the poisson equation for electrostatics · 2014. 12. 20. · derivation from maxwell’s equations...
TRANSCRIPT
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Derivation from Maxwell’s EquationsExample: Laplace Equation in Rectangular Coordinates
Uniqueness TheoremsBibliography
The Poisson Equation for Electrostatics
Yesé J. Felipe
University of Puerto Rico - Mayagüez
Yesé J. Felipe The Poisson Equation for Electrostatics
-
Derivation from Maxwell’s EquationsExample: Laplace Equation in Rectangular Coordinates
Uniqueness TheoremsBibliography
Table of Contents
1 Derivation from Maxwell’s Equations
2 Example: Laplace Equation in Rectangular Coordinates
3 Uniqueness Theorems
4 Bibliography
Yesé J. Felipe The Poisson Equation for Electrostatics
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Derivation from Maxwell’s EquationsExample: Laplace Equation in Rectangular Coordinates
Uniqueness TheoremsBibliography
Derivation from Maxwell’s Equations
Maxwell’s Equations for Electrodynamics in differential form are:
∇ · ~E = 1�0ρ(~r , t) (Gauss’s Law for Electricity) (1a)
∇ · ~B = 0 (Gauss’s Law for Magnetism)(1b)
∇× ~E = −∂~B
∂t(Faraday’s Law of Induction) (1c)
∇× ~B = µ0~J(~r , t) + µ0�0∂~E
∂t(Ampère-Maxwell’s Law) .(1d)
Yesé J. Felipe The Poisson Equation for Electrostatics
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Derivation from Maxwell’s EquationsExample: Laplace Equation in Rectangular Coordinates
Uniqueness TheoremsBibliography
Using the Helmholtz Theorem and that ~B is divergenceless, themagnetic 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 beexpressed as:
~E = −∇V − ∂~A
∂t(3)
Yesé J. Felipe The Poisson Equation for Electrostatics
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Derivation from Maxwell’s EquationsExample: Laplace Equation in Rectangular Coordinates
Uniqueness TheoremsBibliography
Substiting into Gauss’ Law, Eq. (1a), results in:
∇2V + ∂∂t
(∇ · ~A) = − 1�0ρ (4)
In the electrostatic case, it reduces to
∆V ≡ ∇2V = − 1�0ρ(~r), (5)
which is Poisson’s equation. For a region of space in which there isno charge, we obtain Laplace’s equation:
∆V ≡ ∇2V = 0 (6)
Yesé J. Felipe The Poisson Equation for Electrostatics
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Derivation from Maxwell’s EquationsExample: Laplace Equation in Rectangular Coordinates
Uniqueness TheoremsBibliography
Example: Laplace Equation in Rectangular Coordinates
The Laplace equation in rectangular coordinates is
∂2V
∂x2+∂2V
∂y2+∂2V
∂z2= 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
X (x)
d2X
dx2+
1
Y (y)
d2Y
dy2+
1
Z (z)
d2Z
dz2= 0 (9)
Yesé J. Felipe The Poisson Equation for Electrostatics
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Derivation from Maxwell’s EquationsExample: Laplace Equation in Rectangular Coordinates
Uniqueness TheoremsBibliography
In order for the result to hold for arbitrary values of thecoordinates, each of the terms must be individually constant:
1
X (x)
d2X
dx2= −α2 (10a)
1
Y (y)
d2Y
dy2= −β2 (10b)
1
Z (z)
d2Z
dz2= γ2, (10c)
where α2 + β2 = γ2.
Yesé J. Felipe The Poisson Equation for Electrostatics
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Derivation from Maxwell’s EquationsExample: Laplace Equation in Rectangular Coordinates
Uniqueness TheoremsBibliography
If α2 and β2 are arbitrarily chosen to be positive, the solutions tothe set of ODEs are then:
X (x) = e±iαx (11a)
Y (y) = e±iβy (11b)
Z (z) = e±√α2+β2z . (11c)
Yesé J. Felipe The Poisson Equation for Electrostatics
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Derivation from Maxwell’s EquationsExample: Laplace Equation in Rectangular Coordinates
Uniqueness TheoremsBibliography
In order to determine α and β, we impose the boundary conditionson the potential. An example is shown in the figure.
