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Superposition of Electric Fields

A charge Q ...

Question

A charge $Q$ located at a point $\to r$ is in equilibrium under the combined electric field of three charges $q_{1}, q_{2}, q_{3}$ . If the charges $q_{1}, q_{2}$ are located at points ${\to r}_{1}$ and ${\to r}_{2}$ respectively, find the direction of the force on $Q$ , due to $q_{3}$ in terms of $q_{1}, q_{2}, {\to r}_{1} {\to r}_{2}$ and $\to r$ .

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Solution

${\to F}_{1} + {\to F}_{2} + {\to F}_{3} = 0$

$\Rightarrow \frac{1}{4 π ϵ_{0}} \frac{Q q_{1}}{| \to r - {\to r}_{1} |^{3}} (\to r - {\to r}_{1}) + \frac{1}{4 π ϵ_{0}} \frac{Q q_{2}}{| \to r - {\to r}_{2} |^{3}} (\to r - {\to r}_{2}) + \frac{1}{4 π ϵ_{0}} \frac{Q q_{3}}{| \to r - {\to r}_{3} |^{3}} (\to r - {\to r}_{3}) = 0$

$\Rightarrow \frac{q_{1}}{| \to r - {\to r}_{1} |^{3}} (\to r - {\to r}_{1}) + \frac{q_{2}}{| \to r - {\to r}_{2} |^{3}} (\to r - {\to r}_{2}) = - \frac{q_{3}}{| \to r - {\to r}_{3} |^{3}} (\to r - {\to r}_{3})$

$\Rightarrow \to r - {\to r}_{3} = - \frac{| \to r - {\to r}_{3} |^{3}}{q_{3}} [\frac{q_{1} (\to r - {\to r}_{1})}{| \to r - {\to r}_{1} |^{3}} + \frac{q_{2} (\to r - {\to r}_{2})}{| \to r - {\to r}_{2} |^{3}}]$

Direction of force on $Q$ due to $q_{3}$ along $(\to r - {\to r}_{3})$ is given by

$_{3}$ = $- K Q [\frac{q_{1}}{| \to r - {\to r}_{1} |^{3}} ({\to r}_{1} - \to r) + \frac{q_{2}}{| \to r - {\to r}_{2} |^{3}} ({\to r}_{2} - \to r)]$ .

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