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-induction 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 practical constant and B = µ₀ nI

∴ Magnetic flux linked with the loop, dΦ = µ₀n IA

Magnetic flux linked with entire solenoid Φ = nl (µ₀ n IA) = µ₀n²lAI

Since Φ = LI

∴ LI = µ₀n² l I A

L = µ₀n² l A ............(i)

If the coil is wound on a material having relative permittivity µr then L = µ₀ µ_r n² l A