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# A metal ball with diameter of a half a centimeter and hanging from an insulating thread is charged up with $1 \times 10^{10}$ excess electrons. An initially uncharged identical metal ball hanging from an insulating thread is brought in contact with the first ball, then moved away, and they hang so that the distance between their centers is 20 cm. Calculate the electric force one ball exerts on the other, and state whether it is attractive or repulsive. If you have to make any simplifying assumptions, state them explicitly and justify them.

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$\textbf{\textit{\underline{Knowns:}}}$

Since we have a metallic ''conductors'' balls which means that any excess charges are distributed uniformly on the surface, thus when the uncharged metallic ball touched the charged ball the excess charges on the first ball distributed all over the total surface of both spheres, and since both spheres are identical which means that both spheres have the same surface area thus the excess charges would be distributed evenly among the two spheres.

Hence after the contact each of the two sphere would have the same charge which is equal to half the initial charge on the first sphere, in other words '' the charges divided between the two sphere'' so each sphere has an excess of electrons of $\fbox{$0.5\times 10^{10} ~ \rm{e}$}$.

And since the two metallic sphere are conductors, thus the excess charges is distributed uniformly about the surface which means $\textbf{\textit{we can treat the charged sphere as a point charge located at the center of the sphere}}$, where it would produce an electric field the same way the point charge would normally produces.

And the distance between the center of the two sphere is $\fbox{20 cm}$, hence we can calculate the magnitude of the force between the two charged sphere, where it is given by the following equation

$\tag{1} \left| F \right|=q_1 \left| E_2 \right|= q_2 \left| E_1 \right|= \dfrac{1}{4\pi \varepsilon_o} ~ \dfrac{q_1 ~ q_2}{r^2}$

Where,

$q_1:$ is the charge on the first sphere.

$q_2:$ is the charge on the second sphere.

$r:$ is the distance between the centers of the two sphere.

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