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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