Answer to Question #234117 in Electricity and Magnetism for Thato

Question #234117

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?) 


1
Expert's answer
2021-09-07T09:50:26-0400

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

E=ρ/ϵ0\nabla\cdot {\bf E}=\rho/\epsilon_0

E=ϕ{\bf E}=-\nabla \phi

(ϕ)=ρ/ϵ0\nabla\cdot (-\nabla \phi)=\rho/\epsilon_0

2ϕ=ρ/ϵ0\nabla^2\phi=-\rho/\epsilon_0


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