Question

Which of the following is the Laplace equation?

• A. \( \nabla^2 V = -\rho/\varepsilon \)
• B. \( \nabla^2 V = 0 \)
• C. \( \nabla^2 V = -4\pi\rho \)
• D. \( \nabla^2 V = -4\pi\sigma \)

Hint:

Laplace equation is a special case of Poisson’s equation when \( \rho = 0 \). Remember: \( \nabla^2 V = 0 \) in charge-free regions.

Answer 1

The Correct Option is B

Step 1: Understand Laplace's equation
Laplace’s equation is a special case of Poisson’s equation: \[ \nabla^2 V = -\frac{\rho}{\varepsilon} \] Where:
- \( V \) is the electric potential,
- \( \rho \) is the charge density,
- \( \varepsilon \) is the permittivity.
Step 2: Conditions for Laplace's equation
When \( \rho = 0 \) (no free charges in the region), then Poisson’s equation reduces to Laplace’s equation: \[ \nabla^2 V = 0 \] This condition is often found in:
- Electrostatics in vacuum or homogeneous media without charges
- Solving for potential in a charge-free region
\[ \boxed{\nabla^2 V = 0} \]