Problem 2.34

Find the energy stored in a uniformly charged solid sphere of radius R and charge q. Do it three different ways: (a) Use Eq. 2.43. You found the potential in Prob. 2.21. (b) Use Eq. 2.45. Don’t forget to integrate over all space. © Use Eq. 2.44. Take a spherical volume of radius a. What happens as a → ∞?

Problem 2.34 is asking to find the energy stored in a uniformly charged solid sphere of radius R and charge q, using three different equations.

(a) Using Eq. 2.43, the energy stored in a uniformly charged solid sphere can be found by multiplying the potential energy by the charge, which is given by: E = qV Where V is the potential energy found in Prob. 2.21

(b) Using Eq. 2.45, the energy stored in a uniformly charged solid sphere can be found by integrating the square of the electric field over all space. Since the sphere is uniformly charged, the electric field is given by: E = kq/r² (where k is Coulomb’s constant) So the energy stored in the sphere is given by: E = (1/2) * q² * (4πR³) / (3*εo)

© Using Eq. 2.44, the energy stored in a uniformly charged solid sphere can be found by taking the limit of the energy stored in a spherical volume of radius a as a approaches infinity. The energy stored in a spherical volume of radius a is given by: E = (1/2) * q² * (4πa³) / (3*εo) As a approaches infinity, the energy stored in the sphere becomes infinite, which is not physically possible. So, this method is not valid for finding the energy stored in a uniformly charged solid sphere.