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?

  • $\begingroup$ 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 . $\endgroup$ – user93237 Jun 21 '18 at 21:39
  • $\begingroup$ 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. $\endgroup$ – JMLCarter Jun 21 '18 at 21:41
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    $\begingroup$ @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. $\endgroup$ – user93237 Jun 21 '18 at 21:43
  • $\begingroup$ 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?" $\endgroup$ – V.F. Jun 22 '18 at 2:05
  • $\begingroup$ @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?". $\endgroup$ – user93237 Jun 22 '18 at 2:39

The question is

What is the magnetic energy stored in the conductor?

So the answer is right, because here it is only asked for the energy stored inside the conducting cylinder.

BTW: the equation with $0.5 \cdot \mathbf{B}^2$ is wrong! You should multiply it with $1 \over \mu$.

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  • $\begingroup$ 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. $\endgroup$ – probably_someone Jun 25 '18 at 20:39
  • $\begingroup$ @probably_someone: if cgs units are used and you assume $\mu \approx \mu_0$, the energy density should be $\frac{1}{8\pi}\cdot \mathbf{B}^2$. $\endgroup$ – abu_bua Jun 25 '18 at 20:54
  • $\begingroup$ Ah, yeah, good point. $\endgroup$ – probably_someone Jun 25 '18 at 21:23

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