Question

Two positive point charges q and 4q are at x = 0 and x = L, respectively, and free to move. A third charge is placed so that the entire three-charge system is in static equilibrium. What are the magnitude, sign, and x-coordinate of the third charge?

Short answer

The magnitude, sign and x-coordinate of the third charge$x=\frac{L}{3}andQ=-\frac{4q}{9}$

Step-by-step solution

Content Introduction

Suppose take $+neQ$at distance x from the origin. The equilbrium Q must be$-ne$.

Content Explanation

Force at $0.$$F=0$

$$\begin{gathered} k\frac{qQ}{x^2}+k\frac{Q4q}{L^2}=0 \\ Q=-\frac{4q}{L^2}{x}^2 \end{gathered}$$

Now force at P

$$\begin{gathered} k\frac{qQ}{x^2}+k\frac{Q4q}{{(L-x)}^2}=0 \\ (\frac{L}{x}-L)=2 \\ x=\frac{L}{3} \end{gathered}$$

Substitute the value of x

$$\begin{gathered} Q=-\frac{4q}{L^2}+\frac{L^2}{9} \\ Q=-\frac{4q}{9} \end{gathered}$$