Final answer:
To prove that the given electric potential satisfies Laplace's equation, we must substitute the potential into the Laplacian in spherical coordinates and show the result is zero.
Step-by-step explanation:
To demonstrate that the electric potential V=V₀(1-a²/ρ²)ρsinϕ satisfies Laplace's equation, we need to take the Laplacian of V and show that it equals zero. Laplace's equation in spherical coordinates is given by:
∇²V = ∂²V/∂r² + (2/r)∂V/∂r + (1/r²sinϕ)∂/∂ϕ(sinϕ∂V/∂ϕ) + (1/r²sinʸ²)∂²V/∂Φ² = 0
Substituting V into the equation and simplifying should provide a result of zero, confirming that the electric potential function satisfies Laplace's equation. The provided information and expressions related to electric fields and potentials from point charges, wires, capacitors, and uniform electric fields are additional context for understanding electric potentials and fields but are not directly used in the calculation.