Question

Find the inductance of a uniformly wound solenoid having N turns and length l. Assume that l is much longer than the radius of the windings and that the core of the solenoid is air.

What would happen to the inductance if you inserted a ferromagnetic material inside the solenoid?

Answer 1

L = N^2Aμ/L

Step-by-step explanation:

To find the inductance you use the following general formula:

`L=N(\Phi_B)/(I)\\\\\Phi_B=BA`

N: turns of the solenoid

I: current in the solenoid

ФB: magnetic flux

B: magnitude of the magnetic field

A: cross area of the solenoid

Next, you take into account the magnitude of magnetic field inside a solenoid:

`B=(\mu NI)/(L)\\\\\mu=k\mu_o`

where k is the relative permeability of the material inside the solenoid. the last expression for B also takes into account the long solenoid a approximation, which makes possible that magnetic fields effect in the borders of the solenoid is negligible.

Next, you replace the equation for B in the equation for L:

`L=N(A\mu_oNI)/(IL)=N^2(A\mu_o)/(L)`

where you have used k=1 (for air).

If a ferromagnetic material is inserted in the solenoid the value of k is different of 1 and you have for L:

`L=N^2(Ak\mu_o)/(L)`