Yesé J. Felipe The Poisson Equation for Electrostatics
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Derivation from Maxwell’s EquationsExample: Laplace Equation in Rectangular Coordinates
Uniqueness TheoremsBibliography
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(√α2 + β2z), (12c)
From V = 0, for x = a, and y = b, we must have that α(a) = nπ,and β(b) = mπ. Therefore,
αn =nπ
a(13a)
βm =mπ
b(13b)
γnm = π
√n2
α2+
m2
β2, (13c)
Yesé J. Felipe The Poisson Equation for Electrostatics
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Derivation from Maxwell’s EquationsExample: Laplace Equation in Rectangular Coordinates
Uniqueness TheoremsBibliography
So, the partial potential that satisfies the above boundaryconditions is
Vnm = sin(αnx)sin(βmy)sinh(γnmz) (14)
The potential can then be expanded in terms of Vnm with arbitrarycoefficients that will be chosen to fulfill the final boundarycondition:
V (x , y , z) =∞∑
n,m=1
Anmsin(αnx)sin(βmy)sinh(γnmz) (15)
Yesé J. Felipe The Poisson Equation for Electrostatics
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Derivation from Maxwell’s EquationsExample: Laplace Equation in Rectangular Coordinates
Uniqueness TheoremsBibliography
Evaluating the final boundary condition, V (x , y , c) = V0(x , y) atz = c :
V0(x , y) =∞∑
n,m=1
Anmsin(αnx)sin(βmy)sinh(γnmc) (16)
Which is a double Fourier series for the function V0(x , y). Herethe coefficients Anm are given by
Anm =4
ab sinh γnmc
∫ a0
dx
∫ b0
dyV0(x , y)sin(αnx)sin(βmy) (17)
Yesé J. Felipe The Poisson Equation for Electrostatics
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Derivation from Maxwell’s EquationsExample: Laplace Equation in Rectangular Coordinates
Uniqueness TheoremsBibliography
If the box has potentials different from zero on all six sides, thesolution for the potential inside the box can be obtained by linearsuperposition of six solutions, one for each side, equivalent toEqs.(17) and (15). In the case of the potential inside the box witha charge distribution inside, Poisson’s equation with prescribedboundary conditions on the surface, requires the construction ofthe appropiate Green function, whose discussion shall be ommited.
Yesé J. Felipe The Poisson Equation for Electrostatics
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Derivation from Maxwell’s EquationsExample: Laplace Equation in Rectangular Coordinates
Uniqueness TheoremsBibliography
Uniqueness Theorems
First uniqueness theorem: The solution to Laplace’s equation insome volume τ is uniquely determined if V is specified on theboundary surface S .This theorem guarantees that the solution found for the previousexample is unique.Corollary: The potential in a volume τ is uniquely determined if(a) the charge density throughout the region and (b) the value ofV on all boundaries, are specified.This corollary is crucial for the validity of the solutions obtainedwith another method used for finding the electric potential, knownas The Method of Images.
Yesé J. Felipe The Poisson Equation for Electrostatics
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Derivation from Maxwell’s EquationsExample: Laplace Equation in Rectangular Coordinates
Uniqueness TheoremsBibliography
Second uniqueness theorem: In a volume τ surrounded byconductors and containing a specified charge density ρ, the electricfield is uniquely determined if the total charge, Qtot =
∑i Qi , on
each conductor is given (as ilustrated below).
Yesé J. Felipe The Poisson Equation for Electrostatics
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Derivation from Maxwell’s EquationsExample: Laplace Equation in Rectangular Coordinates
Uniqueness TheoremsBibliography
Bibliography
J.D. JacksonClassical Electrodynamics.John Wiley and Sons, Inc., 3rd edition, 1998.
D.J. GriffithsIntroduction to Electrodynamics.Prentice-Hall, 3rd edition, 1999.
Yesé J. Felipe The Poisson Equation for Electrostatics
Derivation from Maxwell's EquationsExample: Laplace Equation in Rectangular CoordinatesUniqueness TheoremsBibliography