Problem 2.49

The energy can be calculated as:

where the volume element dV = 4/3 π r³. After integrating, we get:

the energy of the charged sphere can also be calculated using Gauss’s law, which states that the flux of the electric field through a closed surface is proportional to the charge enclosed within the surface.

Let’s consider a spherical surface of radius r centered at the origin, with the charged sphere inside it. The electric field E at a point outside the sphere is given by:

The flux of the electric field through the spherical surface is given by:

Φ = E * 4πr²

From Gauss’s law, we have:

So, the flux can be expressed as:

Comparing the expressions for Φ, we get:

Solving for k, we get:

The energy of the charged sphere can be calculated as:

where dV = 4πr² dr. After integrating, we get:

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