# Inductance

**Inductance:**

· Inductance is the tendency of an electrical conductor to resist a change in the electrical current passing through it.

· L is used to denote inductance.

· Henry is the SI unit of inductance.

· 1 Henry is defined as the inductance required producing 1 volt of emf in a conductor when the change of current in the conductor is 1 ampere per second.

· An electric current passing through a conductor creates a magnetic field around it. The strength of the field depends on the strength of the current. The resulting magnetic field follows any change in current, and from Faraday's law of induction, we know that changing the field induces an electromotive force in the conductor. With this principle in mind, inductance is defined as the ratio of the induced voltage to the rate of change of the current causing it.

· An electronic component designed to add inductance to a circuit is an inductor.

**Factors affecting inductance **

The inductance of a circuit is affected by the following factors:

· The number of turns of wire in the coil Inductance is greater when the number of turns of wire in the coil is greater.

· More turns of wire indicate greater magnetic field strength for a given coil current. Basement area Inductance is proportional to the area of the coil. The larger the area of the coil, the larger the inductance.

· A larger coil surface area provides less resistance to magnetic field flux for a given field strength Basic material.

· The greater the magnetic permeability of the core around which the coil is wrapped, the greater the inductance.

· The longer the length of the coil, the lower the inductance. The shorter the length of the coil, the higher the inductance.

Types of inductance Induction is divided into two types:

1. Self-induction

2. Mutual induction

** Self Induction: **

When the coil current or magnetic flux changes, an electromotive force is induced. This phenomenon is called self-induction.

As the current begins to flow through the coil at some point, it has been observed that the magnetic flux becomes directly proportional to the current flowing through the circuit.

The relationship is presented as follows:

If L is called the coil self-inductance or the self-inductance coefficient,

then the self-inductance depends on the cross-sectional area, the permeability of the material and the number of turns of the coil.

The rate of change of the magnetic flux in the coil is given by:

** Self Inductance Formula**

Where,

· L is the self-inductance in Henries

· N is the number of turns

· Φ is the magnetic flux

· I is the current in ampere

** Mutual Inductance**

Consider two coils: P – coil (Primary coil) and S – coil (Secondary coil). A battery and a key are connected to the P-coil, whereas a galvanometer is connected across the S-coil. When there is a change in the current or magnetic flux linked with the two coils, an opposing electromotive force is produced across each coil, and this phenomenon is termed Mutual inductance.

This phenomenon is given by the relation:

Where M is termed as the mutual inductance of the two coils or the coefficient of the mutual inductance of the two coils.

The rate of change of magnetic flux in the coil is given as,

** Mutual Inductance Formula**

Where,

· μ _{0 }is the permeability of free space

· μ_{r} is the relative permeability of the soft iron core

· N is the number of turns in coil

· A is the cross-sectional area in m^{2}

· l is the length of the coil inductance

** Derivation of Inductance**

Consider a DC source. When the switch is turned on, the current flows from zero to a certain value such that there is a change in the rate of current flowing. Let φ be the change in flux due to current flow. The change in flux is with respect to time which is given as:

** Apply Faraday’s law of electromagnetic induction,**

Where,

· N is the number of turns in the coil

· E is the induced EMF across the coil

From Lenz’s law, we can write the above equation as

The above equation is modified for calculating the value of inductance

N = dΦ = L di

NΦ = Li

Therefore,

Li = NΦ = NBA

Where B is the flux density, A is the area of the coil.

Hl = Ni

Where H is the magnetizing force due to magnetic flux

B = μH

Li = NBA

L = NBA/i = N^{2}BA/Ni

N^{2}BA/Hl = N^{2}μHA/Hl

L = μN^{2}A/l = μN^{2}r^{2}/l

Where,

r is the radius of the coil

**Mutual Inductance of a Coaxial Solenoid**

Consider two coaxial solenoids, of which the outer solenoid S2 has radius r2, and N2 turns, whereas the inner solenoid S1 has radius r1 and N1 turns. Both the solenoids are of equal length.

When there is a current I2 in the solenoid S2, the magnetic induction due to I2 is given by,

The corresponding flux linkage with solenoid S1 is,

On comparing (11) and (12),

Similarly, when a current I1 is set up through S1, then the magnetic flux linked in S2 is given by,

Note that the magnetic flux due to current I1 in S1 is assumed to be confined only inside the solenoid S1. Also, the solenoids are very long compared to their radii, and the flux linkage in S2 is

→ (Area confined within S1.)

From (13) and (14),

**Induced Current:**

Any change in the magnetic field associated with the coil of wire causes an emf in the coil. This emf is called induced emf and when the conductor circuit is closed, current also circulates in the circuit. This current is called induced current.

The magnetic field can be changed in the following ways:

· By moving the magnet towards or away from the coil

· By moving the coils in or out of the magnetic field.

· By changing the area of the coil placed in the magnetic field By rotating the coil relative to the magnet.