Problem on Magnetic force on current carrying wire I recently came across this question which seems quite interesting

Two long wires with negligible resistance are shunted at their end by a large resistor R  and at the other ends are connected to a source of constant voltage. The radius of the cross section of each wire is smaller than the distance between the axes of the two wires by a factor of 20 . If the value of the resistance R , at which resultant force of interaction of the wires vanishes, then R is


I can find the force between the wires due to current I flowing through them but have no idea how to proceed further. I don't understand why the resultant force would become zero too. It would be great if someone could help me out and post a detailed solution.
 A: Since this is a homework-style problem, I will not provide the complete answer, I will just address the conceptual question: "How could the resultant force become zero?"
I suspect the idea is that the combination of the two long wires behaves as a capacitor, with each wire being one "plate". As a result, there is an attractive electric field generated between these two wires (indicated on your diagram by the "$E$") which could -- given some value of resistance -- cancel out the repulsive force due to the magnetic attraction of the wires.
So, in order to solve this problem, you need to find out the capacitance of such a configuration, using the simple formula $$C = \frac{A \epsilon_0}{\ell},$$ where $A$ is the area of the "plate" and $\ell$ the separation between them. (The formula above is usually used when the plates are flat. I do not know of any special formula when the plates are "curved" as in this case, but I assume that since the wires are quite far apart, you could approximate the "cylinders" of radius $r$ as bring cubes of width $r$.)
Given the plates are at a fixed voltage, knowing the capacitance would allow you to calculate the total charge on the plates. From here, it's a simple job to find the attractive electric force between two charged wires, and equate it to the repulsive magnetic force to find the critical value of $R$ at which these two are equal.
Good luck!
