4. Electrostatic Boundary-Value Problems

Introduction and context

Electrostatic problems are governed by a small set of field equations, yet their practical solution may be difficult because the electric field must satisfy both differential equations in the region of interest and boundary conditions imposed by conductors, dielectrics, interfaces, and infinity. This class of problems is called an electrostatic boundary-value problem.

In a source region, the electric field is related to charge density through Gauss’s law,

∇ · D = ρᵥ,

where D is electric flux density and ρᵥ is volume charge density. For a linear, homogeneous, isotropic medium, D = εE, and because electrostatic fields are irrotational,

∇ × E = 0.

Consequently, a scalar potential V may be defined by

E = −∇V.

Combining these relations gives Poisson’s equation,

∇²V = −ρᵥ/ε,

and, in a charge-free region, Laplace’s equation,

∇²V = 0.

The central task is therefore to determine a scalar potential that satisfies the governing equation and the specified boundary conditions. Once V is known, the electric field, flux density, surface charge, capacitance, force, and stored energy can be obtained by differentiation or integration.

Boundary-value techniques are fundamental in high-voltage insulation, capacitive sensors, semiconductor devices, microwave structures, electrostatic actuators, shielding, cable design, and computational electromagnetics. This chapter develops the principal analytical methods: separation of variables, image theory, Green’s functions, uniqueness theorems, conformal mapping, and variational methods.

Learning objectives

After studying this chapter, the reader should be able to:

1. Formulate electrostatic problems using Poisson’s and Laplace’s equations.

2. Distinguish Dirichlet, Neumann, and mixed boundary conditions.

3. Apply uniqueness theorems to establish whether a solution is determined.

4. Solve representative rectangular, cylindrical, and spherical problems by separation of variables.

5. Use image charges to replace selected conductor boundaries with equivalent sources.

6. Interpret Green’s functions as fundamental solutions for potential problems.

7. Use conformal transformations to solve two-dimensional electrostatic fields.

8. Formulate approximate solutions using energy and variational principles.

9. Evaluate the assumptions, limitations, and suitability of each method.

10. Select an analytical or computational approach for practical engineering problems.

Formulation of an electrostatic boundary-value problem

A complete electrostatic problem consists of a region Ω, its material properties, charge distribution, and boundary conditions. In a linear isotropic medium,

∇ · (ε∇V) = −ρᵥ.

If ε is constant, this reduces to Poisson’s equation. If ρᵥ = 0, it reduces to Laplace’s equation. At an interface between two media, the boundary conditions follow from Maxwell’s equations. The tangential electric field is continuous in electrostatics,

n × (E₂ − E₁) = 0,

while the normal component of flux density satisfies

n · (D₂ − D₁) = ρₛ,

where ρₛ is free surface charge density. If no free surface charge is present, the normal components of D are equal.

For a perfect conductor in electrostatic equilibrium, the electric field inside the conductor is zero and the conductor is an equipotential. The field immediately outside is normal to its surface, and

ρₛ = D · n = εEₙ.

Three boundary specifications occur most often:

• Dirichlet condition: V is prescribed on the boundary.
• Neumann condition: the normal derivative ∂V/∂n, or equivalently normal flux density, is prescribed.
• Mixed or Robin condition: a relation involving V and ∂V/∂n is prescribed.

A Dirichlet problem may specify the potential on all boundaries. A Neumann problem may specify normal derivative on all boundaries, but its solution is determined only up to an additive constant. For a Neumann problem to be compatible with Gauss’s law, the total specified outward flux must equal the total enclosed charge. Mixed conditions arise in interfaces, impedance-like mathematical models, and some truncated computational domains.

The potential is continuous across an ideal dielectric interface unless an idealized dipole sheet is present. The normal derivative may be discontinuous because ε or free surface charge may change. These facts are often more useful in solving a problem than the differential equation itself.

