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(D) The electrostatic potential energy $U$ of a system of charges is given by $U = \sum \frac{k q_i q_j}{r_{ij}}$.In the given cube,there are $8$ char

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Depends on sign and value of $q$

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(D) The electrostatic potential energy $U$ of a system of charges is given by $U = \sum \frac{k q_i q_j}{r_{ij}}$.In the given cube,there are $8$ charges at the vertices and $1$ charge $q$ at the center.Let the side length of the cube be $a$.$1$. Interaction between vertex charges: There are $12$ edges of length $a$,$12$ face diagonals of length $\sqrt{2}a$,and $4$ body diagonals of length $\sqrt{3}a$.$2$. Interaction between the center charge $q$ and the $8$ vertex charges: Each vertex charge is at a distance of $\frac{\sqrt{3}a}{2}$ from the center.Given the arrangement of charges at the vertices ($4$ are $+Q$ and $4$ are $-Q$),the total potential energy calculation involves summing all pair interactions.Since the net charge at the vertices is $4Q - 4Q = 0$,and the symmetry of the arrangement leads to a specific potential energy value,the final expression depends on the value of $q$. Therefore,the total electrostatic potential energy of the system depends on the sign and value of $q$.

Nine point charges are placed on a cube as shown in the figure. The charge $q$ is placed at the body centre whereas all other charges are at the vertices. The electrostatic potential energy of the system will be

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