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Question

Two point charge $$q_1$$ and $$q_2$$ are kept r distance apart in a uniform external electric field $$\vec{E}$$. Find the amount of work done in assembling this system of charges.

Solution
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Let the charge $$q_1$$ travels $$r_1$$ distance and $$q_1$$ travels
The work done in bringing the charge $$q_1$$ in the field $$to$$
$$W_1 = F_1 \times r_1$$
$$=q_1E \times r_1$$
the work done in bringing the second charge $$\to$$
$$W_2 = F_2 \times r_2$$
$$= q_2 E \times r_2$$
and the work is also done to overcome the force of the charge on one-another.
$$W_3 = \dfrac{1}{4\pi\in_0}. \dfrac{q_1q_2}{r^2}$$
So, total work $$= q_1Er_1 + q_2Er_2 +\dfrac{1}{4\pi \in_0} \dfrac{q_1q_2}{r^2}.$$

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