Energy stored

Electricity and Magnetism Level 5

Consider the following system of charge distribution of two equally, oppositely and uniformly charged clouds are brought close to overlap each other till only small very thin charged caps are left on either side as shown in figure.

The distance between their centers is \(X\) and \( X << R\)

Find the total electrostatic energy stored in the system 3.

If it is given by the expression \( \dfrac {a {\rho}^{b} R^c X^d \pi } {e \epsilon } \) for positive integers \(a,b,c,d,e\) with \(a,e\) coprime, what is the value of \(a+b+c+d+e\)?

Hint: Energy density in electric field is given by \( \dfrac { \epsilon E^2}{2} \).

Details and Assumptions:

  • \( \rho \) stands for charge per unit volume.

  • \(X\) is the distance between the centers in configuration 3.

  • \(R\) is the radius of each cloud .


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