Question

Deduce an expression for the self inductance of a long solenoid of N turns, having a core of relative permeability µr.

Answer 1

Expression for coefficient of self inductance for a Solenoid:

Consider a solenoid of length l

Let r = radius of solenoid such that r < l

n = no. of turns per unit length

I = strength of current through the solenoid

The magnetic field produced by the current inside the solenoid is practically constant and B = µ₀ nI

∴ Magnetic flux linked with the loop,

dΦ = µ₀ n IA

Magnetic flux linked with entire solenoid

Φ = nl(µ₀ n IA) = µ₀n²lAI

Since Φ = LI

∴ LI = µ₀ n² l A

L = µ₀ n² l A

If the coil is wound on a material having relative permittivity µr

then L = µ₀ µ_r n2 l A