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PHY 712 Electrodynamics 9-9:50 AM MWF Olin 105 Plan for Lecture 3: Reading: Chapter 1 in JDJ Review of electrostatics with one-dimensional examples Poisson and Laplace Equations Green’s Theorem and their use in electrostatics 1/18/2019 PHY 712 Spring 2019 -- Lecture 3

1/18/2019 PHY 712 Spring 2019 -- Lecture 3

Poisson and Laplace Equations We are concerned with finding solutions to the Poisson equation: and the Laplace equation: The Laplace equation is the “homogeneous” version of the Poisson equation. The Green's theorem allows us to determine the electrostatic potential from volume and surface integrals: 1/18/2019 PHY 712 Spring 2019 -- Lecture 3

Poisson equation -- continued 1/18/2019 PHY 712 Spring 2019 -- Lecture 3

General comments on Green’s theorem This general form can be used in 1, 2, or 3 dimensions. In general, the Green's function must be constructed to satisfy the appropriate (Dirichlet or Neumann) boundary conditions. Alternatively or in addition, boundary conditions can be adjusted using the fact that for any solution to the Poisson equation, other solutions may be generated by use of solutions of the Laplace equation 1/18/2019 PHY 712 Spring 2019 -- Lecture 3

“Derivation” of Green’s Theorem 1/18/2019 PHY 712 Spring 2019 -- Lecture 3

“Derivation” of Green’s Theorem 1/18/2019 PHY 712 Spring 2019 -- Lecture 3

1/18/2019 PHY 712 Spring 2019 -- Lecture 3

1/18/2019 PHY 712 Spring 2019 -- Lecture 3

1/18/2019 PHY 712 Spring 2019 -- Lecture 3

Comment about the example and solution This particular example is one that is used to model semiconductor junctions where the charge density is controlled by introducing charged impurities near the junction. The solution of the Poisson equation for this case can be determined by piecewise solution within each of the four regions. Alternatively, from Green's theorem in one-dimension, one can use the Green's function 1/18/2019 PHY 712 Spring 2019 -- Lecture 3

Notes on the one-dimensional Green’s function 1/18/2019 PHY 712 Spring 2019 -- Lecture 3

Construction of a Green’s function in one dimension 1/18/2019 PHY 712 Spring 2019 -- Lecture 3

Summary 1/18/2019 PHY 712 Spring 2019 -- Lecture 3

One dimensional Green’s function in practice This expression gives the same result as previously obtained for the example r(x) and more generally is appropriate for any neutral charge distribution. 1/18/2019 PHY 712 Spring 2019 -- Lecture 3

1/18/2019 PHY 712 Spring 2019 -- Lecture 3

1/18/2019 PHY 712 Spring 2019 -- Lecture 3

1/18/2019 PHY 712 Spring 2019 -- Lecture 3

1/18/2019 PHY 712 Spring 2019 -- Lecture 3