329lect07 - 7 Poissons and Laplaces equations Summarizing...

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Unformatted text preview: 7 Poissons and Laplaces equations Summarizing the properties of electrostatic fields we have learned so far, they satisfy the constraints D = and E = 0 where D = o E ; in addition E =- V as a consequence of E = 0 . Combining the equations above, we can re-write Gausss law as D = o E =- o ( V ) = , from which it follows that 2 V =- o , Poissons eqn where 2 V 2 V x 2 + 2 V y 2 + 2 V z 2 is known as Laplacian of V . A special case of Poissons equation corresponding to having ( x,y,z ) = everywhere in the region of interest is 2 V = 0 Laplaces eqn . 1 Focusing our attention first on Laplaces equation, we note that the equation can be used in charge free-regions to determine the electrostatic potential V ( x,y,z ) by matching it to specified potentials at boundaries as illustrated in the following examples: z x y z = d = 2 m V ( d ) =- 3 V V (0) = 0 z = 0 V ( z ) =? z V ( z ) V ( z ) = Az + B Example 1: Consider a pair of parallel conducting metallic plates of infinite extents in x and y directions but separated in z direction by a finite distance of d = 2 m (as shown in the margin). The conducting plates have non-zero surface charge densities (to be determined in Example 2), which are known to be responsible for an electrostatic field E = zE z measured in between the plates. Each plate has some unique and constant electrostatic potential V since neither E ( r ) nor V ( r ) can dependent the coordinates x or y given the geometry of the problem. Using Laplaces equation determine V ( z ) and E ( z ) between the plates if the potential of the plate at z = 0 is 0 (the ground), while the potential of the plate at z = d is- 3 V. Solution: Since the potential function V = V ( z ) between the plates is only dependent on z , it follows that Laplaces equation simplifies as 2 V = 2 V x 2 + 2 V y 2 + 2 V z 2 = 2 V z 2 = 0 ....
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329lect07 - 7 Poissons and Laplaces equations Summarizing...

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