 # Self Inductance

## What is Self Inductance?

Self-inductance is the property of the current-carrying coil that resists or opposes the change of current flowing through it. This occurs mainly due to the self-induced emf produced in the coil itself. In simple terms, we can also say that self-inductance is a phenomenon where there is the induction of a voltage in a current-carrying wire.

The self-induced emf present in the coil will resist the rise of current when the current increases and it will also resist the fall of current if the current decreases. In essence, the direction of the induced emf is opposite to the applied voltage if the current is increasing and the direction of the induced emf is in the same direction as the applied voltage if the current is falling.

The above property of the coil exists only for the changing current which is the alternating current and not for the direct or steady current. Self-inductance is always opposing the changing current and is measured in Henry (SI unit).

Induced current always opposes the change in current in the circuit, whether the change in the current is an increase or a decrease. Self-inductance is a type of electromagnetic induction.

## Self-inductance Formula

We can derive an expression for the self-inductance of a coil from Faraday’s law of electromagnetic induction.

VL = −N (dϕ / dt)

Where:

VL = induced voltage in volts

N = number of turns in the coil

dφ / dt = rate of change of magnetic flux in webers / second

Alternatively, the induced voltage in an inductor may also be expressed in terms of the inductance (in henries) and the rate of change of current.

VL = −L (di / dt)

Or

E = −L (di / dt)

## Self-inductance of a Solenoid

We will understand the concept with the help of an example. We will take a solenoid having N turns, Let its length be ‘l’ and area of cross-section be ‘A’ where current I is flowing through it. There will be a magnetic field ‘B’ at any given point in the solenoid. Therefore, the magnetic flux per turn will be equal to B × area of each turn.

However, B = (μ0NI)/l

Therefore, magnetic flux per turn = (μ0NIA)/l

Now, the total magnetic flux (φ) that is connected with the solenoid will be given by the product of flux present through every turn and the total number of turns.

Φ = (μ0NIA) x N /l

That is, Φ = (μ0N2IA) /l ….(eq 1)

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

Φ = LI …..(eq 2)

Combining the equations (1) and (2) from above, we get

L = (μ0N2A) /l

If you have a core that is made up of a magnetic material of permeability μ, then

L = (μN2A) /l

## Mechanical Equivalent of Self-inductance

The self-induced emf is also called the back emf as it opposes any change in the current in a circuit. Physically, the self-inductance acts as inertia. In mechanics, it is also the electromagnetic analogue of mass. So, work needs to be done against the back emf (VL) in establishing the current. This work done is stored as magnetic potential energy and is given by:

W = (1 / 2) LI2.

This is the energy required to build up a current I in the inductor.

This expression reminds us of mv 2 /2 for the (mechanical) kinetic energy of a particle of mass m and shows that L is analogous to m (i.e., L is electrical inertia and opposes both the growth and decay of current in a circuit).

## Mutual Inductance

Mutual inductance is the opposition to the change of current in one coil due to the presence of a second coil. , if a medium of relative permeability μr is present, the mutual inductance would be M =μr μ0 n1n2π r21 l, where n1 and n2 are the numbers of turns of the two coils per unit length, and r1 is the radius of the inner coil and l is the length of the coil.

## Differences Between Mutual Inductance and Self Inductance

 Self-inductance Mutual Inductance Changing current in a coil induces an emf in itself and opposes the change in current. Changing current in one coil induces emf in another coil, and opposes the change in current. L = μr μ0 n2 A l M = μr μ0 n1n2π r21 l Self-inductance plays the role of inertia.

## Uses of Self Inductance

The major function of an inductor is to store electrical energy in the form of a magnetic field. Inductors are used in the following:

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

## Limitations of Inductors

1) An inductor is limited in its current-carrying capacity by its resistance, and the ability to dissipate heat. So, it usually has a maximum rated current.

2) Inductors in pure form are difficult to manufacture due to stray effects and size, whereas, capacitors are relatively easy to manufacture with negligible stray effects.

3) Inductors may affect the nearby components in the circuit with their magnetic fields.

## Solved Questions

Question 1:

Mutual inductance between two coils is given as:

The mutual inductance between the two coils is given by the relation

where k is the coefficient of coupling

M is the mutual inductance between the coils.

Question 2:

Self-inductance of an inductor is given by ________

a) L = N⁄S

b) L = 1⁄S

c) L = N2⁄S

d) L = N2

The self-inductance of an inductor is obtained by making use of the relation

L = N2⁄S

where N is the number of turns

S is the reluctance of the coil (A/Wb).

Question 3:

Self-inductance depends on ________

a) permeability

b) permittivity

c) plank’s constant

d) rydberg constant

The self-inductance of an inductor is obtained by making use of the relation

L = N2⁄S

where N is the number of turns

S is the reluctance of the coil (A/Wb)

As reluctance depends on the permeability, the self-inductance of a coil depends on the permeability.

Question 4:

The phase angle is always ________ for an RL circuit.

a) Positive

b) Negative

c) 0

d) 90

For a series resistance and inductance circuit, the phase angle is always a negative value because the current will always lag the voltage.

Question 5:

What is φ in terms of voltage?

a) Φ = cos-1V/VR

b) Φ = cos-1V*VR

c) Φ = cos-1VR/V

d) Φ = tan-1V/VR

From the voltage triangle, we get cosφ = VR/V.

Hence φ = cos-1VR/V.

Question 6:

What is sinϕ from impedance triangle?

a) XL/R

b) XL/Z

c) R/Z

d) Z/R