Reading: Chapter in JDJ

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Presentation transcript:

Reading: Chapter 1 - 3 in JDJ PHY 712 Electrodynamics 9-9:50 AM MWF Olin 105 Plan for Lecture 4: Reading: Chapter 1 - 3 in JDJ Electrostatic potentials One, two, and three dimensions (Cartesian coordinates) Mean value theorem for the electrostatic potential 1/23/2019 PHY 712 Spring 2019 -- Lecture 4

1/23/2019 PHY 712 Spring 2019 -- Lecture 4

1/23/2019 PHY 712 Spring 2019 -- Lecture 4

Poisson Equation 1/23/2019 PHY 712 Spring 2019 -- Lecture 4

Poisson equation for one-dimensional system 1/23/2019 PHY 712 Spring 2019 -- Lecture 4

Beautiful method; but only works in one dimension. General procedure for constructing Green’s function for one-dimensional system using 2 independent solutions of the homogeneous equations Beautiful method; but only works in one dimension. 1/23/2019 PHY 712 Spring 2019 -- Lecture 4

1/23/2019 PHY 712 Spring 2019 -- Lecture 4

Orthogonal function expansion -- continued 1/23/2019 PHY 712 Spring 2019 -- Lecture 4

Example 1/23/2019 PHY 712 Spring 2019 -- Lecture 4

Example -- continued 1/23/2019 PHY 712 Spring 2019 -- Lecture 4

Example -- continued 1/23/2019 PHY 712 Spring 2019 -- Lecture 4

Orthogonal function expansions in 2 and 3 dimensions (See Eq. 3.167 in Jackson for example.) 1/23/2019 PHY 712 Spring 2019 -- Lecture 4

Combined orthogonal function expansion and homogeneous solution construction of Green’s function in 2 and 3 dimensions. For the two and three dimensional cases, we can use this technique in one of the dimensions in order to reduce the number of summation terms. These ideas are discussed in Section 3.11 of Jackson. 1/23/2019 PHY 712 Spring 2019 -- Lecture 4

Green’s function construction -- continued 1/23/2019 PHY 712 Spring 2019 -- Lecture 4

1/23/2019 PHY 712 Spring 2019 -- Lecture 4

Example: b a Two dimensional box with sides a and b with boundary conditions: F(0,y)=F(a,y)=F(x,0)=F(x,b)=0 Know this term Don’t know this term 1/23/2019 PHY 712 Spring 2019 -- Lecture 4

Example: b a Two dimensional box with sides a and b with boundary conditions: F(0,y)=F(a,y)=F(x,0)=F(x,b)=0 1/23/2019 PHY 712 Spring 2019 -- Lecture 4

, 1/23/2019 PHY 712 Spring 2019 -- Lecture 4

Green’s function construction -- continued b a 1/23/2019 PHY 712 Spring 2019 -- Lecture 4

b a 1/23/2019 PHY 712 Spring 2019 -- Lecture 4

1/23/2019 PHY 712 Spring 2019 -- Lecture 4

1/23/2019 PHY 712 Spring 2019 -- Lecture 4

1/23/2019 PHY 712 Spring 2019 -- Lecture 4

Summary: Mean value theorem 1/23/2019 PHY 712 Spring 2019 -- Lecture 4

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