Uniqueness and the logic of solution methods

The uniqueness theorems provide the mathematical foundation for electrostatic solution techniques. The first uniqueness theorem states that, in a volume containing specified charge density, the potential is uniquely determined if its value is specified over the entire boundary. The second uniqueness theorem states that the potential is uniquely determined if the charge distribution and normal component of electric flux density are specified over the entire boundary, subject to an arbitrary additive constant.

A related result applies to interfaces: if the governing equation, material properties, and appropriate boundary conditions are satisfied in every subregion, the resulting solution is the physical solution. This means that a proposed solution need not be derived in the most direct way. It is sufficient to verify that it satisfies the equation and all boundary conditions.

Uniqueness explains why image charges work. The image configuration is not required to reproduce the actual field everywhere outside the conducting boundary; it is required to produce the correct potential in the physical region and satisfy the boundary condition there. By uniqueness, the resulting field is the actual field in that region.

It also explains why numerical approximations can be assessed effectively. If an approximate solution satisfies the boundary conditions and produces a small residual in the governing equation, it can be compared with a unique exact solution when one exists. In design practice, uniqueness prevents ambiguity between apparently different mathematical constructions.

Separation of variables: constructing solutions from modes

Separation of variables is especially effective when the boundaries coincide with coordinate surfaces. The method assumes that the potential can be represented as a product of functions, each depending on one coordinate. Substitution into Laplace’s equation converts one partial differential equation into several ordinary differential equations. Superposition then reconstructs the general solution.

Rectangular coordinates

In Cartesian coordinates,

∂²V/∂x² + ∂²V/∂y² + ∂²V/∂z² = 0.

For a two-dimensional region, let V(x,y) = X(x)Y(y). Substitution gives

X''/X = −Y''/Y = constant.

Depending on the boundary conditions, the resulting functions are trigonometric or hyperbolic. Consider a rectangle 0 < x < a, 0 < y < b with V = 0 on x = 0, x = a, and y = 0, while V(x,b) = f(x). The solution is

V(x,y) = Σₙ₌₁∞ Aₙ sin(nπx/a) sinh(nπy/a),

where the coefficients are determined by the Fourier sine series of f(x):

Aₙ = [2/(a sinh(nπb/a))] ∫₀ᵃ f(x) sin(nπx/a) dx.

The eigenfunctions are selected by the homogeneous boundary conditions. The nonhomogeneous boundary is then satisfied through Fourier expansion. This procedure is typical: first enforce zero conditions on several boundaries, then expand the remaining prescribed function in the corresponding eigenfunctions.

For a rectangular domain with finite length in the z-direction, products of sine, cosine, and hyperbolic functions arise. The choice between sine and cosine follows from whether the relevant boundary condition is zero potential or zero normal derivative.

Cylindrical coordinates

For a charge-free region with no dependence on z, Laplace’s equation is

(1/r)∂/∂r(r∂V/∂r) + (1/r²)∂²V/∂φ² = 0.

Writing V(r,φ) = R(r)Φ(φ) gives angular functions cos(nφ), sin(nφ), and radial functions rⁿ, r⁻ⁿ. The general two-dimensional cylindrical solution is

V = A₀ + B₀ ln r + Σₙ₌₁∞ [rⁿ(Aₙ cos nφ + Bₙ sin nφ) + r⁻ⁿ(Cₙ cos nφ + Dₙ sin nφ)].

Terms that become infinite in a region containing r = 0 must be discarded. Terms that become infinite as r approaches infinity must be discarded when the exterior region is being considered, unless they are required to represent a specified applied field. This regularity requirement is an essential part of the solution.

For an infinitely long coaxial capacitor with inner radius a at potential V₀ and outer radius b at zero potential, cylindrical symmetry gives

V(r) = V₀ ln(b/r)/ln(b/a),

and

Eᵣ = V₀/[r ln(b/a)].

The charge per unit length and capacitance per unit length follow from Gauss’s law:

C' = 2πε/ln(b/a).

