Name: Self Induction
Uploaded: 2019-07-19
Duration: 5 min 48 s
Description: This video describes the principle of self induction. It defines self-inductance and gives the formula for the same and derives its SI unit as well.

Question

Derive expression for the self-inductance of a long solenoid. What factors affect its value?

Accepted Answer

Self Inductance of a Current Carrying Solenoid: Consider a solenoid of radius r metre, length l metre, having the total number of turns N and carrying a current I ampere. The magnetic field produced on its axis inside the solenoid is

$B = \frac{μ_{0} NI}{l} {NA}^{- 1} m^{- 1}$

If it is assumed that the solenoid is very long, the intensity of magnetic field B can be assumed to be uniform inside it at each point, then the total magnetic flux linked with the solenoid is

$B = \frac{μ_{0} m}{4 π} [\frac{d^{2} + l^{2} + 2 d l - (d^{2} + l^{2} - 2 d l)}{{(d - 1) (d + l)}^{2}}]$

$ϕ = B \times t o t a l e f f e c t i v e a r e a o f s o l e n o i d ϕ = B \times (N A)$

Where, $A = {πr}^{2}$ is the area of cross section of solenoid.

$∴ ϕ = \frac{μ_{0} N I}{l} \times N A ϕ = \frac{μ_{0} N^{2} A I}{l} w e b e r$

But by definition, $ϕ = L I$ , where L is the self-inductance of solenoid, therefore,

Self inductance, $L = \frac{ϕ}{I} = \frac{μ_{0} N^{2} A}{l} h e n r y$

If inside the solenoid instead of air or vacuum there is a substance of relative permeability

$L = \frac{μ_{0} μ_{r} N^{2} A}{l} h e n r y$

Thus, the self inductance of a solenoid depends on the following factors:

(i) On the number of turns in the solenoid: The self inductance increases on increasing the number of turns.
(ii) On the area of cross-section of solenoid (i.e., on the radius of solenoid): The self inductance increases on increasing the area of cross-section.
(iii) On the length of solenoid: On increasing the length of solenoid, its self inductance decreases.
(iv) On the relative permeability of the substance placed inside the solenoid: If a soft iron rod is placed inside the solenoid, its self inductance increases.
Remember that if the number of turns per unit length in the solenoid is n, then n=N/l and the self inductance of solenoid is

$L = μ_{0} n^{2} A l h e n r y$

Derive expression for the self-inductance of a long solenoid. What factors affect its value?

Important Questions on Electromagnetic Induction

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