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(C) The electrical potential energy $U$ of a charged sphere can be calculated by considering the work done in bringing small charge elements $dq$ from

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$\frac{1}{2} \frac{k Q^2}{R}$

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(C) The electrical potential energy $U$ of a charged sphere can be calculated by considering the work done in bringing small charge elements $dq$ from infinity to the surface of the sphere.Alternatively,we can use the average potential method.Initially,when there is no charge on the sphere,the potential $V_1 = 0$.When the total charge $Q$ is on the sphere,the potential $V_2 = \frac{k Q}{R}$.The average potential $V$ during the charging process is $V = \frac{V_1 + V_2}{2} = \frac{0 + \frac{k Q}{R}}{2} = \frac{k Q}{2 R}$.The total potential energy $U$ is the work done to charge the sphere,which is given by $U = \int_0^Q V(q) dq = \int_0^Q \frac{k q}{R} dq = \frac{k}{R} \left[ \frac{q^2}{2} \right]_0^Q = \frac{1}{2} \frac{k Q^2}{R}$.

$A$ total electric charge $Q$ is distributed on the surface of a sphere of radius $R$. What is the electrical potential energy of this system?

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