Question

4.1 What are the significant physical differences between Poisson’s and Laplace’s equations.

4.2 The flux of an electric field is given by Gauss’s Law: ∇ ∙ 𝑬 = 𝜌 𝜀0 . Use this to determine Poisson’s equation. (Hint: what is the Laplacian of the potential field 𝜙?) 

Answer 1

4.1 The Laplace’s equation describes electric field in free space without charges. The Poisson’s equation describes electric field in space at the presence of charges.

4.2

$$\nabla\cdot {\bf E}=\rho/\epsilon_0$$

$${\bf E}=-\nabla \phi$$

$$\nabla\cdot (-\nabla \phi)=\rho/\epsilon_0$$

$$\nabla^2\phi=-\rho/\epsilon_0$$