Final answer:
The student must apply Laplace's equation in spherical coordinates to the given potential, calculate the second derivatives with respect to r, θ, and φ, and then show if the sum of these derivatives equals zero, confirming that it satisfies Laplace's equation.
Step-by-step explanation:
The question asks to show that the electric potential V given by V=V0 (1 - ²/p²) ρ sinϕ satisfies Laplace's equation. To verify this, one must apply the Laplace's equation in spherical coordinates since the potential seems to be expressed in these coordinates. The general form of Laplace's equation in spherical coordinates is:
- ∇2V = ∂2V/∂r2 + (2/r)∂V/∂r + (1/r2)∂2V/∂θ2 + (cotθ/r2)∂V/∂θ + (1/r2sin2θ)∂2V/∂φ2 = 0
To prove the given potential satisfies the equation, we need to calculate the second derivatives of V with respect to r, θ (theta), and φ (phi), and insert them into Laplace's equation to see if the equation holds true. In this case, since there's no explicit dependence on θ (theta), and the potential is of the form V=V0 f(r) sinφ, many terms will disappear, simplifying the demonstration considerably.