid

 Length of solenoid S2 = l

 The radius of solenoid S2 = r2

 The number of turns per unit length of solenoid S2 = n2 

 Number of turns of solenoid S2 = N2

 The mutual inductance produced in solenoid S2 because of solenoid S1 = M21

The mutual inductance M12 is produced in solenoid S1 with respect to to solenoid S2 depending on the following factors.

1. Number of turns per unit length of solenoid S1

2. Number of turns per unit length of solenoid S2

3. Area of cross-section of solenoid S1, A1 = π (r1)2 

4. Length of the solenoid S1 is l

N1 Φ1 = (n1.l) (π(r1)2) (µ0 n2 I2) = µ0 n1 n2 π(r1)2 ll2

M12  = µ0 n1 n2 A1 l

µ0 is the proportionality constant of mutual inductance. 

Mutual inductance M21 is the inductance produced in solenoid S2 with respect to solenoid S1, depending on the following factors. 

1. Number of turns per unit length of solenoid S2

2.  Number of turns per unit length of solenoid S1

3. Area of cross-section of solenoid S2,  A2 = π(r2)2 

4. Length of solenoid S2, l 

N2 Φ2 = (n2.l) (π(r1)2) (µ0 n1 I1) = µ0 n1 n2 π(r1)2 lI2

M21 = µ0 n1 n2 A1 l

The mutual inductance in solenoid S1 because of S2 (M12) and that in solenoid S2 because of S1 (M21), are identical.

M12 = M21 = M

Reciprocity Theorem

The equality for long coaxial solenoids is termed as reciprocity theorem. Reciprocity theorem implies that the mutual inductance in both the coils is equal. Similar results are obtained when we derive mutual inductance between coaxial solenoids of coaxially placed coils using Ampere’s law and Biot Savart law. 

The inner solenoid is much shorter than and is completely encircled by the outer solenoid. We can easily calculate the flux linked, N1 Φ1, as the inner solenoid is effectively immersed in a uniform magnetic field produced by the outer solenoid. It would be extremely difficult to calculate the flux linked with the outer solenoid due to the inner solenoid, M21, as the magnetic field varies across the length and area of the cross-section of the outer solenoid. 

 Hence, the reciprocity theorem makes it valuable and easy to determine the mutual inductance of two coaxial placed coils or solenoids.

 If these two coaxial replaced solenoids are present in a medium other than air, then an additional factor called relative permeability will determine the mutual inductance. It can be represented by

M12 = µ0 µR n1 n2 A1 l

µR= relative permeability in which the solenoid is present.

Conclusion 

Mutual induction displays the property of two coils or solenoids under which ban Koyal opposes any change in the magnitude of current flowing through the Other coil. The electromotive force in its magnetic flux of one coil is linked with another. 

This EMF can be calculated using the following formula- 

E2 = -M dI1/dt

Mutual inductance between two coils depends upon the geometry of two coils and their geometrical orientation with respect to each other. To find mutual inductance for a given arrangement we assume that a current is flowing through one of the coils or solenoid and calculate the flux through the other coil.