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Physics Guru is an online education and competitive examinations preparation portal founded by Manish Verma, an IIT Madras alumnus, providing learning resources for competitive examinations.

The emphasis is on conceptual understanding, analytical thinking and problem solving rather than simply memorising formulas and standard methods.

The objective is to help students understand what they are learning and develop the ability to solve problems they have not seen before.

Potential Energy of a Charged Spherical Shell

Problem
What is the self-potential energy of a thin spherical shell of radius \(R\), mass \(M\), and charge \(Q\), assuming uniform surface distributions of mass and charge?
Solution

The total self-potential energy is the sum of gravitational and electrostatic self-energies.

1. Gravitational self-energy

Using the standard result for a thin spherical shell,

\[ U_g=-\frac{GM^2}{2R} \]
2. Electrostatic self-energy

Using the capacitance of an isolated spherical shell,

\[ C=4\pi\epsilon_0R \]

The electrostatic self-energy is

\[ \begin{aligned} U_e&=\frac{Q^2}{2C}\\[6pt] &=\frac{Q^2}{8\pi\epsilon_0R} \end{aligned} \]
3. Total self-potential energy

Adding the gravitational and electrostatic contributions,

\[ U=U_g+U_e \]
Final Answer
\[ \boxed{ U=\frac{1}{2R} \left( \frac{Q^2}{4\pi\epsilon_0}-GM^2 \right) } \]

In the animation below, the gravitational potential energy illustration is shown first in which particles assemble to form spherical shell coming from infinity. Subsequently, the electrostatic potential energy illustration is shown in which charges assemble on the spherical shell coming from infinity.