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## Homework Statement

So I came across with following problem:

> Consider a cylindrical conductor of infinite length and circular section of radius a and that is traversed by a stationary current I. What is the magnetic energy stored in the conductor.

## Homework Equations

3. The Attempt at a Solution [/B]

So my question is more of a conceptual one.

I proceeded to apply Ampere's law to calculate the B field which is

\begin{cases} \frac{\mu_0Ir}{2\pi a^2} \,,\, r<a\\ \\ \\ \frac{\mu_0I}{2\pi r} \,,\, r>a\end{cases}

I checked the resolution of the problem and they seem to only calculate the magnetic energy on the conductor. But according to

$$\iiint_{all\,space} 0.5 B^2 \,dV$$

Shouldn't we take it all space? Because B isn't zero outside of the conductor. I'm so confused on why they just considered the conductor, am i misunderstanding something.

Also let me add that the integral in the cylinder gave us

$$\frac{\mu_0 I^2}{16 \pi}$$

I also have know idea on how to compute the integral outside the conductor. What limits of integration should I take?

I'm really confused, can someone help me?