This example illustrates that separation of variables includes simple one-dimensional reductions when symmetry eliminates angular and axial dependence.

Spherical coordinates

For spherical symmetry, Laplace’s equation reduces to

(1/r²)d/dr(r² dV/dr) = 0,

so that V = A + B/r. More generally, the axisymmetric solution is expressed in Legendre polynomials:

V(r,θ) = Σₙ₌₀∞ (Aₙrⁿ + Bₙr⁻ⁿ⁻¹)Pₙ(cos θ).

The terms rⁿ are regular at the origin, whereas r⁻ⁿ⁻¹ terms are appropriate for exterior fields but singular at the origin. For a conducting sphere of radius a placed in a uniform field E₀ directed along z, the exterior potential is

V(r,θ) = −E₀r cos θ + E₀a³ cos θ/r².

The second term is the induced dipole potential. At r = a, V = 0, confirming that the sphere is an equipotential. The field enhancement at the poles is three times the applied field, a result relevant to corona inception and high-voltage electrode design.

Image theory: replacing boundaries by equivalent sources

Image theory replaces a conductor boundary by fictitious charges positioned outside the physical region. The images are selected so that the potential satisfies the required boundary condition on the conductor. The image charges are not physical charges in the original problem; they are a mathematical construction.

For a point charge q at height h above an infinite grounded conducting plane, an image charge −q is placed at depth h below the plane. The potential in the upper half-space is

V = (q/4πε)[1/R₁ − 1/R₂],

where R₁ is the distance from the real charge and R₂ from the image. On the plane, R₁ = R₂, so V = 0. The force on the charge is attractive and has magnitude

F = q²/(16πεh²).

The force must be calculated using the field due to the image alone, not by adding the singular self-field of the real charge.

For a point charge near a grounded conducting sphere, a single image charge can also be used, but its magnitude and location depend on the charge position and sphere radius. The method is more specialized because the image is not generally equal and opposite to the real charge.

Image theory is powerful for planes, spheres, cylinders, and combinations that preserve a suitable geometry. It becomes difficult or impossible for irregular boundaries, multiple nearby conductors, or spatially varying permittivity. In those cases, boundary-element or finite-element methods are normally preferred.

Green’s functions: a systematic source-response framework

A Green’s function expresses the potential produced by a unit point source while satisfying the boundary condition of the domain. For a Dirichlet problem, the Green’s function G(r,r′) satisfies

∇²G(r,r′) = −δ(r − r′)/ε,

with G = 0 on the boundary. The potential due to a volume charge distribution is then

V(r) = ∫Ω G(r,r′)ρᵥ(r′) dτ′,

with additional surface terms if nonzero boundary potentials are present.

Green’s functions are generalized influence functions. Once G is known for a geometry, any charge distribution can be handled by integration. In free space,

G = 1/(4πε|r − r′|).

A grounded conducting plane is represented by the free-space Green’s function plus an image term of opposite sign. Thus image theory is a special construction of a Green’s function.

Green’s second identity provides the integral representation of a harmonic or Poisson potential. It also clarifies the distinction between source effects and boundary effects. In practical computational electromagnetics, Green’s functions form the basis of boundary integral and method-of-moments formulations. Unknown surface charge is discretized into elements, and the potential equation becomes a matrix equation relating element charges to sampled potentials.

The principal difficulties are singular kernels, complex geometry, and the need to impose global charge constraints. Numerical quadrature must treat the self-interaction of an element carefully. Nevertheless, boundary methods are attractive when the field region is unbounded, because only the boundary—not the entire surrounding volume—needs to be discretized.

Conformal mapping in two-dimensional electrostatics

A two-dimensional electrostatic potential in a charge-free region is harmonic. If z = x + jy and w = u + jv are related by an analytic conformal transformation w = f(z), the potential can be transferred between the two planes. Harmonic functions remain harmonic under such transformations, and angles between field lines and equipotentials are preserved.

