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Unformatted text preview: 7 Poisson’s and Laplace’s equations Summarizing the properties of electrostatic fields we have learned so far, they satisfy the laws of electrostatics shown in the margin and, in addition, Laws of electrostatics: ∇· E = ρ/ o ∇× E = 0 E =-∇ V as a consequence of ∇× E = 0 . • Using these relations, we can re-write Gauss’s law as ∇· E =-∇· ( ∇ V ) = ρ o , 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 – A special case of Poisson’s equation corresponding to having ρ ( x, y, z ) = 0 everywhere in the region of interest is ∇ 2 V = 0 . (Laplace’s eqn) 1 Focusing our attention first on Laplace’s 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 = ˆ 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 , it follows that Laplace’s 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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This note was uploaded on 06/20/2011 for the course ECE 329 taught by Professor Kim during the Spring '08 term at University of Illinois, Urbana Champaign.
- Spring '08