Amount of magnetic energy stored in a cylinder

[Physics lover]

Homework Statement

Find the amount of energy stored in thin cylinder of length l,around a conductor carrying current I as shown.

The figure is in Attempt at a solution.

Relevant Equations

E=0.5LI^2
Ampere's Circuital Law
Inductance=Flux/Current

I first calculated Magnetic field at a distance x using Ampere's law.Then i took an element and wrote its inductance using L=Flux/Current.
Then i integrated from a to b.
But i arrived at nothing.
Help.

 

[mfb]

What is the energy density stored in a magnetic field of a given strength?

 

[Physics lover]

What is the energy density stored in a magnetic field of a given strength?

It is Energy per unit volume,so should i first find energy for a part by energy density and integrate it?

 

[mfb]

You should find a relation that links energy density of a magnetic field (what you need) to the field strength (what you calculated).

 

[Physics lover]

You should find a relation that links energy density of a magnetic field (what you need) to the field strength (what you calculated).

yeah i think it is energy density=B²$/$2u₀

 

[Physics lover]

You should find a relation that links energy density of a magnetic field (what you need) to the field strength (what you calculated).

Thanks i got the answer.Earlier i thought that this result was valid for only solenoid.But it is for all cases is it?

 

[mfb]

The formula for the energy density is valid for all cases (in vacuum, technically, but it is a good approximation in air). How the field strength varies in space depends on the geometry of the cables.