Answer to Question #234117 in Electricity and Magnetism for Thato

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

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

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

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

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