Question

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 ∮cB⃗ ⋅dl→=μ0∬(Je→+ϵ0∂E⃗ ∂t)⋅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 Ii
, as opposed to the original current, I0 Here's where the recursive issue comes in: I0 will induce a current, Ii
. But Ii is a current on its own merits, no different than the original current, so Ii will also induce another current, Iii, which will then produce ANOTHER current, Iiii ..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.

Answer 1

Maxwell-Ampère's law and the wider framework of Maxwell's equations already account for the effects of self-inductance, resolving the issue of recursive induction.

In practical scenarios with steady currents in long straight wires, the effects are encapsulated in Ampère's law without the need to separately account for multiple induced currents.

Step-by-step explanation:

When using Maxwell-Ampère's law to find the magnetic field due to a long straight wire with a time-varying current, it's important to understand that the induced currents, due to self-inductance (which you've referred to as Ii, Iii, etc.), are already accounted for in the complete formulation of Maxwell's equations.

The recursive induction problem you are referring to is implicitly resolved in the formulation of the laws themselves. Specifically, the Ampère-Maxwell law, which is a part of Maxwell's equations that you are using, already incorporates both the effects of traditional currents and the displacement current term which relates to changing electric fields in its statement.

The displacement current serves to prevent the recursive issue of constantly inducing new currents. Instead of separately considering each induced current in isolation, the full scenario, including the original current and the effects of induction, are encapsulated in the equations.

Remember that when you are dealing with an infinitely long, straight wire carrying a steady current, the situation is much simpler, and you can use Ampère's law without the displacement current term as the electric field is not changing in time.

Ultimately, for practical calculations in typical introductory physics scenarios like a steady current in a long straight wire, the recursive induction effects are simply part of the static magnetic field calculation using Ampère's law without having to iterate through multiple layers of self-induction.

The symmetric nature of Maxwell's equations shows the equivalence and interdependence of electric and magnetic fields, thus enabling the linking of Maxwell's displacement current to the properties of electromagnetic waves.