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Method of Moments Solution for Charged Conducting Plates in Electrodynamics

This lecture outlines the Method of Moments (MoM) solution for a charged square conducting plate situated on the z=0 plane. The discussion centers around deriving the potential due to the surface charge density of the plate, which has no thickness. Key equations and considerations are demonstrated, including the integral equation for potential, the division of the plate into subsections, and evaluating the capacitance of the system. The importance of robust basis functions and the inner product in deriving the necessary matrix equations are emphasized, providing a comprehensive guide for students in computational electrodynamics.

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Method of Moments Solution for Charged Conducting Plates in Electrodynamics

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  1. EEE 431Computational Methods in Electrodynamics Lecture 17 By Dr. Rasime Uyguroglu Rasime.uyguroglu@emu.edu.tr

  2. Charged Conducting Plate Moment Method Solution

  3. Charged Conducting Plate/ MoM Solution • Consider a square conducting plate 2a meters on a side lying on the z=0 plane with center at the origin.

  4. Charged Conducting Plate/ MoM Solution • Let represent the surface charge density on the plate. • Assume that the plate has zero thickness.

  5. Charged Conducting Plate/ MoM Solution • Then, V(x,y,z): • Where;

  6. Charged Conducting Plate/ MoM Solution • Integral Equation: • When • This is the integral equation for

  7. Charged Conducting Plate/ MoM Solution • Method of Moment Solution: • Consider that the plate is divided into N square subsections. Define: • And let:

  8. Charged Conducting Plate/ MoM Solution • Substituting this into the integral equation and satisfying the resultant equation at the midpoint of each , we get:

  9. Charged Conducting Plate/ MoM Solution • Where: • is the potential at the center of due to a uniform charge density of unit amplitude over

  10. Charged Conducting Plate/ MoM Solution • Let : • denote the side length of each • the potential at the center of due to the unit charge density over its own surface.

  11. Charged Conducting Plate/ MoM Solution • So,

  12. Charged Conducting Plate/ MoM Solution • The potential at the center of can simply be evaluated by treating the charge over as if it were a point charge, so,

  13. Charged Conducting Plate/ MoM Solution • So, the matrix equation:

  14. Charged Conducting Plate/ MoM Solution • The capacitance:

  15. Charged Conducting Plate/ MoM Solution • The capacitance (Cont.):

  16. Charged Conducting Plate/ MoM Solution The charge distribution along the width of the plate Harrington, Field Computation by Moment Methods

  17. Moment Method/ Review • Consider the operator equation: • Linear Operator. • Known function, source. • Unknown function. • The problem is to find g from f.

  18. Moment Method/ Review • Let f be represented by a set of functions • scalar to be determined (unknown expansion coefficients. • expansion functions or basis functions.

  19. Moment Method/ Review • Now, substitute (2) into (1): • Since L is linear:

  20. Moment Method/ Review • Now define a set of testing functions or weighting functions • Define the inner product (usually an integral). Then take the inner product of (3) with each and use the linearity of the inner product:

  21. Moment Method/ Review • It is common practice to select M=N, but this is not necessary. • For M=N, (4) can be written as:

  22. Moment Method/ Review • Where,

  23. Moment Method/ Review • Or,

  24. Moment Method/ Review • Where,

  25. Moment Method/ Review • If is nonsingular, its inverse exists and . • Let

  26. Moment Method/ Review • The solution (6) may be either approximate or exact, depending upon on the choice of expansion and testing functions.

  27. Moment Method/ Review • Summary: • 1)Expand the unknown in a series of basis functions. • 2) Determine a suitable inner product and define a set of weighting functions. • 3) Take the inner products and form the matrix equation. • 4)Solve the matrix equation for the unknown.

  28. Moment Method/ Review • Inner Product: • Where:

  29. Moment Method/ Review • Inner product can be defined as:

  30. Moment Method/ Review • If u and v are complex:

  31. Moment Method/ Review • Here, a suitable inner product can be defined:

  32. Moment Method/ Review • Example: • Find the inner product of u(x)=1-x and v(x)=2x in the interval (0,1). • Solution: • In this case u and v are real functions.

  33. Moment Method/ Review • Hence:

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