Two parallel infinite line charges with linear charge densities $+\lambda \mathrm{C} / \mathrm{m}$ a

Question

Two parallel infinite line charges with linear charge densities $+\lambda \mathrm{C} / \mathrm{m}$ and $-3\lambda \mathrm{C} / \mathrm{m}$ are placed at a distance of $R$ in free space. At which point is the net electric field zero?

Professor PC Thomas Classes · Answer

(a) : The net electric field can only be zero outside the region between the wires and closer to the weaker charge ($+\lambda$). Let the null point be at a distance $x$ from the $+\lambda$ wire. The distance from the $-3\lambda$ wire will be $R+x$. Equating the magnitudes of the electric fields: $\frac{\lambda}{2\pi\varepsilon_0 x} = \frac{3\lambda}{2\pi\varepsilon_0 (R+x)}$. This yields $\frac{1}{x} = \frac{3}{R+x}$, so $R+x = 3x$, which gives $2x = R$, or $x = R/2$.