Poisson Equation - Edubirdie

University:
Santa Fe College
Course:
ETS 2527C | Electromechanical Components and Mechanisms
Academic year:

2024
Views:

64

Pages:

2
Author:

Jabin F.

Poisson eqn. ∇⋅E = ρ/ε₀ ∇⋅E = 0 ⇒ E = -∇V ⇒ ∇²V = -ρ/ε₀ If ρ = 0 (region is charge free) ∇²V = 0 → Laplace equation When V or φ ∝ 1/r ⇒ E ∝ 1/r² ∇⋅(1/r) = 4πδ³(r) ∇²(1/r) = -4πδ³(r) (charge density) at r = 0 ∫δ³(r) = 1 ∫δ³(r) = 0 at r ≠ 0 ∫δ³(r) = 0, ∫δ³(r) = 1 outside the plane r ≠ 0 δ³(r) = 0 ∫δ³(r) = 1 f(r)δ³(r) = f(0)δ³(r) V = 1/4πε₀ ∫ (ρ/r') dτ' ∇²V = -ρ/ε₀ E = -∇V V = -∫ E⋅dl E' = 1/(4πε₀r³) DE= ∇ XE=0 60 Ide J = 1/(4πε₀) ∇²V = -ρ/ε₀

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