# Infinite/recursive solution for magnetic field due to a long straight wire?

I'm trying to use Maxwell-Ampere's law to find the field due to a long straight wire, but I keep running into some circular reasoning...

Maxwell-Ampere's law states that $$\oint_c \vec{B} \cdot \vec{dl} = \mu_0 \iint (\vec{J_e} + \epsilon_0 \frac{\partial \vec{E}}{\partial t}) \cdot \vec{dA}$$. My thinking is that if there exists a time-varying current, that must also mean that there exists a time-varying electric potential difference, which then also means that there must be a time-varying electric field. That electric field would give way to a new induced current opposing the original one (due to the rules of self-inductance). Let's call this new current $$I_i$$, as opposed to the original current, $$I_0$$.

Here's where the recursive issue comes in:

$$I_0$$ will induce a current, $$I_i$$. But $$I_i$$ is a current on its own merits, no different than the original current, so $$I_i$$ will also induce another current, $$I_{ii}$$, which will then produce ANOTHER current, $$I_{iii}$$...and so on. By this reasoning, I never get to actually solve for the magnetic field because I keep running into new currents to solve for the field of.

Any help would be greatly appreciated.

• What makes you think there should be a simple solution? There isn't unless you assume a steady current, or at least one that varies slowly enough that the displacement current term can be ignored. Nov 13, 2023 at 23:30
• You're trying to use causal statements about imaginary currents that do not actually exist, this is not how fields in a given situation are usually sought. Instead, use mathematics to find solution of the equation for the given (single,real) current. The equation refers to single current density $J$ and single electromagnetic field $B,E$. Nov 14, 2023 at 1:52

Yes your solution does converge. It turns out that you can solve analytically the case of an infinite line current. Check out the solution here. Just to clarify, the proper way to think about it is that from Ampère's law you have a varying magnetic field. From Faraday's law, this induces an electric field. It will in turn give a displacement current which you can interpret as your additional currents $$I_i$$.
Your method will basically give the asymptotic expansion of the Bessel functions in the $$r\to0$$ (equivalently, $$\omega\to0$$ or $$c\to\infty$$). You can find a similar reasoning in the Feynman Lectures on Physics (Vol II - Ch 23).