Consider two long wire with negligible resistance closed at one end of the resistance R (say a light bulb), and the other end connected to a battery (DC voltage). Cross-sectional radius of each wire is in $x = 100$ times smaller than the distance between the axes of the wires.
Question:
At which value of resistance R (in ohms) the resultant force of the interaction of parallel conductors disappears?
Answer
The magnetic force should be straightforward, you probably have a formula for it, or you can use the field strength from one wire, cross the current in the other to compute the force. The charge is more difficult, there you have two oppositely charged cyllinders separated by a distance. You can need to figure out the voltage difference between them as a function of the charge denisty per unit length of the system. As a first approximation assume the charge is evenly distributed around the circumference of your wire, then you should be able to compute the voltage difference between the two wires by integrating the electric field between them. I'm not going to do this (if it really is homework, you need to do it yourself). You should be able to use symmetry to reduce the amount of detailed math you need. If you are really ambitious, you could calculate the effect of uneven charge distribution on each wire, i.e. there will be more charge on the surface closer to the other wire than on the outside -or at least show the approximate size of the correction factor. Should be a fun exercise.
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