The complex potential is commonly written as

W(z) = V(x,y) + jΨ(x,y),

where V is the potential and Ψ is the electric flux function. The real and imaginary parts satisfy the Cauchy–Riemann equations and are orthogonal families of curves. Equipotential lines and electric field lines therefore intersect at right angles.

A transformation may convert a difficult geometry into a parallel-plate capacitor, for which the potential is elementary. For example, logarithmic transformations can map circular or wedge-shaped boundaries into straight lines. If a known solution in the w-plane is V(u,v), then V[ f(z) ] gives the corresponding solution in the z-plane, subject to proper identification of boundaries and scaling.

Conformal mapping also gives field enhancement information. The local field magnitude is affected by the derivative of the mapping:

|E_z| = |E_w| |dw/dz|.

A large derivative indicates local field concentration. This is useful in analyzing sharp electrode edges, slots, wedges, and two-dimensional cable or capacitor geometries. Ideal sharp corners may produce mathematically unbounded fields, signaling that a real design must include a finite radius, insulation grading, or a field-control electrode.

The method is limited to two-dimensional or effectively two-dimensional problems and requires an analytic mapping. It is therefore elegant but geometry-dependent.

Variational and energy methods

Many electrostatic boundary-value problems cannot be solved conveniently in closed form. Variational methods provide approximate solutions and powerful design estimates. The electrostatic energy functional for a prescribed-potential problem is

Π[V] = ∫Ω [ (1/2)ε|∇V|² − ρᵥV ] dτ,

with V fixed on the Dirichlet boundary. The physical potential makes this functional stationary. Taking its first variation yields Poisson’s equation and the associated natural boundary condition.

If ρᵥ = 0, the functional reduces to field energy,

Π[V] = (1/2)∫Ω ε|∇V|² dτ.

Among admissible potentials satisfying the prescribed potential boundary conditions, the exact solution minimizes the electrostatic energy. This principle is the basis of the finite-element method. The approximate potential is expanded in shape functions,

V ≈ Σᵢ Nᵢ(x,y,z)Vᵢ.

Substitution into the functional produces a matrix equation

Kv = f,

where the stiffness matrix contains integrals of ε∇Nᵢ · ∇Nⱼ. The entries of v are nodal potentials. Once solved, the electric field is obtained from the gradient of the interpolated potential.

A trial function method can be used analytically. The trial function must satisfy essential boundary conditions, while unknown coefficients are chosen by minimizing the functional or making the residual orthogonal to selected weighting functions. The Rayleigh–Ritz method uses trial functions; the weighted-residual and Galerkin methods generalize this approach.

Variational methods are particularly valuable for capacitance, field concentration, and optimization. They also provide a physical interpretation of approximation error: an admissible approximation generally overestimates the minimum energy in a Dirichlet problem. However, a coarse mesh can miss sharp field gradients, singular corners, thin dielectric layers, or small gaps. Mesh refinement and convergence studies are therefore essential.

Selecting a solution method

Method selection should follow geometry, material distribution, source complexity, and the required output. Separation of variables is preferred when boundaries align with standard coordinates and the region is simple. Image theory is efficient for isolated conductors with plane or spherical boundaries. Green’s functions and boundary-element methods are useful for unbounded domains and surface-charge problems. Conformal mapping is appropriate for two-dimensional geometries with a known analytic transformation. Variational and finite-element methods are the most flexible for irregular geometries and inhomogeneous materials.

A practical workflow is:

1. Identify symmetry and choose coordinates.

2. Define the physical region and all interfaces.

3. Write Poisson’s or Laplace’s equation in each subregion.

4. State conductor, interface, infinity, and charge conditions.

5. Check compatibility and uniqueness.

6. Select an analytical or numerical representation.

7. Verify dimensions, limiting cases, boundary residuals, and total charge.

8. Derive engineering quantities such as capacitance, force, energy, or peak field.

A solution that satisfies the differential equation but violates one boundary condition is not an electrostatic solution. Similarly, a numerically smooth potential may still be inaccurate near a singularity or a narrow gap.

