The four conducting spheres of equal radii r were charged and arranged and interconnected firmly by 2 linear rigid wires with common switcher S according to the Pic.(see attach.). At what resistance R of each wire the upper and lower pairs of the spheres with the wires repel one from another just after switch in of the S?
Is this a puzzle? The puzzle is explained by both Coulomb's Law and Ampere's Law.
The spheres attract, but should never repel. The wires repel, and should never attract. But this goes on for a short amount of time, after the switch is closed. Then all attraction (of spheres) and repulsion (of wires) ceases. Assuming the spheres have the same amount of charge, and the wires have the same resistive characteristics.
As soon as you close the switch, current will flow from the + to the - charged sphere, along each wire. That current flow is in opposite directions, in the two wires. Current decrease in time, negative exponential, and will stop flowing, when the charges have equalized.
While the current is flowing, the charged spheres will continue to attract one another, because the sphere on top and the sphere on the bottom will still have opposite charge. The attraction will diminish, as the charges equalize. Soon there will be no attraction (well, practically speaking "soon" anyway), as the charges have equalized, and the current will cease to flow.
As to the wires, the right hand rule shows in which direction the magnetic field surrounding each wire will rotate. Because current flows in opposite directions, the B field around the top wire rotates opposite the B field of the bottom wire. The B fields of the two wires therefore approach with the same polarity, and the wires must repel. As the charges equalize, current diminishes, the B fields diminish, repulsion diminishes, and will soon cease.
At what resistance R of each wire the upper and lower pairs of the spheres with the wires repel one from another just after switch in of the S?
If R is the same, I can't see how the upper and lower spheres can ever repel. If the upper R is infinite, and the lower R is zero, or vice versa, you will also never have repulsion of the spheres. The spheres connected by the wire with zero resistance will soon discharge, so you've ended any attraction, and never achieved repulsion.
Is this a trick, where you are postulating negative R?
Albert,"just after", that is in the first moment. What will happen further is not subject of question.
Viktor, okay, so in the first moment, you have max current flow, and the spheres are still fully charged. Maximum attraction of spheres, maximum repulsion of wires!
The problem may seem very simple and banal. If it were thus.! For those who simply have found the spheres potential and then expressed initial currents in the long wires and calculated Ampere forces - I give the following advice. Turn the lower wire 90 deg around its left sphere in the plane clockwise and then determine the Ampere forces' directions at least. They will be perpendicular each other. III-rd Newton's law, forever!
To save the III-rd Newton's law we are to take into account displacement current due to the discharges of the spheres in couple. But this current for any sphere shouldn't generate magnetic field due to spherical symmetry of decreasing electric field of the sphere and superposition principle for the fields involved. It means that spherical symmetry should disappear just like in case of the magnetic field of quasi-spherical Earth.
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