I'm trying to test my recently installed MEEP program for a very simple AC current. I know that for DC current, Ampere's law dictates that the magnetic fields must drop off as 1/r. How does this change with AC current as we depart from magnetostatics? I'm explicitly testing these predictions on scales where electromagnetic wavelengths become important, so we're not allowed to assume that the electric field is constant or large compared to light. Is there a better way to solve this problem than explicitly crunching our Maxwell's equations? Thanks!

P.S. I was thinking about whether energy flux would be helpful?

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    $\begingroup$ Your requirements sound a bit contradictory. On one hand you want something simple, on the other hand you are not happy with what simple can get you. You want the results of Maxwell's equations without solving Maxwell's equations. If that was possible, why would anybody have to write the software you are using, to begin with? Does the program come with a test suite? Did you try those examples? $\endgroup$ – CuriousOne Oct 20 '14 at 1:35

As CuriousOne says, look carefully for a test suite within your software installation. MEEP is widespread, notwithstanding the LISP interface (gotta love MIT's confidence in its own creations), so if you seek carefully, you are bound to find MEEP analysed examples on the web.

As for your proposed simple test: it is a good idea, and there are many, well known full solutions to Maxwell's equations that you might test. Such tests include:

  1. Model the current element antenna: the short, uniform current element with a ball capacitor storage at either end (so that $I(t) = {\rm d_t} Q(t)$, where $I(t)$ is the uniform current in the element, and $\pm Q(t)$ the charge at either end. See the "Elementary Doublet" section of the "Dipole Antenna" Wikipedia page

  2. The half wave dipole, wherein the current in the wire is assumed to be sinusoidally distributed; See the "Half-Wave Dipole" section of the "Dipole Antenna" Wikipedia page

  3. Another easy one is the infinite current. This is a two dimensional problem and the only independent variable is the radial co-ordinate, i.e. the distance from an infinite current. Simply use the retarded potential idea and you find that the field components varies as:

$$\vec{E}(r) = E_0\,H_0^{(1)}(k\,r)\,\hat{z}$$ $$\vec{B}(r) = -i\,c\,E_0\,H_1^{(1)}(k\,r)\,\hat{\phi}$$ $$\vec{A}(r) = -i\,\frac{c}{k}\,E_0\,H_0^{(1)}(k\,r)\,\hat{z}$$

where $H_n^{(1)}$ is the Hankel function ($H_n^{(1)}(z) = J_n(z) + i\,Y_n(z)$) and has the property that $H_n^{(1)}(z)\to\sqrt{\frac{2}{\pi\,z}}\,e^{i\,z}\exp\left(-i\,\left(n\frac{\pi}{2}+\frac{\pi}{4}\right)\right)$ as $z\to\infty$, so that the cylindrical waves described in example (3) become more and more like plane waves as $r\to\infty$.

In example 3), if you want to take account of a conductor of nonzero radius, you need to split the problem into two regions: that within and that outside the conductor. Inside the conductor, we have solutions like those in example 3) aside from that:

  1. The Hankel function is replaced by the entire function $J_0(k\,r)$ and
  2. $k$ is replaced by $\omega\,\sqrt{\mu\,\left(\epsilon - i\frac{\sigma}{\omega}\right)}$, where $\mu$ and $\epsilon$ are, respectively, the conductor's magnetic and electric constant and $\sigma$ its conductivity.

You then choose the scaling constants of the solutions in both regions so that the electromagnetic boundary conditions are fulfilled. That is, the amplitude of $\vec{B}$ cannot change discontinuously at the conductor's surface.


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