Illustrative case study: field grading in a high-voltage bushing

Background

Consider an illustrative epoxy-insulated high-voltage bushing containing a cylindrical conductor. The conductor is maintained at high potential, while a grounded metallic flange surrounds the lower portion of the insulation. The design objective is to prevent excessive electric-field concentration at the conductor exit and at the flange edge.

Situation and problem

An initial two-dimensional axisymmetric model shows a high field near the sharp flange corner. The calculated maximum field is substantially greater than the average radial field. Such concentration may initiate partial discharge, accelerate aging, and reduce reliability.

Analysis

The field is governed by Laplace’s equation in the charge-free epoxy and surrounding air, with continuity conditions at the dielectric interface. The conductor and flange are equipotential boundaries. A local sharp corner behaves similarly to a wedge, for which conformal mapping predicts strong field concentration. The ideal zero-radius corner may even produce a mathematical singularity; therefore, the exact peak field at the ideal corner is not a physically meaningful design quantity.

A finite-element formulation is selected because the geometry contains curved interfaces, a finite corner radius, and multiple dielectric regions. The potential is prescribed on the conductor and flange, and an outer boundary is placed sufficiently far away with a suitable approximation of the open region. The mesh is refined near the flange, conductor exit, and dielectric interface.

Approach and solution

The design team evaluates three modifications: increasing the flange radius, adding a semiconductive stress-control layer, and introducing a graded dielectric contour. The finite-element model is checked against the analytical cylindrical-capacitor field in a simplified region. Mesh convergence is tested by monitoring total capacitance and the maximum field away from the idealized corner.

The selected design combines a larger radius with a field-grading layer. The potential changes more gradually along the insulation surface, and the field becomes more evenly distributed. The result is then assessed under manufacturing tolerances, because small deviations in radius and layer thickness can alter local field values.

Outcome

The redesigned geometry reduces field concentration in the physically relevant region and produces a more robust field distribution. The study also demonstrates why the maximum value at an ideal mathematical corner should not be accepted without examining geometry, mesh dependence, and physical radius.

Learning points

The case illustrates four principles. First, boundary conditions and material interfaces are as important as the governing equation. Second, conformal-mapping insight can explain a numerical field pattern. Third, a numerical model requires verification and convergence testing. Finally, electrostatic design is governed not only by average field or voltage rating but by local field concentration and manufacturing reality.

Limitations, errors, and professional considerations

Common analytical errors include using the wrong sign in E = −∇V, retaining singular terms in the wrong region, confusing free charge with bound charge, and applying a conductor boundary condition to a dielectric interface. Another frequent mistake is to impose both potential and normal derivative arbitrarily on the same boundary; generally, these data cannot be prescribed independently.

The assumption of a linear isotropic dielectric may fail in ferroelectric, nonlinear, anisotropic, or field-dependent materials. Space charge, leakage current, humidity, temperature, and frequency-dependent polarization can also invalidate a purely electrostatic model. At sufficiently high fields, corona, partial discharge, charge injection, and breakdown introduce physics beyond Laplace’s equation.

In computational work, apparent accuracy can be misleading. Singular corners, thin layers, poorly shaped elements, insufficient outer boundaries, and incorrect interface conditions can produce plausible but wrong results. Results should be checked through global charge balance, energy-capacitance consistency, symmetry, limiting cases, and mesh or basis refinement.

Professional practice also requires clear documentation of assumptions, material data, voltage polarity, reference potential, geometry tolerances, and post-processing definitions. Safety considerations are essential when applying electrostatic results to high-voltage systems. Numerical field maps should not be used as a substitute for insulation coordination, testing, or appropriate safety margins.

Current developments and best practices

Modern electrostatic analysis increasingly combines analytical insight with numerical simulation. Finite-element solvers provide detailed volumetric fields, while boundary-element methods efficiently treat open regions. Adaptive meshing uses estimated error or field gradients to refine only critical regions. Reduced-order models and surrogate approximations support parameter studies and optimization.

