The Poisson Equation for Electrostatics

Transcription

1 University of Puerto Rico - Mayagüez

2 Table of Contents 1 Derivation from Maxwell s Equations 2 3 4

3 Derivation from Maxwell s Equations Maxwell s Equations for Electrodynamics in differential form are: E = 1 ɛ 0 ρ( r, t) B = 0 E = B t B = µ 0 E J( r, t) + µ0 ɛ 0 t (Gauss s Law for Electricity) (1a) (Gauss s Law for Magnetism)(1b) (Faraday s Law of Induction) (1c) (Ampère-Maxwell s Law) (1d).

4 Using the Helmholtz Theorem and that B is divergenceless, the magnetic field can be expressed in terms of a vector potential, A: B = A (2) From this and Faraday s Law, Eq. (1c), the electric field can be expressed as: E = V A (3) t

5 Substiting into Gauss Law, Eq. (1a), results in: In the electrostatic case, it reduces to 2 V + t ( A) = 1 ɛ 0 ρ (4) V 2 V = 1 ɛ 0 ρ( r), (5) which is Poisson s equation. For a region of space in which there is no charge, we obtain Laplace s equation: V 2 V = 0 (6)

6 The Laplace equation in rectangular coordinates is 2 V x V y V z 2 = 0 (7) To solve by separation of variables we assume that: V (x, y, z) = X (x)y (y)z(z) (8) After substituting and diving, this results in: 1 d 2 X X (x) dx d 2 Y Y (y) dy d 2 Z Z(z) dz 2 = 0 (9)

7 In order for the result to hold for arbitrary values of the coordinates, each of the terms must be individually constant: where α 2 + β 2 = γ 2. 1 d 2 X X (x) dx 2 = α 2 (10a) 1 d 2 Y Y (y) dy 2 = β 2 (10b) 1 d 2 Z Z(z) dz 2 = γ 2, (10c)

8 If α 2 and β 2 are arbitrarily chosen to be positive, the solutions to the set of ODEs are then: X (x) = e ±iαx (11a) Y (y) = e ±iβy (11b) Z(z) = e ± α 2 +β 2z. (11c)

9 Derivation from Maxwell s Equations In order to determine α and β, we impose the boundary conditions on the potential. An example is shown in the figure. Yese J. Felipe

10 Since V = 0, for x = 0, y = 0, z = 0, we obtain that X (x) = sin(αx) (12a) Y (y) = sin(βy) (12b) Z(z) = sinh( α 2 + β 2 z), (12c) From V = 0, for x = a, and y = b, we must have that α(a) = nπ, and β(b) = mπ. Therefore, α n = nπ a β m = mπ b n γ nm = π 2 α 2 + m2 β 2, (13a) (13b) (13c)

11 So, the partial potential that satisfies the above boundary conditions is V nm = sin(α n x)sin(β m y)sinh(γ nm z) (14) The potential can then be expanded in terms of V nm with arbitrary coefficients that will be chosen to fulfill the final boundary condition: V (x, y, z) = n,m=1 A nm sin(α n x)sin(β m y)sinh(γ nm z) (15)

12 Evaluating the final boundary condition, V (x, y, c) = V 0 (x, y) at z = c: V 0 (x, y) = n,m=1 A nm sin(α n x)sin(β m y)sinh(γ nm c) (16) Which is a double Fourier series for the function V 0 (x, y). Here the coefficients A nm are given by A nm = 4 ab sinh γ nm c a 0 dx b 0 dyv 0 (x, y)sin(α n x)sin(β m y) (17)

13 If the box has potentials different from zero on all six sides, the solution for the potential inside the box can be obtained by linear superposition of six solutions, one for each side, equivalent to Eqs.(17) and (15). In the case of the potential inside the box with a charge distribution inside, Poisson s equation with prescribed boundary conditions on the surface, requires the construction of the appropiate Green function, whose discussion shall be ommited.

14 First uniqueness theorem: The solution to Laplace s equation in some volume τ is uniquely determined if V is specified on the boundary surface S. This theorem guarantees that the solution found for the previous example is unique. Corollary: The potential in a volume τ is uniquely determined if (a) the charge density throughout the region and (b) the value of V on all boundaries, are specified. This corollary is crucial for the validity of the solutions obtained with another method used for finding the electric potential, known as The Method of Images.

15 Derivation from Maxwell s Equations Second uniqueness theorem: In a volume τ surrounded by conductors and containing a specified charge density ρ,p the electric field is uniquely determined if the total charge, Qtot = i Qi, on each conductor is given (as ilustrated below). Yese J. Felipe

16 J.D. Jackson Classical Electrodynamics. John Wiley and Sons, Inc., 3rd edition, D.J. Griffiths Introduction to Electrodynamics. Prentice-Hall, 3rd edition, 1999.