Self Inductance – Definition, Coefficient, Factors Affecting, Applications

When a coil made of copper wire is placed inside a magnetic field, the magnetic flux gets added to the coil. Faraday found that when the magnetic flux attached to the coil is charged, an electric current is introduced into the coil, provided that the coil is closed. If the coil is open, then an electromotive force is established in the coil. Thus generated current and emf are called induced current and induced emf respectively. The induced current and the electromotive force in the coil move only as long as the magnetic flux associated with the coil changes.

Inductance

The property of a conductor or circuit due to which a change in the flow of electric current produces an electromotive force. The flux of a close wound coil is directly proportional to the linkage current, that is ∅_B ∝ I.

For an N-turn closed wound coil, the same magnetic flux is connected across all the windings. As the flux  ∅_B through the coil changes, each turn contributes to the induced emf. Therefore, flux linked with the coil (flux linkage) is equal to N×∅_B.

Then, total flux, N∅_B ∝ I. 

The constant of proportionality (I) in the above reaction is called inductance. 

Self-Induction

The phenomenon according to which an inversely induced electromotive force occurs as a result of a change in electric current or magnetic flux in a coil is called self-induction. or we can define it as the phenomenon of production of induced emf in a coil changes when an electric current is passed through it. 

Can also be expressed in words because the potency of the current in the coil, current in the coil also changes. In such a situation an emf is induced in the coil as well. Such emf is called self-induced emf and this phenomenon is known as self-induction.

Self-inductance is the property of a coil due to which, the coil opposes any change in the strength of the current flowing through it by inducing an emf in itself.

Induced emf is also called back emf. Self-induction opposes the increase of current when current is turned on in a coil and self-induction opposes the loss of current when the current is turned off. Self-induction is well known as the inertia of electricity.

Self-Induction L of a coil depends upon-

1. The size and shape of the coil.
2. The number of turns N.
3. The magnetic property of the medium within the coil in which the flux is present.

Note: Self-induction L does not depend on the current I.

Coefficient of Self-inductance formula: Total flux linked with the coil, N∅ ∝ I

N∅ = LI

where ∅ = flux linked with each turn and L = coefficient of self-induction of self-inductance.

Also, induced emf,

 e= -(d∅/ d × t) = -L dI/dt

where 

L = ε/(dl / d × t)

1 Henry (H) = 1V – s / A or 1 T × m ² / A or ohm-s.

Self-inductance

Self-Inductance of Long Solenoid 

A long solenoid is one whose length is much larger than the radius of the cross-section.

Derivation for self-inductance of Long Solenoid formula-

The magnetic field (B) at any point inside alike a solenoid is realistically constant and is specified by-

B = µ°NI/L                                                                                                                                                                                              ……(1)

where,

• μ_°= absolute magnetic permeability of free space, 
• N= total number of turns in the solenoid 
• l =length of the solenoid. 

Magnetic flux using every inflexion of the solenoid, 

∅=B × area of each turn

∅ = µ_°NI/L × A

where A  area of all inflexion of the solenoid.

Total magnetic flux fascinated the solenoid = flux through every turn × total number of turns

N∅ = µ_°I × (N/L) × AI × N

If L is the coefficient of self-inductance of the solenoid, then 

N*∅ = LI                                                                                                                                                                                                       …….(2)

From Equations 1 and 2, we get

L = (µ_°I N² A) / l

It gives the self-Inductance of al long solenoid of length l, area of cross-section. A is having a number of turns per unit length equal to N. If the core is of any other magnetic material u is placed, then 

μ= μ_° μ_r

L = (µ_° µ_r N² A) / l

The magnitude of emf is given by

e = L(dI × dt)

Multiplying (I) to both sides, we get 

el × dt = LI × dl                                                                                                                                                                                             …….(3)

But

I × dt = dq

Also, work done (dW) = voltage (e) x charge (dq) or dW=e x dq=el × dt

deputize these values in equation 3, we get

dW = LI × dl     .                                                                                                                                                                                           …….(4)

Total work done in increasing the current from zero to I_° we have By integrating both sides of equation 4 we get

W = ½ × LI_° ²

This work done in increasing the current flowing through the inductor is stored as the potential energy (U) in the magnetic field of the inductor.

U =  ½ × LI_° 2

Factors affecting the strength of the magnetic field around a solenoid

• Number of turns (twisting) the greater the number of turns the stronger the magnetic field of the solenoid and vice versa.
• Amount of current flowing.
• The greater the flux, the stronger the magnetic field and vice versa.
• Placing a soft iron core along the axis of the solenoid increases the magnetic field strength.

Applications of self-inductance

Some of the Applications of self-inductance are:

• Transformers
• Tuning circuits
• Induction motors
• Sensors
• Store energy in a device
• Ferrite beads
• Chokes
• Filters
• Inductors used as relays

Sample Questions

Question 1: What is Inertia of Electricity?

Answer:

When a source of emf sends current in a coiled electric circuit, the self-inductance of the inductor (coil) behaves in the same way as the mass of a body, when a force causes a change in its position. , We know that the greater the mass of a body, the more it opposes the change in the state of the body. Similarly, the mass of a body gives a measure of its inertia. Similarly, the greater the self-inductance of a coil, the greater is its resistance to the change in current through the coil. That’s why self-inductance is also known as the inertia of electricity.

Question 2: Explain why induction coils are made of copper.

Answer:

The ohmic resistance of an induction coil made of copper will be very low. Due to the change in magnetic flux, a large induced current will be generated in such an inductance, offering considerable opposition to the flow of current due to the applied emf.

Question 3: What factors control the magnitude of emf in an electric circuit?

Answer:

The magnitude of the induced emf in an electric circuit is proportional to the rate of change of the magnetic flux associated with the circuit.

Question 4: Two circular balls of the same size, one of metal and the other of glass, are allowed to fall freely from the same height above the ground. Which of the two will reach first and why?

Answer:

The glass bob will reach earlier on the ground as acceleration due to gravity is independent of the mass of the falling bodies. Being an insulator, no induced current is developed in it due to the earth’s magnetic field.

Question 5: How can the self-inductance of a given coil be increased by N number of turns, area, and length/ of cross-section A?

Answer:

The self-inductance can be increased with the help of electric fields. It does not depend on the current through the circuit but depends upon the permeability of the material from which the core is made up off.