Two mutually perpendicular long straight conducting rods carrying uniformly distributed charges of linear charge densities λ1 and λ2 are positioned at a distance a from each other. If the force between the rods is found to be Fnet =λ1λ2nε0 find the value of n.

Let us make two-dimensional view of the situation. because of symmetry we can say the force on rod2 λ2 will be along negative y direction. Let us consider an element of length dx charged. having charge dq.Force on the element, dF=E1dq=λ2πε0a2+x2λ2dxThe net force on rod ‘2’ Fnet =∫dFcos⁡θ    =∫λ12πε0a2+x2λ2dx⋅cos⁡θwhere cos⁡θ=aa2+x2⇒ Fnet=λ1λ22πε0a∫−∞∞ dxa2+x2=λ1λ2a2πε0×1atan−1⁡xa−∞∞⇒ Fnet=λ1λ22ε0