The Poisson Equation for Electrostatics

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Derivation from Maxwell’s Equations Example: Laplace Equation in Rectangular Coordinates Uniqueness Theorems Bibliography

Yes´eJ. Felipe

University of Puerto Rico - Mayag¨uez

Yes´eJ. Felipe The Poisson Equation for Electrostatics Derivation from Maxwell’s Equations Example: Laplace Equation in Rectangular Coordinates Uniqueness Theorems Bibliography Table of Contents

1 Derivation from Maxwell’s Equations

2 Example: Laplace Equation in Rectangular Coordinates

3 Uniqueness Theorems

4 Bibliography

Yes´eJ. Felipe The Poisson Equation for Electrostatics Derivation from Maxwell’s Equations Example: Laplace Equation in Rectangular Coordinates Uniqueness Theorems Bibliography Derivation from Maxwell’s Equations

Maxwell’s Equations for Electrodynamics in diﬀerential form are: 1 ∇ · E~ = ρ(~r, t) (Gauss’s Law for Electricity) (1a) 0 ∇ · B~ = 0 (Gauss’s Law for Magnetism)(1b) ∂B~ ∇ × E~ = − (Faraday’s Law of Induction) (1c) ∂t ∂E~ ∇ × B~ = µ ~J(~r, t) + µ (Amp`ere-Maxwell’s Law) (1d). 0 0 0 ∂t

Yes´eJ. Felipe The Poisson Equation for Electrostatics Derivation from Maxwell’s Equations Example: Laplace Equation in Rectangular Coordinates Uniqueness Theorems Bibliography

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: ∂A~ E~ = −∇V − (3) ∂t

Yes´eJ. Felipe The Poisson Equation for Electrostatics Derivation from Maxwell’s Equations Example: Laplace Equation in Rectangular Coordinates Uniqueness Theorems Bibliography

Substiting into Gauss’ Law, Eq. (1a), results in:

∂ 1 ∇2V + (∇ · A~ ) = − ρ (4) ∂t 0 In the electrostatic case, it reduces to 1 ∆V ≡ ∇2V = − ρ(~r), (5) 0 which is Poisson’s equation. For a region of space in which there is no charge, we obtain Laplace’s equation:

∆V ≡ ∇2V = 0 (6)

The Laplace equation in rectangular coordinates is

∂2V ∂2V ∂2V + + = 0 (7) ∂x2 ∂y 2 ∂z2 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 d2X 1 d2Y 1 d2Z + + = 0 (9) X (x) dx2 Y (y) dy 2 Z(z) dz2

Yes´eJ. Felipe The Poisson Equation for Electrostatics Derivation from Maxwell’s Equations Example: Laplace Equation in Rectangular Coordinates Uniqueness Theorems Bibliography

In order for the result to hold for arbitrary values of the coordinates, each of the terms must be individually constant:

1 d2X = −α2 (10a) X (x) dx2 1 d2Y = −β2 (10b) Y (y) dy 2 1 d2Z = γ2, (10c) Z(z) dz2

where α2 + β2 = γ2.

Yes´eJ. Felipe The Poisson Equation for Electrostatics Derivation from Maxwell’s Equations Example: Laplace Equation in Rectangular Coordinates Uniqueness Theorems Bibliography

If α2 and β2 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± α2+β2z . (11c)

Yes´eJ. Felipe The Poisson Equation for Electrostatics Derivation from Maxwell’s Equations Example: Laplace Equation in Rectangular Coordinates Uniqueness Theorems Bibliography

In order to determine α and β, we impose the boundary conditions on the potential. An example is shown in the ﬁgure.

Yes´eJ. Felipe The Poisson Equation for Electrostatics Derivation from Maxwell’s Equations Example: Laplace Equation in Rectangular Coordinates Uniqueness Theorems Bibliography

Since V = 0, for x = 0, y = 0, z = 0, we obtain that

X (x) = sin(αx) (12a) Y (y) = sin(βy) (12b) p Z(z) = sinh( α2 + β2z), (12c)

From V = 0, for x = a, and y = b, we must have that α(a) = nπ, and β(b) = mπ. Therefore, nπ α = (13a) n a mπ β = (13b) m b s n2 m2 γ = π + , (13c) nm α2 β2

Yes´eJ. Felipe The Poisson Equation for Electrostatics Derivation from Maxwell’s Equations Example: Laplace Equation in Rectangular Coordinates Uniqueness Theorems Bibliography

So, the partial potential that satisﬁes the above boundary conditions is

Vnm = sin(αnx)sin(βmy)sinh(γnmz) (14)

The potential can then be expanded in terms of Vnm with arbitrary coefficients that will be chosen to fulfill the final boundary condition: ∞ X V (x, y, z) = Anmsin(αnx)sin(βmy)sinh(γnmz) (15) n,m=1

Yes´eJ. Felipe The Poisson Equation for Electrostatics Derivation from Maxwell’s Equations Example: Laplace Equation in Rectangular Coordinates Uniqueness Theorems Bibliography

Evaluating the ﬁnal boundary condition, V (x, y, c) = V0(x, y) at z = c: ∞ X V0(x, y) = Anmsin(αnx)sin(βmy)sinh(γnmc) (16) n,m=1

Which is a double Fourier series for the function V0(x, y). Here the coeﬃcients Anm are given by

4 Z a Z b Anm = dx dyV0(x, y)sin(αnx)sin(βmy) (17) ab sinh γnmc 0 0

Yes´eJ. Felipe The Poisson Equation for Electrostatics Derivation from Maxwell’s Equations Example: Laplace Equation in Rectangular Coordinates Uniqueness Theorems Bibliography

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.

Yes´eJ. Felipe The Poisson Equation for Electrostatics Derivation from Maxwell’s Equations Example: Laplace Equation in Rectangular Coordinates Uniqueness Theorems Bibliography Uniqueness Theorems

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.

Yes´eJ. Felipe The Poisson Equation for Electrostatics Derivation from Maxwell’s Equations Example: Laplace Equation in Rectangular Coordinates Uniqueness Theorems Bibliography

Second uniqueness theorem: In a volume τ surrounded by conductors and containing a speciﬁed charge density ρ, the electric P ﬁeld is uniquely determined if the total charge, Qtot = i Qi , on each conductor is given (as ilustrated below).

Yes´eJ. Felipe The Poisson Equation for Electrostatics Derivation from Maxwell’s Equations Example: Laplace Equation in Rectangular Coordinates Uniqueness Theorems Bibliography Bibliography

J.D. Jackson Classical Electrodynamics. John Wiley and Sons, Inc., 3rd edition, 1998. D.J. Griﬃths Introduction to Electrodynamics. Prentice-Hall, 3rd edition, 1999.

Yes´eJ. Felipe The Poisson Equation for Electrostatics