Lecture 7 - 7 Poisson's and Laplace's equations Summarizing...

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7Po isson ’sandLap lace ’sequat ions Summarizing the properties of electrostatic Felds we have learned so far, they satisfy the constraints ∇· D = ρ 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 Gauss’s law as ∇· D = ± o ∇· E = - ± o ∇· ( V )= ρ, from which it follows that 2 V = - ρ ± o , (Poisson’s eqn) where 2 V 2 V ∂x 2 + 2 V ∂y 2 + 2 V ∂z 2 is known as Laplacian of V . Poisson’s eqn: 2 V = - ρ ± o Laplace’s eqn: 2 V =0 Aspec ia lcaseo fPo isson ’sequat ioncorrespond ingtohav
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Focusing our attention frst on Laplace’s equation, we note that the equation can be used in charge ±ree-regions to determine the electrostatic potential V ( x,y,z ) by matching it to specifed potentials at boundaries as illustrated in the ±ollowing examples: z x y z = d =2m V ( d )= - 3V 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 inFnite extents in x and y directions but separated in z direction by a Fnite distance of d =2 m(asshowninthemarg in) .Theconduct ingp lateshavenon-zerosurfacecharge densities (to be determined in Example 2), which are known to be responsible for an electrostatic Feld 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 Laplace’s 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 ,itfo l lowsthatLap lace ’sequat ions imp l iFesas 2 V = 2 V ∂x 2 + 2 V ∂y 2 +
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This note was uploaded on 05/05/2010 for the course ECE ece 442 taught by Professor Kim during the Spring '10 term at University of Illinois at Urbana–Champaign.

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Lecture 7 - 7 Poisson's and Laplace's equations Summarizing...

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