Question

# Three identical positive point charges are located at fixed points in space. Then charge $q_2$ is moved from its initial location to a final location as shown in the figure. Four different paths, marked (a) through (d), are shown. Path (a) follows the phortest line; path (b) takes $q_2$ around $q_3$; path (c) takes $q_2$ around $q_3$ and $q_1 ;$ path (d) takes $q_2$ out to infinity and then to the final location. Which path requires the least work? a) path (a) b) path (b) c) path (c) d) path (d) e) Thework is the same for all the paths.

Solution

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We will consider that in this physical situation, the electric potential due to a point charge at a distance of $r$ from the charge is given by

\begin{align*} V &= k ~ \dfrac{q}{r} \end{align*}

Therefore, for the system of point charges, the electric potential due to the system is given by

\begin{align*} V &= k ~ \sum_{i=1}^{n} ~ \dfrac{q_{i}}{r_{i}} \end{align*}

Also, we know that the work done is given by

\begin{align*} W &= - q ~ \Delta V \\ \\ &= - F \cdot d \end{align*}