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PHY 712 Electrodynamics 9-9:50 AM MWF Olin 105 Plan for Lecture 3:

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Presentation on theme: "PHY 712 Electrodynamics 9-9:50 AM MWF Olin 105 Plan for Lecture 3:"— Presentation transcript:

1 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 Lecture 3

2 1/18/2019 PHY 712 Spring Lecture 3

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 Lecture 3

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

5 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 Lecture 3

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

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

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11 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 Lecture 3

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

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

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

15 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 Lecture 3

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