# Direct calculation of time varying Aharonov Bohm effect

## TL;DR

This paper investigates the time-varying Aharonov-Bohm effect using direct calculation, revealing a frequency-dependent phase shift and clarifying the roles of electric fields and path changes in the phenomenon.

## Contribution

It provides a novel direct calculation approach to the time-dependent Aharonov-Bohm effect, showing how electric fields influence the phase shift and how the effect varies with frequency.

## Key findings

- The electric field does not enter the action integral but affects the electron path.
- A frequency-dependent Aharonov-Bohm phase shift is observed.
- At high frequencies, the phase effect diminishes due to rapid oscillations.

## Abstract

The Aharonov-Bohm effect (ABE) for steady magnetic fields is a well known phenomenon. However, if the current in the infinite solenoid that creates the magnetic field is time-dependent, that is in the presence of both magnetic and electric fields, there is no agreement whether the effect would be present. In this note, we try to investigate time varying ABE by a direct calculation in a set-up with a weak time dependent magnetic field. We find that the electric field arising out of the time-varying magnetic field in the path of the electrons does not enter the action integral but only changes the path of the electron from the source to the slits and then on to the detector. We find a frequency dependent AB phase shift. At low frequencies the result smoothly approaches the one for a constant field as the frequency tends towards zero. On the other hand, for high frequencies such that the AB-phase induced in the path of the wave packet oscillates rapidly, the net effect will be very small which is borne out by our results.