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Two parallel conducting wires exert a force on each other since they are in a magnetic field. However, do these wires exert an electric force on each other? How is Coulomb's law of electrostatics applicable here if so as 1) the charges are in motion and 2) the conductors are electrically neutral.

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The two wires do not exert any electric forces between them. The two wires are electrically neutral. If you consider any electron in any wire, and calculate the net force due to all the positive and negative charges in both the wires, it will turn out to be zero. The electrons in the wires move only due to the applied voltage of battery(or any other). Thus the final answer is, Coulomb's law can be applied, whose result will be zero. I think this solved you problem.

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