# 5: Electrostatics

- Page ID
- 3929

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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)*Electrostatics* is the theory of the electric field in conditions in which its behavior is independent of magnetic fields, including

- The electric field associated with fixed distributions of electric charge
*Capacitance*(the ability of a structure to store energy in an electric field)- The
*energy*associated with the electrostatic field *Steady current*induced in a conducting material in the presence of an electrostatic field (essentially, Ohm’s Law)

The term “static” refers to the fact that these aspects of electromagnetic theory can be developed by assuming sources are time-invariant; we might say that electrostatics is the study of the electric field at DC. However, many aspects of electrostatics are relevant to AC, radio frequency, and higher-frequency applications as well.

- 5.1: Coulomb’s Law
- Consider two charge-bearing particles in free space. Let the charges borne by these particles be q1 and q2 , and let R be the distance between them. If the particles bear charges of the same sign, then the particles repel; otherwise, they attract. This repulsion or attraction can be quantified as a force experienced by each particle. Physical observations reveal that the magnitude of the force is proportional to the product of charges, and inversely proportional to R2 . For particle 2 we f

- 5.2: Electric Field Due to Point Charges
- The electric field resulting from a set of charged particles is equal to the sum of the fields associated with the individual particles.

- 5.3: Charge Distributions
- In principle, the smallest unit of electric charge that can be isolated is the charge of a single electron. This is very small, and we rarely deal with electrons one at a time, so it is usually more convenient to describe charge as a quantity that is continuous over some region of space. In particular, it is convenient to describe charge as being distributed in one of three ways: along a curve, over a surface, or within a volume.

- 5.4: Electric Field Due to a Continuous Distribution of Charge
- It is common to have a continuous distribution of charge as opposed to a countable number of charged particles. In this section, we extend the discrete perspective of charge distributions into the concept of continuous distribution of charge so that we may address this more general class of problems.

- 5.5: Gauss’ Law - Integral Form
- Gauss’ Law is one of the four fundamental laws of classical electromagnetics, collectively known as Maxwell’s Equations. Gauss’ Law states that the flux of the electric field through a closed surface is equal to the enclosed charge.

- 5.6: Electric Field Due to an Infinite Line Charge using Gauss’ Law
- One application of Gauss’ Law is to find the electric field due to a charged particle. In this section, we present another application – the electric field due to an infinite line of charge. The result serves as a useful “building block” in a number of other problems, including determination of the capacitance of coaxial cable.

- 5.7: Gauss’ Law - Differential Form
- However, even the Coulomb’s Law / direct integration approach has a limitation that is very important to recognize: It does not account for the presence of structures that may influence the electric field. For example, the electric field due to a charge in free space is different from the electric field due to the same charge located near a perfectly-conducting surface. In fact, these approaches do not account for the possibility of any spatial variation in material composition.

- 5.8: Force, Energy, and Potential Difference
- The force Fe experienced by a particle at location r bearing charge q in an electric field intensity E is Fe=qE(r)(5.8.1) If left alone in free space, this particle would immediately begin to move. The resulting displacement represents a loss of potential energy. This loss can quantified using the concept of work, W . The incremental work ΔW done by moving the particle a short distance Δl , over which we assume the change in Fe is negligible.

- 5.9: Independence of Path
- The integral of the electric field over a path between two points depends only on the locations of the start and end points and is independent of the path taken between those points.

- 5.10: Kirchoff’s Voltage Law for Electrostatics - Integral Form
- Kirchoff’s Voltage Law for Electrostatics states that the integral of the electric field over a closed path is zero.

- 5.11: Kirchoff’s Voltage Law for Electrostatics - Differential Form
- The integral form of Kirchoff’s Voltage Law for electrostatics states that an integral of the electric field along a closed path is equal to zero. In this section, we derive the differential form of this equation. In some applications, this differential equation, combined with boundary conditions imposed by structure and materials and can be used to solve for the electric field in arbitrarily complicated scenarios.

