Final answer:
To find the total stored energy of a sphere with a uniform volume charge density, use the formula Wₑ = 2/3 * P₀ * πa³, where P₀ is the charge density and a is the radius of the sphere.
Step-by-step explanation:
To find the total stored energy of a sphere with a uniform volume charge density, we can use the formula given as Wₑ = 1/2∫ᵥₒₗ ρᵥVdv, where ρ is the charge density.
Since the sphere has a uniform volume charge density, we can express it as ρ = P₀, where P₀ is the charge density in C/m³.
The integral in the formula represents integrating the charge density over the volume of the sphere. Since the charge density is constant, we can take it out of the integral and integrate over the volume of the sphere.
The volume of a sphere is given by V = (4/3)πa³, where a is the radius of the sphere.
Substituting the values into the formula, we get Wₑ = 1/2 * P₀ * V
Wₑ = 1/2 * P₀ * (4/3)πa³
Wₑ = 2/3 * P₀ * πa³