Question: (II) You are given two unknown point charges, \({Q_1}\) and \({Q_2}\). At a point on the line joining them, one-third of the way from \({Q_1}\) to \({Q_2}\), the electric field is zero (Fig. 16–59). What is the ratio Q₁/Q₂ ?

FIGURE 16–59 Problem 34.

The electric field at any location in a particular region due to one or more charges is described as the force working on a unit charge located at that point.

The point where the distance is zero is at a distance\({r_1} = \frac{l}{3}\)from charge \({Q_1}\) and a distance \({r_2} = l - \frac{l}{3} = \frac{{2l}}{3}\) from charge \({Q_2}\).

If the electric field is zero, then you can write:

\(\begin{aligned}{c}\frac{{k{Q_1}}}{{r_1^2}} &= \frac{{k{Q_2}}}{{r_2^2}}\\\frac{{{Q_1}}}{{{{\left( {\frac{l}{3}} \right)}^2}}} &= \frac{{{Q_2}}}{{{{\left( {\frac{{2l}}{3}} \right)}^2}}}\\\frac{{{Q_1}}}{{{Q_2}}} &= \frac{{{{\left( {\frac{l}{3}} \right)}^2}}}{{{{\left( {\frac{{2l}}{3}} \right)}^2}}}\\\frac{{{Q_1}}}{{{Q_2}}} = \frac{1}{4}\end{aligned}\)

Thus, the ratio of Q₁/Q₂ is 1/4.