# Magnetic energy stored in a conductor

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.

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?

• Yeah, you should integrate over all of space, but I think that I see a problem. If $B=\mu_o I/2\pi r$ outside the conductor, then the integral of $B^2$ over all volume from r=a to infinity becomes an integral over $(\mu_o I/2\pi r)^2 2\pi r dr dz$ rom r=a to infinity, and that doesn't seem to converge since the integrand goes as 1/r .
– user93237
Jun 21, 2018 at 21:39
• So the conductor is of infinite length, therefore the energy stored in it's B field is infinite. I think they meant to ask for the energy/unit length. Jun 21, 2018 at 21:41
• @JMLCarter - The problem is that the integral doesn't seem to converge even if you consider the problem in terms of energy per unit length.
– user93237
Jun 21, 2018 at 21:43
• To be sure, the question was "What is the magnetic energy stored in the conductor?", not "What is all the magnetic energy associated with the current in the conductor?"
– V.F.
Jun 22, 2018 at 2:05
• @V.F. - That may be it, but if so IMHO that's a poorly worded question. If that's the intent of the question, then it could have been made completely unambiguous by asking something like "What is the magnetic energy stored $\textit{within the volume}$ of the conductor?".
– user93237
Jun 22, 2018 at 2:39

BTW: the equation with $0.5 \cdot \mathbf{B}^2$ is wrong! You should multiply it with $1 \over \mu$.
• If they're using cgs units, with $\mu\approx\mu_0$ (as is typical for most real conductors that aren't Mu-metal, Metglas, or ferromagnetic), what is written is correct. Jun 25, 2018 at 20:39
• @probably_someone: if cgs units are used and you assume $\mu \approx \mu_0$, the energy density should be $\frac{1}{8\pi}\cdot \mathbf{B}^2$. Jun 25, 2018 at 20:54