Electrostatics and Green's Theorem in Electrodynamics
Explore electrostatics, Poisson and Laplace equations, and Green's Theorem in electromagnetism. Learn how these principles help determine electrostatic potential and generate solutions for various equations.
Download Presentation
Please find below an Image/Link to download the presentation.
The content on the website is provided AS IS for your information and personal use only. It may not be sold, licensed, or shared on other websites without obtaining consent from the author. If you encounter any issues during the download, it is possible that the publisher has removed the file from their server.
You are allowed to download the files provided on this website for personal or commercial use, subject to the condition that they are used lawfully. All files are the property of their respective owners.
The content on the website is provided AS IS for your information and personal use only. It may not be sold, licensed, or shared on other websites without obtaining consent from the author.
E N D
Presentation Transcript
PHY 712 Electrodynamics 9-9:50 AM MWF Olin 103 Plan for Lecture 3: Reading: Chapter 1 in JDJ 1. Review of electrostatics with one- dimensional examples 2. Poisson and Laplace Equations 3. Green s Theorem and their use in electrostatics 1/21/2015 PHY 712 Spring 2015 -- Lecture 3 1
1/21/2015 PHY 712 Spring 2015 -- Lecture 3 2
1/21/2015 PHY 712 Spring 2015 -- Lecture 3 3
Poisson and Laplace Equations We are concerned with finding solutions to the Poisson equation: 2 ( ) P r r ( ) = 0 and the Laplace equation: = 2 ( ) r 0 L 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: 3 1 ( ) ( ) ( , ) 4 1 ( , ) 4 = + r r r r d r G V 0 2 r r r ( ) r ( ) r ( , ) r r . d r G G S 1/21/2015 PHY 712 Spring 2015 -- Lecture 3 4
General comments on Greens theorem 1 ( ) 4 1 4 = ( ) ( , ) G r + 3 r r r d r V 0 2 r ( , ) r r ( ) r ( ) r ( , ) r r . d r G G S 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 ( ) ( ) ( ),for any constant . P L C = + r r r r ( ) P C 1/21/2015 PHY 712 Spring 2015 -- Lecture 3 5
Derivation of Greens Theorem r r ( ) = 2 Poisson equation: ( ) 0 = 2 3 ( , ) r r r r Green's relation: ' 4 ( ' ). G = 3 2 A A r Divergence theorm: d r d r V g S f ( ) ( ) ( ) ( ) ) = = A r r r r Let f g ( ( ) ( ) r ( ) r ( ) r ( ) r ( ) r ( ) r ( ) r ( ) r 3 2 r d r f g g f d r f g g f V S ( ) ( ) r ( ) r ( ) r ( ) r 3 2 2 d r f g g f V 1/21/2015 PHY 712 Spring 2015 -- Lecture 3 6
Derivation of Greens Theorem r r ( ) = 2 Poisson equation: ( ) 0 = 2 3 ( , ) r r r r Green's relation: ' 4 ( ' ). G ( ) ( ) ( ) r ( ) r ( ) r ( ) r ( ) r ( ) r ( ) r ( ) r = 3 2 2 2 r d r f g g f d r f g g f V S ( ) r ( ) r ( ) = ( ) r , ' r r f g G 1 = ( ) ( , ) G r + 3 ( ) r r r d r 4 V 0 1 2 r ( , ) r r ( ) r ( ) r ( , ) r r . d r G G 4 S 1/21/2015 PHY 712 Spring 2015 -- Lecture 3 7
1/21/2015 PHY 712 Spring 2015 -- Lecture 3 8
1/21/2015 PHY 712 Spring 2015 -- Lecture 3 9
1/21/2015 PHY 712 Spring 2015 -- Lecture 3 10
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 = = ( ) x ( , ) ( ) G x x where ( , ) G x x 4 x dx x 4 0 should be take as the smaller of and '. x x x 1/21/2015 PHY 712 Spring 2015 -- Lecture 3 11
Notes on the one-dimensional Greens function The Green's function for the one-dimensional Poisson equation can be defined as a solution to the equ ( , ) Here the factor o is not really necessary, b f ensures con sis tency with your text's treatment of the 3-dimensional case. The meaning of this expression is that ' is held fixed while taking the derivative with respect to . x = 2 ation: 4 ( ) G x x x x 4 ut x 1/21/2015 PHY 712 Spring 2015 -- Lecture 3 12
Construction of a Greens function in one dimension Co nsider two independent solution = = s to the h omogeneous equation 2 ( ) 0 1 or 2. Let 4 W ix where i = ( , ) G x x ( ) ( ). x x 1 2 This notation means that smaller of and ' and x sho sho uld be taken as the uld be taken as the larger. x x x is defined as the "Wronskin": ( ) ( ) x dx W ( ) x dx d x d ( ) x . 1 2 W 2 1 1/21/2015 PHY 712 Spring 2015 -- Lecture 3 13
Summary = 2 ( , ) G x x 4 ( ) x x 4 W x = ) ( ( , ) G x x ( ) x x 1 2 ( ) dx ( ) x dx d d ( ) x ( ) x 1 2 W 2 1 ( , ) dx ( , ) dx dG x x dG x x = t t 4 x x = + x x = 1/21/2015 PHY 712 Spring 2015 -- Lecture 3 14
One dimensional Greens function in practice 1 = ( ) x ( , ') ( ') G x x ' x dx 4 0 1 x + = ( , ) ( ) G x x ( , ) ( ) G x x x dx x dx 4 x 0 For the one-dimensional Poisson equation, we can construct the Green's function by choosing ( ) 0 This expression gives the same result as previously obtained for the example (x) and more generally is appropriate for any neutral charge distribution. = = = and ( ) x 1; 1: x x W 1 2 1 x = ( ) x dx + ( ) x dx ( ) . x x x x 1/21/2015 PHY 712 Spring 2015 -- Lecture 3 15
1/21/2015 PHY 712 Spring 2015 -- Lecture 3 16
1/21/2015 PHY 712 Spring 2015 -- Lecture 3 17
1/21/2015 PHY 712 Spring 2015 -- Lecture 3 18
1/21/2015 PHY 712 Spring 2015 -- Lecture 3 19
