Electrostatics

JEE Main level, single correct, medium

206 of 311

Question 206 of 311

Four equal point charges QQ each are placed in the xyxy plane at (0,2),(4,2),(4,−2)\left(0, 2\right),\left(4, 2\right),\left(4, -2\right) and (0,−2)\left(0, -2\right) . The work required to put a fifth charge QQ at the origin of the coordinate system will be:

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Answer: (C) Q24πϵ0(1+15)\dfrac{{Q}^{2}}{4\pi {ϵ}_{0}}\left(1+\dfrac{1}{\sqrt{5}}\right)

Work done in bringing the fifth charge at origin is equal to change in the potential energy of the system of charges.
Wext=ΔU{W}_{ext}=\Delta U=Uf−Ui={U}_{f}-{U}_{i} if Ui=0{U}_{i}=0 (assumed)

Wext=Uf−0{W}_{ext}={U}_{f}-0=[KQ22+KQ22+KQ225+KQ225]=\left[\dfrac{K{Q}^{2}}{2}+\dfrac{K{Q}^{2}}{2}+\dfrac{K{Q}^{2}}{2\sqrt{5}}+\dfrac{K{Q}^{2}}{2\sqrt{5}}\right]
Hence, the work done in bringing the fifth charge at the origin is
Wext=Q24πϵ0[1+15]{W}_{ext}=\dfrac{{Q}^{2}}{4\pi {\epsilon }_{0}}\left[1+\dfrac{1}{\sqrt{5}}\right].

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