Electrostatic Potential and Capacitance - Result Question 20

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22. Each corner of a cube of side $l$ has a negative charge, $-q$. The electrostatic potential energy of a charge $q$ at the centre of the cube is [2002]

======= ####22. Each corner of a cube of side $l$ has a negative charge, $-q$. The electrostatic potential energy of a charge $q$ at the centre of the cube is [2002]

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(b) $\frac{\sqrt{3} q^{2}}{4 \pi \varepsilon_0 l}$

(c) $\frac{4 q^{2}}{\sqrt{2} \pi \varepsilon_0 l}$

(d) $-\frac{4 q^{2}}{\sqrt{3} \pi \varepsilon_0 l}$

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Answer:

Correct Answer: 22. (d)

Solution:

  1. (d) Length of body diagonal $=\sqrt{3} \ell$

$\therefore$ Distance of centre of cube from each corner

$r=\frac{\sqrt{3}}{2} \ell$

P.E. at centre

$=8 \times$ Potential Energy of the charge $(+q)$ due to charge $(-q)$ at one corner

$=8 \times \frac{K q \times(-q)}{r}$

$=8 \times \frac{1}{4 \pi \varepsilon_0 \sqrt{3} l} \times 2 \times q \times(-q)=\frac{-4 q^{2}}{\sqrt{3} \pi \varepsilon_0 l}$