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Question

Four charges $$ ^{+}q ,^{-}q $$ and $$ ^{-}q $$ are to be arranged respectively at the four corners of a square ABCD of side 'a'
(i) Find the work required to put together this arrangement. 
(ii) A charge qo is brought to the centre of the square, the four charges being held fixed. How much extra work is needed to do this


Solution

Work done in bringing charge +q at point A
$$ W_{A}= 0 $$
Work done in bringing charge -q to the point B

$$ W_{B}=W_{AB}= -q\times \dfrac{1}{4\pi \varepsilon _o }\dfrac{q}{a} $$ $$ = -q\times \dfrac{1}{4\pi \varepsilon _{o}}\dfrac{q^{2}}{a} $$
Work done in bring the charge +q to the point C
$$ W_{C}=W_{AC}+W_{BC} $$

$$ = q\times \dfrac{1}{4\pi \varepsilon _{o}}.\dfrac{q}{a\sqrt{2}}+q\times = (\dfrac{1}{4\pi \varepsilon _{o}}\dfrac{q}{a}) $$

$$  = \dfrac{1}{4\pi \epsilon _{o}}.\dfrac{q^{2}}{a\sqrt{2}}-\dfrac{1}{4\pi \varepsilon _{o}}.\dfrac{q^{2}}{a} $$

Work done in bringing a change -q to the point D
$$ W_{D}=W_{AD}+W_{BD}+W_{CD} $$

$$ = -q\times \dfrac{1}{4\pi \epsilon _{o}}\dfrac{q}{a}+(-q)(\dfrac{1}{4\pi\varepsilon _{0}}\dfrac{-q}{a\sqrt{2}}) $$  $$ + (-q)\times \dfrac{1}{4\pi \varepsilon _{o}}.\dfrac{q}{a} $$

Total work done $$W= W_{A} $$ +$$ W_{B} $$+$$ W_{C} $$+$$ W_{D} $$

$$ = 2\times \dfrac{1}{4\pi \varepsilon _{o}}\dfrac{q^{2}}{a\sqrt{2}}-4\times \dfrac{1}{4\pi \varepsilon _{o}}\dfrac{q^{2}}{a}(\sqrt{2}-4) $$

Physics

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