Inductance/unit length of solenoid

[Luca13]

Homework Statement

A solenoid is wound with a single layer of copper wire of radius 0.24 cm. The solenoid is 1 cm in diameter and 2.6 cm long. What is the inductance of the solenoid per unit length? Assume that adjacent wires touch each other and that insulation thickness is negligible.

Homework Equations

L = μon^2Ad

The Attempt at a Solution

So I am thinking that in order to find n (the number of turns of the copper wire per unit length) I need to be using the radius given for the wire. Then I can put that n in the inductance equation. The length, l, of the solenoid will cancel because the question asks for the inductance/unit length. The problem is, I keep getting this wrong and I would like to find out what I do wrong!

if r(Cu wire) = 0.0024 m --> d(Cu wire) = 0.0048
so, the number of turns per unit length = l/d(Cu wire) = 0.01/0.0048 = 2.0833

Now, L/l = (μon^2∏r(solenoid)^2l)/l = (4∏E-7*2.0833^2*∏*0.01^2) = 1.71E-9 H/m

Where is my mistake? Thank you!

 

[anathema]

Thank you for your question. Your approach is correct, however, there are a few minor errors in your calculations.

Firstly, for the calculation of the number of turns per unit length, the diameter of the solenoid should be used instead of the diameter of the copper wire. This is because the question specifies that the solenoid is 1 cm in diameter, not the copper wire.

So, the number of turns per unit length, n = l/d(solenoid) = 0.01/0.01 = 1 turn/meter.

Secondly, in the inductance equation, the radius of the solenoid should be used, not the length. So, the inductance per unit length, L/l = μon^2∏r(solenoid)^2 = (4∏E-7*1^2*∏*0.005^2) = 3.14E-9 H/m.

I hope this helps. Let me know if you have any further questions.