EMLecture3 - Lecture 3 Notes 95.657 Electromagnetic Theory...

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Lecture 3 Notes, 95.657 Electromagnetic Theory I, Fall 2011 Dr. Christopher S. Baird, UMass Lowell 1. Method of Images - Use the method of images when one or more point charges are in the presence of boundary surfaces with constant potentials across them. - The method of images is very important because it can be used to find the Green function and then the Green function solution can be applied when the potential is not constant across the boundary. - Replace the boundary surfaces with image charges external to the region of interest in locations recommended by symmetries. - Make adjustable parameters out of the unknowns such as the image charge magnitude and location. - Vary the adjustable parameters until the boundary condition at the surface is met. - The solution to the original problem is the solution to the real charges and image charges. 2. Point Charge in the Presence of a Grounded Sphere - Center the sphere of radius a at the origin, the real charge q at y and the observation point at x . - The grounded conducting sphere has zero potential at the surface:  x = a = 0 - If we sketch the field lines of the sphere and point, we see they look like the fields created by two charges and no surfaces. This gives us motivation that the method of images will work. - Place the image charge q ' inside the sphere at the point y '. - The potential due to the two charges using Coulomb's law is: = 1 4  0 q x y 1 4   0 q' x y ' - Split up vectors in terms of magnitude and directions. Symmetry dictates that y and y ' point in the same direction, so that y ' = y ' y :  x = 1 4  0 q x x y y 1 4  0 q' x x y ' y - Apply the boundary condition:  x = a = 0 = 1 4  0 q a x y y 1 4  0 q' a x y ' y for all angles of the vector x q a x y y = q' a x y' y - This must be true for all directions of the vector x , so we can pick out two different directions to derive two independent equations which we can then solve for our two unknowns. - Let us first choose the vector x so that it points in the same direction as the vector y: x = y q a y y y = q' a y y ' y a q y y ' q' x
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q y a = q' a y' - Now pick the vector x to point perpendicular to the vector y and expand the magnitudes according to
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EMLecture3 - Lecture 3 Notes 95.657 Electromagnetic Theory...

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