Applied Electromagnetics Using QuickField and MATLAB

By James R. Claycomb

Chapter 2 - Solutions To Laplace's Equation: Separation of Variables

In This Chapter

Separation of Variables
Conformal Mapping
Finite Difference Methods
Monte Carlo Methods
Finite Element Method

Cartesian Coordinates

In Cartesian coordinates, Laplace’s equation S(x, y, z) = 0 is written as

where −∞ < x < ∞, −∞ < y < ∞, −∞ < z < ∞ . Applying separation of variables we assume a product solution

Substituting this form of S into Laplace's equation and dividing by S gives

Each term in the above equation must be equal to a constant if the sum is zero for all x, y, and z, since these variables may vary independently

where α² + β² = γ² . The solutions to these differential equations may be written as linear combinations of sine, cosine, and exponential functions so that we may construct a product solution

with constants determined by the boundary conditions.

As an example of the application of separation of variables, consider a unit square region with boundary condition S = 0 on three sides and S = 1 on the other side. Since there is no z dependence α² + β² equation (2.5) become

Applying the boundary conditions S(0,y) = S(1,y) = 0, where the sine function is zero given α = nπ and relabeling the constants and writing the solution as a sum

The condition S(x, 1) = 0 gives B_n = -A_n cosh(nπ)/sinh(nπ) and

Finally, A_nis determined by applying the boundary condition at S(x, 1) = 1

Multiplying both sides of this expression by sin(nπx) and integrating

Making use of orthogonality, where only terms with n = n′ remain, and using the identity sin²(nπ) = (1 − cos 2nπ)/2, we have

where A_nis zero for even n so the sum is over odd n

The following MATLAB program calculates equation (2.12) and plots the solution over the unit square:

In the following coordinate systems we consider Laplace's equation with axial symmetry and no variation in the φ-direction, or with planar symmetry and no variation in the z-direction. These results will be useful for comparison to QuickField problems that have either axial or x-y symmetry.

Cylindrical Coordinates

In cylindrical coordinates with axial symmetry, Laplace's equation S(r, z) = 0 is written as

where r ≥ 0, −∞ < z < ∞. Substituting S(r, z) = R(r)Z(z) with separation constant k² gives the differential equations

so that we may construct our solution

where J₀(kr) and N₀(kr) are Bessel functions of zero order. Laplace's equation in cylindrical coordinates without variation in the z-direction S(r,φ) = 0 is written as

where r ≥ 0, 0 < φ ≤ 2π. Substituting S(r,φ) = R(r)Φ(φ)

with terms that may include a₀ln(r) + b₀φ + c₀, where a₀, b₀and c₀ are constants.

Bicylindrical Coordinates

The Laplacian S(α, β) is written in bicylindrical coordinates

where −∞ < α < ∞, 0 ≤ β ≤ 2π. Substituting S(α, β) = Λ(α)Ω(β) with separation constant p² gives

Elliptic Cylindrical Coordinates

The Laplacian S(α, β) =0 is written in elliptic cylindrical coordinates

where 0 ≤ α, 0 ≤ β ≤ 2π. Substituting S(α, β) = Λ(α)Ω(β) with separation constant p²gives

Spherical Coordinates

In spherical coordinates with axial symmetry, Laplace's equation S(r,θ) = 0 is written as

Substituting S(r,θ) = R(r)Θ(θ) with separation constant l(l + 1) we obtain

where P_land Q_lare Legendre functions of the first and second kind, respectively.

As an example of a spherically symmetric solution to (2.22) consider a spherical surface of radius R with boundary conditions S(R,θ) = V₀and S(r → ∞,θ) = 0. Since there is no θ dependence we take l = 0 so that (2.23) becomes

The boundary condition at infinity gives A = 0. The boundary condition at r = R gives B = RV₀and we have

Note that the same result may be obtained by setting the radial part of (2.22) equal to zero and integrating twice.

Prolate Spheroidal Coordinates

Laplace's equation in prolate spheroidal coordinates with axial symmetry S(ξ,η) = 0 is written as

where ξ ≥ 0, 0 ≤ η ≤ π.. Substituting S(ξ,η) = Y(ξ)Θ(η) with separation constant l(l+1) gives

We now consider a football shaped prolate spheroidal surface ξ = ξ₀with potential V₀. There is no η dependence so that (2.27) becomes

where P₀(x) = 1. Applying the boundary conditions S(ξ → ∞) = 0 and S(ξ → ξ₀) = V₀ we have

This is equivalent to

which may also be obtained by setting the first term of (2.26) equal to zero and integrating twice. Fortified by this simple result it might be tempting to find the potential between two opposing electrodes in the shape of hyperboloids of revolution by setting the second term in (2.26) equal to zero and integrating twice. However, this procedure would not yield a solution over the region between the electrodes, where £ is varying. Instead we must use (2.27)

where the Q_l(cosh ξ) are divergent at ξ = 0 and are discarded. The constants A_land B_lare determined by boundary conditions at η = η₀, making use of the orthogonality of P_l(cosh η) and Q_l(cosh η).