Homogeneous Charge Density
For the special case of a homogeneous charge density ρ0, independent of position i.e. constant throughout the region of the material, the equation simplifies to:
The proof of this is immediate. Start with the definition of the charge of any volume:
Then, by definition of homogeneity, ρq(r) is a constant denoted by ρq, 0 (to differ between the constant and non-constant densities), and so by the properties of an integral can be pulled outside of the integral resulting in:
so,
The equivalent proofs for linear charge density and surface charge density follow the same arguments as above.
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