Current Loop with a Rotating Test Particle: Conversion of Magnetic Forces to Electric Forces by Special Relativity Transform To properly envision the circular loop when the test particle rotates around it, think of a train with a very large number of cars on a circular track. If the test particle moves an appreciable fraction of the speed of light, the car length and track segments just adjacent to it will be observed to be contracted by the factor 1/gamma. The cars that are at 90 degrees from the charge will not be contracted. So the charge will view the train and track as an ellipse. If the train cars are moving (negative charge flow or current for our problem) along the track, then they too will be contracted but they are constrained to stay on the track so they occupy the same ellipse as the stationary cars did. In this case, we need to convert the shape of the circle to an ellipse with minor/major axis ratio 1/gamma. In this animation we have negative charge flow counter-clockwise in a circular loop of wire as well as a negatively charged test particle, qtest, moving clockwise at speed vtest. Since the test charge is displaced out of the screen by a small distance, dz, and on the right side of the wire loop, the magnetic field from the wire would normally cause a magnetic force on the test charge in the downward direction. However, if we use special relativity to transform both the positive and negative charges in the wire so that the test charge is stationary, then the magnetic force on it will disappear. If, on the other hand, the test particle is held stationary and the loop rotates (select Loop Rotating button), the magnetic force is replaced by an electric force that, when transformed back to the laboratory frame, is exactly the same as the previous magnetic force. This demonstration of the way that magnetic forces can be transformed to electric forces is usually done by considering an infinitely long straight wire. A problem with that approach is that there is no way of seeing how the total numbers of positive and negative charges are conserved. For this animation discrete integer numbers of positive and negative charges are dealt with and charge conservation is obvious. To keep the problem as simple as possible, the number of negative charges is the same as the number of positive charges thereby making the average charge density of the wire 0.
Feynman Electric Field Due to a Charge Electric Forces Due to a Current in a Rotating Loop rotatingCurrentLoop.htm

betaTest betaI nC0 Single Step Show Test Charge Rotating Show Loop Rotating