- 5.12: Electric Potential Field Due to Point Charges
- The electrical potential at a point is defined as the potential difference measured beginning at a sphere of infinite radius and ending at the point r . The potential obtained in this manner is with respect to the potential infinitely far away.

- 5.13: Electric Potential Field due to a Continuous Distribution of Charge
- It is more common to have a continuous distribution of charge as opposed to a countable number of charged particles. We now consider how to compute V(r) three types of these commonly-encountered distributions.

- 5.14: Electric Field as the Gradient of Potential
- We defined the scalar electric potential field V(r) as the electric potential difference at r relative to a datum at infinity. In this section, we address the “inverse problem” – namely, how to calculate E(r) given V(r) . Specifically, we are interested in a direct “point-wise” mathematical transform from one to the other.

- 5.15: Poisson’s and Laplace’s Equations
- The electric scalar potential field V(r) is useful for a number of reasons including the ability to conveniently compute potential differences and the ability to conveniently determine the electric field by taking the gradient. In this section, we develop an alternative approach to calculating V(r) that accommodates these boundary conditions, and facilitates the analysis of the scalar potential field. This alternative approach is based on Poisson’s Equation.

- 5.16: Potential Field Within a Parallel Plate Capacitor
- This section presents a simple example that demonstrates the use of Laplace’s Equation to determine the potential field in a source free region. The example pertains to an important structure in electromagnetic theory – the parallel plate capacitor. Here we are concerned only with the potential field V(r) between the plates of the capacitor; you do not need to be familiar with capacitance or capacitors to follow this section.

- 5.17: Boundary Conditions on the Electric Field Intensity (E)
- In homogeneous media, electromagnetic quantities vary smoothly and continuously. At an interface between dissimilar media, however, it is possible for electromagnetic quantities to be discontinuous. These discontinuities can be described mathematically as boundary conditions and used to to constrain solutions for the associated electromagnetic quantities. In this section, we derive boundary conditions on the electric field intensity E .

- 5.18: Boundary Conditions on the Electric Flux Density (D)
- In this section, we derive boundary conditions on the electric flux density D . The considerations are quite similar to those encountered in the development of boundary conditions on the electric field intensity (E).

- 5.19: Charge and Electric Field for a Perfectly Conducting Region
- In this section, we consider the behavior of charge and the electric field in the vicinity of a perfect electrical conductor (PEC).

- 5.20: Dielectric Media
- Dielectric is particular category of materials that exhibit low conductivity because their constituent molecules remain intact when exposed to an electric field, as opposed to shedding electrons as is the case in good conductors. Subsequently, dielectrics do not effectively pass current, and are therefore considered “good insulators” as well as “poor conductors.” An important application of dielectrics in electrical engineering is as a spacer material in printed circuit boards & coaxial cables.

- 5.21: Dielectric Breakdown
- All practical dielectrics fail with sufficiently strong electric field, which is abrupt and is observed as a sudden, dramatic increase in conductivity, signaling that electrons are being successfully dislodged from their host molecules. The threshold value of the electric field intensity at which this occurs is known as the dielectric strength, and the sudden change in behavior observed in the presence of an electric field greater than this threshold value is known as dielectric breakdown.

- 5.22: Capacitance
- Capacitance is the ability of a structure to store energy in an electric field.

- 5.23: The Thin Parallel Plate Capacitor
- This section determines the capacitance of a common type of capacitor known as the thin parallel plate capacitor. This capacitor consists of two flat plates, each having area A , separated by distance d .

- 5.24: Capacitance of a Coaxial Structure
- This section determines the capacitance of coaxially-arranged conductors. Among other applications, this information is useful in the analysis of voltage and current waves on coaxial transmission line.

- 5.25: Electrostatic Energy
- Assuming the conductors are not free to move, potential energy is stored in the electric field associated with the surface charges. We now ask the question, what is the energy stored in this field? The answer to this question has relevance in several engineering applications. For example, when capacitors are used as batteries, it is useful to know to amount of energy that can be stored.

Thumbnail: Electric field lines due to a point charge in the vicinity of PEC regions (shaded) of various shapes. (CC BY SA 4.0; K. Kikkeri).