Automation can generate meshes, sweep geometric parameters, and identify regions of high field. Data-driven methods may assist geometry optimization, but they do not replace the governing equations, boundary-condition verification, or physical validation. A reliable workflow retains an analytical benchmark wherever possible and uses numerical results within a documented verification and validation process.

Best practice is to begin with a simplified model, exploit symmetry, establish scales and limiting behavior, and then add geometric and material complexity. The final model should distinguish mathematical singularities from physically meaningful peak fields and should report uncertainty associated with geometry, material properties, and discretization.

Chapter Summary

Electrostatic boundary-value problems determine the potential and field in a region from charge distributions, material properties, and boundary conditions. Poisson’s equation applies in charged regions, while Laplace’s equation applies in charge-free regions. Dirichlet conditions prescribe potential; Neumann conditions prescribe normal flux; mixed conditions combine them.

Uniqueness theorems establish when a solution is determined and justify alternative constructions such as image charges. Separation of variables provides series solutions in rectangular, cylindrical, and spherical coordinates. Image theory replaces selected conductor boundaries with fictitious sources, while Green’s functions express the response to arbitrary charge distributions. Conformal mapping transforms suitable two-dimensional geometries into simpler ones. Variational methods form the basis of approximate analysis and finite-element computation.

Engineering practice requires method selection based on geometry and materials, careful enforcement of interface and conductor conditions, and verification through symmetry, limiting cases, charge balance, energy, and convergence. Field concentration, singularities, nonlinear materials, and computational modeling assumptions must be interpreted physically rather than accepted mechanically.

Key Terms

Poisson’s equation: The potential equation ∇²V = −ρᵥ/ε in a uniform medium containing volume charge.

Laplace’s equation: The charge-free potential equation ∇²V = 0.

Dirichlet boundary condition: A boundary condition that specifies the value of potential.

Neumann boundary condition: A boundary condition that specifies the normal derivative of potential or normal electric flux.

Mixed boundary condition: A condition involving both potential and its normal derivative.

Uniqueness theorem: A theorem establishing that specified governing data and boundary conditions determine one physical potential solution.

Equipotential: A surface or region on which electric potential is constant.

Image charge: A fictitious charge introduced to satisfy a conductor boundary condition.

Green’s function: The potential response of a domain to a unit point source under specified boundary conditions.

Separation of variables: A method that represents a multivariable solution as products of single-variable functions.

Eigenfunction: A coordinate-dependent function selected by a homogeneous boundary-value problem.

Conformal mapping: An analytic coordinate transformation that preserves angles and harmonic character in two dimensions.

Complex potential: A complex function whose real and imaginary parts represent potential and flux functions.

Variational principle: A formulation in which the physical field makes an energy or residual functional stationary.

Rayleigh–Ritz method: An approximate variational method using a finite set of admissible trial functions.

Finite-element method: A numerical method that approximates the field over interconnected elements using shape functions.

Field enhancement: The increase of local electric-field magnitude caused by geometry, especially sharp curvature or small gaps.

Boundary integral method: A method that represents the field using unknown quantities distributed on boundaries rather than throughout the volume.

Electrostatic energy: Stored field energy given by (1/2)∫ ε|E|² dτ for a linear dielectric.

Surface charge density: Free charge per unit area on a conductor or interface.

Review Questions

Short Answer Questions

1. State Poisson’s equation and identify the condition under which it becomes Laplace’s equation.

2. Distinguish between Dirichlet and Neumann boundary conditions.

3. Why must the electrostatic field inside a perfect conductor be zero?

4. What is the physical meaning of an image charge?

5. Why are singular terms discarded when they diverge inside a region containing the origin?

Long Answer Questions

1. Derive the potential and capacitance per unit length of a coaxial cylindrical capacitor.

2. Explain the first and second uniqueness theorems and discuss their engineering significance.

3. Develop the separation-of-variables solution for a rectangular region with three grounded sides and a prescribed potential on the fourth side.

