A Copper Probe Finds the Edge of a Magnetic Bubble ✛

Original: "Induced electromotive force of a thin metal rod in the alternating electromagnetic field of Helmholtz coil: experimental results and theoretical analysis"
arXiv:2609.06548v1 · 2026-09-06 · CC BY 4.0 · 1 min
A copper rod moving across the field of two coils reads zero voltage at the center and the strongest signal at the bubble's edge.

A copper rod as long as your palm moves slowly across an invisible magnetic field created by two identical coils. At the very center of the field, the voltmeter reads zero. Shift the rod just a little to the side, and the needle comes alive: the voltage climbs, hits a peak, and then gently falls. The rod acts like a probe that feels out the edge of an invisible bubble where the field stops being uniform.

The signal maximum occurs not at the coil's boundary but slightly earlier: when the rod's middle is still inside the field while its ends have already slipped out.

Why zero at the center? The alternating magnetic field induces a voltage in the rod. At the center, the upper and lower halves of the rod sense the same field change, so their signals cancel out. When the rod shifts, the halves experience different conditions, creating a difference, that is, a voltage. But if you go too far, the field beyond the coil's edge points the opposite way and interferes with the signal. So the curve first rises and then falls.

The experiment showed that the peak sits at about ninety-three hundredths of the coil radius from the center. That means the rod has almost entirely left the field, but its middle is still inside. This knowledge helps calibrate magnetic field sensors: at the edge the signal is stronger, so the instrument can be tuned more precisely. A simple copper rod becomes a tool that makes an invisible field tangible.

The EMF maximum occurs when the rod is shifted from the center by 93% of the coil radius, meaning its center is almost at the field boundary while its ends are already outside. At this point the signal is nearly three times stronger than at half the radius.

\varepsilon_{AB} = B_0 \frac{dB_t(t)}{dt} y l
The EMF grows linearly with distance y from the center while the rod is entirely inside the uniform field.
\varepsilon_{AB} = B_0 \frac{dB_t(t)}{dt} \left[ y \sqrt{R^2 - y^2} + R^2 \left( \arctan\frac{l}{y} - \arctan\frac{\sqrt{R^2 - y^2}}{y} \right) \right]
This formula describes the rise and subsequent fall of the EMF in the transition zone where part of the rod crosses the field boundary.
\varepsilon_{AB} = B_0 \frac{dB_t(t)}{dt} R^2 \arctan\frac{l}{y}
The EMF decreases monotonically with distance because the whole rod is in the zero-field region, but the induced electric field outside does not vanish.
Concepts
Faraday's law of electromagnetic induction external magnetic field thermal hall effect ferron hall effect