Physics 604 Electromagnetic Theory I G A Krafft

Physics 604 Electromagnetic Theory I G. A. Krafft Jefferson Lab Professor of Physics Old Dominion University 09 -09 -10 Physics 604 Electromagnetic Theory I

Work • Will seldom use this expression to find potential: Solve a differential equation! • Moving a charge q from location to requires • Falling through a potential difference causes energy gain to a positive charge • Potential measured in volts [V] or [Nt m/C] Physics 604 Electromagnetic Theory I

Equations for Potential • Poisson Equation • Laplace Equation (ρ = 0) • Verify Solution with vanishing Boundary Conditions at infinity Physics 604 Electromagnetic Theory I

• Equation for scalar potential • Now • And Physics 604 Electromagnetic Theory I

Solution to Poisson Equation • Representation • First Example of a Green Function. Made rigorous in the mathematical theory of distributions. Physics 604 Electromagnetic Theory I

Green’s Identities • Green’s First Identity • Green’s Second Identity Physics 604 Electromagnetic Theory I

Formal Solution Physics 604 Electromagnetic Theory I

Uniqueness of Solutions • Dirichlet Boundary Conditions • Neumann Boundary Conditions • Consider two different solutions Φ 1 and Φ 2, and their difference U = Φ 2 - Φ 1 Physics 604 Electromagnetic Theory I

Solution with Specified BCs • Use Green function method to solve Poisson Equation with specified BCs. Physics 604 Electromagnetic Theory I

Dirichlet Conditions Physics 604 Electromagnetic Theory I