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Question

Consider the configuration of a system of four charges each of value $$ + q.$$ Find the work done by external agent in changing the configuration of the system from figure (a) to figure (b).
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Solution

$${U_1} = K{q^2}\left\{ {\dfrac{1}{a} + \dfrac{1}{{\sqrt 2 a}} + \dfrac{1}{a} + \dfrac{1}{a} + \dfrac{1}{{\sqrt 2 a}} + \dfrac{1}{a}} \right\}$$
$$ = K{q^2}\left\{ {\dfrac{4}{a} + \dfrac{2}{{\sqrt 2 a}}} \right\}$$
$${U_2} = K{q^2}\left\{ {\dfrac{1}{{\sqrt 2 a}} + \dfrac{1}{{2a}} + \dfrac{1}{{\sqrt 2 a}} + \dfrac{1}{{\sqrt 2 a}} + \dfrac{1}{{2a}} + \dfrac{1}{{\sqrt 2 a}}} \right\}$$
$${U_2} = K{q^2}\left\{ {\dfrac{2}{{2a}} + \dfrac{4}{{\sqrt 2 a}}} \right\}$$
$$W =  - \left\{ {{U_1} - {U_2}} \right\} =  - K{q^2}\left\{ {\dfrac{{4 + \sqrt 2 }}{a}} \right\} - K{q^2}\left\{ {\dfrac{{1 + 2\sqrt 2 }}{a}} \right\}$$
$$W =  - K{q^2}\left\{ {\dfrac{{4 + \sqrt 2  - 1 - 2\sqrt 2 }}{a}} \right\}$$
$$\boxed{W =  - K{q^2}\left\{ {\dfrac{{3 - \sqrt 2 }}{a}} \right\}}$$

Physics

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