Inductance and Magnetic Energy of a Straight Wire

[AJKing]

Homework Statement

A long wire carries a current i uniformly distributed over a cross section of the wire.

Show that the magnetic energy of a length l equals μ_0 i² l / 16π. Describe why this doesn't depend on diameter.

Show that the inductance for a length of the wire l associated with the flux inside the wire is μ_0 l / 8π

Homework Equations

u_B = B²/2μ_0

U = L i² /2

Biot-Savart law for Field in a long wire:

B = μ_0 i / 2 π r

The Attempt at a Solution

Solve for u_B as

u_B = B²/2μ_0

= (μ_0 i / 2 π r ) ² / 2μ_0

Multiply by volume of wire to find U:

U = ( (μ_0 i / 2 π r ) ² / 2μ_0 ) * π r^2 l

= μ_0 i² l / 8π

I'm off by a factor 1/2. Any suggestions?

 

[TSny]

You need to find B inside the wire.

If u_B is not uniform inside, integration will be necessary to find the total energy.