4. Explain Green’s functions and show how image theory can be interpreted as a Green’s-function construction.

5. Discuss the variational formulation of electrostatics and its connection with the finite-element method.

Analytical / Application Questions

1. A point charge is located a distance h above a grounded infinite plane. Construct the image system, determine the force on the charge, and explain why the self-field is excluded from the force calculation.

2. A conducting sphere is placed in a uniform electric field. Use the spherical harmonic form of the potential to determine the induced dipole term and the field at the sphere’s poles.

3. An axisymmetric high-voltage bushing has a sharp metallic flange edge. Recommend an analytical and numerical investigation strategy, including boundary conditions, mesh checks, field interpretation, and possible geometry modifications.

22. Multiple Choice Questions with Answers

Question 1

Which equation governs the electrostatic potential in a homogeneous region containing volume charge density ρᵥ?

A. ∇²V = 0

B. ∇²V = ρᵥ/ε

C. ∇²V = −ρᵥ/ε

D. ∇²V = −ε/ρᵥ

Correct Answer: C. ∇²V = −ρᵥ/ε

Question 2

A Neumann boundary condition specifies which quantity?

A. The potential itself

B. The normal derivative of potential or normal flux

C. The tangential derivative of potential only

D. The volume charge density everywhere

Correct Answer: B. The normal derivative of potential or normal flux

Question 3

Why does the method of images produce the correct field in the physical region?

A. Image charges are actual charges inside the conductor

B. The image configuration satisfies the governing equation and boundary condition, and uniqueness applies

C. The conductor has zero permittivity

D. The electric field is always symmetric about a conductor

Correct Answer: B. The image configuration satisfies the governing equation and boundary condition, and uniqueness applies

Question 4

For a coaxial capacitor with inner radius a and outer radius b, the potential varies with radius as

A. V proportional to r²

B. V proportional to 1/r²

C. V proportional to ln r

D. V proportional to sin r

Correct Answer: C. V proportional to ln r

Question 5

Which term must be excluded from an interior spherical solution because it is singular at r = 0?

A. A constant term

B. A term proportional to r

C. A term proportional to r²

D. A term proportional to 1/r

Correct Answer: D. A term proportional to 1/r

Question 6

A Green’s function for a Dirichlet problem is constructed to

A. Have prescribed zero potential on the boundary

B. Have infinite potential everywhere

C. Eliminate the source term from the equation

D. Specify the dielectric constant at every point

Correct Answer: A. Have prescribed zero potential on the boundary

Question 7

What property makes conformal mapping useful for two-dimensional electrostatics?

A. It converts all three-dimensional fields into uniform fields

B. It preserves angles and maps harmonic solutions into harmonic solutions

C. It eliminates the need for boundary conditions

D. It applies only to magnetic materials

Correct Answer: B. It preserves angles and maps harmonic solutions into harmonic solutions

Question 8

In a variational finite-element formulation, the unknown nodal quantities for a basic electrostatic problem are usually

A. Nodal potentials

B. Magnetic flux densities

C. Circuit currents only

D. Image-charge magnitudes only

Correct Answer: A. Nodal potentials

Question 9

At a dielectric interface with no free surface charge, which condition is correct?

A. The normal component of E is always continuous

B. The tangential component of E is continuous and the normal component of D is continuous

C. Both E and D must be discontinuous

D. The potential must be discontinuous

Correct Answer: B. The tangential component of E is continuous and the normal component of D is continuous

Question 10

Why can the calculated field at an ideal sharp conductor corner be misleading?

A. Laplace’s equation cannot be used near conductors

B. The mathematical field may be singular, whereas real conductors have finite radius and manufacturing limits

C. The potential is never continuous at a conductor

D. The field is always zero at a corner

Correct Answer: B. The mathematical field may be singular, whereas real conductors have finite radius and manufacturing limits