Lec3 - Winter 2008 Lecture 3 Electric Forces and Fields due...

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Winter 2008 Lecture 3 Page 1 of 5 Lecture 3 Electric Forces and Fields due to surface charge distributions Surface Charges : In most practical circuits and systems, there will be metallic surfaces (for example, microstrip lines for IC and high frequency circuits, wires for low frequency systems, and metallic tubes for waveguides). To model a surface charge we need to define the distribution (or density function) of the charge on the line. The notation is typically ( 29 2 m C S r for surface charge density. This may be a constant or a function of position along the line. In this case we can replace the point charge with a piece of the line charge (the piece will have an area of dS ). This leads to dS dQ S r = . Since the charge is a continuous distribution, it will be more effective to integrate the contributions (summing up each contribution for a continuous function is the job of an integral). This allows us to write the electric field as = 3 4 1 R dS R E S r pe . As in the previous case of a line charge, the electric field due to a surface charge will depend upon the geometry of the problem as well as the distribution of the charge. The following example will illustrate the use of the above relationships. EXAMPLE A circular (radius “a”) of charged surface with uniform charge density S r is centered on the origin. Compute the electric field at a point ( 0, 0 ,z o ). Figure 1 The geometry for the surface charge SOLUTION: The first step is to set up the relationship. The charge distribution is a patch of surface so that the elemental charge is given by f r r r r d d dS dQ S S = = . The vector linking the charge (0 0 z) to the observer at (x o y o z o ) is ( 29 ( 29 ( 29 z o y x a z a y a x R ˆ ˆ ˆ + - + - = = z o a z a ˆ ˆ + - r r .
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