# Self Inductance

## Self Inductance

- Consider the figure given below

- When we establish a current through an inductor or coil, it generates a magnetic field and this result in a magnetic flux passing through the coil as shown in figure 1(a).

- If we vary the amount of current flowing in the coil with time, the magnetic flux associated with the coil also changes and an emf ξ is induced in the coil.

- According to the Lenz's law, the direction of induced emf is such that it opposes its cause i.e. it opposes the change in current or magnetic flux.

- This phenomenon of production of opposing induced emf in inductor or coil itself due to time varying current in the coil is known as self induction.

- If I is the amount of current flowing in the coil at any instant then emf induced in the coil is directly proportional to the change in current i.e.

where L is a constant known as coefficient of self induction.

- If (-dI/dt)=1 then ξ=L

Hence the coefficient of self induction of a inductor or coil is numerically equal to the emf induced in the coil when rate of change of current in the coil is unity.

- Now from the faraday's and Lenz's laws induced emf is

comparing equation 1 and 2 we have,

or Φ=LI

- Again for I=1, Φ=L

hence the coefficient of self induction of coil is also numerically equal to the magnetic flux linked with the inductor carrying a current of one ampere

- If the coil has N number of turns then total flux through the coil is

Φ_{tot}=NΦ

where Φ is the flux through single turn of the coil .So we have,

Φ_{tot}=LI

or L=NΦ/I

for a coil of N turns

- In the figure given below consider the inductor to be the part of a circuit and current flowing in the inductor from left to right

- Now when a inductor is used in a circuit, we can use Kirchhoff’s loop rule and this emf(Self induced emf) can be treated as if it is a potential drop with point A at higher potential and B at lower potential when current flows from a to b as shown in the figure

- We thus have

V_{ab}=LdI/dt

## Self induction of a long solenoid

- Consider a long solenoid of length l, area of cross-section A and having N closely wound turns.

- If I is the amount of current flowing through the solenoid them magnetic field
**B** inside the solenoid is given by,

- Magnetic flux through each turn of the solenoid is,