Finding the energy of a charged sphere

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SUMMARY

The discussion centers on calculating the electric potential energy of a uniformly charged sphere. The correct potential inside the sphere is not constant; instead, it varies with radius, leading to an incorrect integral result when assuming a constant potential. The potential at a distance r from the center of the sphere is given by V(r) = KQ/(2R) for r < R, where K is Coulomb's constant, Q is the total charge, and R is the radius of the sphere. The error arises from the assumption of uniform potential throughout the sphere's volume.

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  • Understanding of electrostatics and electric potential
  • Familiarity with calculus, specifically integration techniques
  • Knowledge of Coulomb's law and electric field concepts
  • Basic principles of charge distribution in spherical bodies
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Eitan Levy
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Homework Statement
Let's say we have a charged sphere with a radius R and total charge Q with constant density over the sphere. Find the energy of the sphere.
Relevant Equations
V=KQ/R
In class we were taught that for spherical bodies we may use the formula below where the integral is done over the volume of the body. However, if we assume that the potential in infinity is 0, the potential inside the sphere is constant and equals KQ/R, where Q is the total charge of the sphere. If I try to do the integral from r=0 to r=R, while plugging the constant density Q/(4/3*pi*R^3)) and dV=4*pi*r^2*dr, I get a result of KQ^2/(2R). Online I can see it is not right.

What is the problem?
 

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If the charge is spread uniformly throughout the volume of the sphere, then V is not constant inside the sphere.
 
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