Charge redistribution

Consider a neutral conducting sphere of radius \(R=1~\text{cm}\). It is clear that if there are no other charges nearby, the electrostatic energy of the sphere is zero. However, if a charge \(q\) is placed in the vicinity of the sphere, the charges on its surface redistribute and, as a consequence, the sphere acquires an electrostatic potential energy \(\Delta{U}\). Determine the interaction energy \(\Delta{U}\) in Joules among the charges on the surface of the sphere if a charge \(q=1~\mu \text{C}\) is placed at a distance \(d=3~\text{cm}\) from its center.

Details and assumptions

\[ k=\frac{1}{4\pi \epsilon_{0}}= 9\times 10^{9}~\text{m/F}\]

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