# Induced current in parallel wires

Consider two parallel wires of finite radius. When a current is applied to one of the wire for a short period of time, what is the current induced in the other wire?

Applying Maxwell's equations, it seems that there is a change in magnetic field perpendicular to the second wire, and as a result, the induced current has a nontrivial distribution which averages somewhat to zero. Is this correct? Intuitively, I had instead expected a simpler result similar to the case of two coils of wires placed side by side.

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## 1 Answer

I think there is a non-zero induced current.

1) During the rise (and fall) of the current pulse in wire 1, a changing, azimuthal, magnetic field is generated (calculable by Ampere's law).

2) Per Maxwell, that changing B-field induces an electric field parallel to the wires, which:

3) causes a current to flow in wire 2 (assuming a wire resistivity $\rho$ and applying Ohm's law).

Thus, if one applies a short pulse of current to wire 1, you'd see two even shorter pulses in wire 2, one positive and one negative, aligned with the rise and fall of the wire 1 current.

There is some variation in E with respect to the radial dimension, which would complicate an exact solution, but the variation is small (and it doesn't result in cancellation).

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