Electromagnetics: Electrostatics

Electrostatics is the study of electric charges at rest and the fields and potentials they produce. Despite its “static” label, electrostatics is foundational for much of electromagnetics, from how sensors work to how cables, capacitors, and insulating materials are designed. The core ideas are simple: charges exert forces, forces define an electric field, fields relate to flux through surfaces, and fields can be described compactly using electric potential. Those ideas become especially powerful when combined with boundary value methods that let you solve real geometries, not just textbook point charges.

Coulomb’s Law: The Starting Point

The fundamental interaction between two stationary point charges is given by Coulomb’s law. If charges $q_1$ and $q_2$ are separated by distance $r$ in a uniform medium, the magnitude of the force is proportional to the product of the charges and inversely proportional to $r^2$. In vector form, the force on $q_2$ due to $q_1$ points along the line joining them: repulsive for like charges, attractive for unlike.

In vacuum, Coulomb’s constant is $1/(4\pi {\epsilon }_0)$, where ${\epsilon }_0$ is the permittivity of free space. In materials, the permittivity is typically $\epsilon ={\epsilon }_r{\epsilon }_0$, and the same structure holds but with the appropriate $\epsilon$. This is not a small detail: permittivity is what makes the same physical geometry behave differently when you fill it with air, oil, ceramic, or polymer.

Coulomb’s law is most directly useful for a small number of discrete charges. Real engineering problems involve continuous charge distributions, and for those we generalize via integration.

Electric Field: From Force to a Field Description

The electric field $\mathbf{E}$ is defined as force per unit test charge: $\mathbf{E}=\mathbf{F}/q$. This definition turns a two-body force law into a field that exists everywhere in space, determined by the source charges.

For a continuous charge distribution, you typically work with one of these densities:

• Volume charge density $\rho (\mathbf{r})$ (C/m³)
• Surface charge density $\sigma (\mathbf{r})$ (C/m²)
• Line charge density $\lambda (\mathbf{r})$ (C/m)

The electric field is then found by superposition, adding the contribution from each differential charge element. In practice, direct integration works well when symmetry is high (spheres, infinite lines, planes). When symmetry is limited, field computation is often replaced by potential methods and boundary value problem techniques.

Field Lines and Physical Interpretation

Field lines are a visualization tool, not a physical substance. They point in the direction of $\mathbf{E}$ and their density suggests field magnitude. They originate on positive charge and terminate on negative charge, or extend to infinity if there is a net charge.

A key physical fact in electrostatics is that conductors in electrostatic equilibrium have zero electric field inside the conducting bulk. Any excess charge resides on the surface, arranging itself so the internal field cancels. This property drives many boundary conditions used in practical solutions.

Gauss’s Law: Flux, Symmetry, and a Powerful Shortcut

Gauss’s law connects the electric field to the total enclosed charge:

$${\oint }_S\mathbf{E}\cdot d\mathbf{A}=\frac{Q_{\text{enc}}}{\epsilon }$$

The left side is the electric flux through a closed surface $S$. The right side depends only on the net charge inside.

Gauss’s law is always true in electrostatics, but it only becomes a “shortcut” when symmetry lets you argue that $\mathbf{E}$ is constant in magnitude on parts of the surface and aligned with the surface normal. Classic examples include:

• Spherical symmetry (charged sphere or point charge): choose a concentric spherical Gaussian surface.
• Cylindrical symmetry (infinite line charge): choose a coaxial cylindrical surface.
• Planar symmetry (infinite sheet of charge): choose a pillbox surface straddling the sheet.

For less symmetric shapes, Gauss’s law still constrains solutions, but it will not directly give $\mathbf{E}$ without additional analysis.

Electric Potential: A Scalar Description of the Field

Electrostatics is conservative, meaning the work done by the electrostatic field around any closed path is zero. This allows defining an electric potential $V$ such that

$$\mathbf{E}=-\nabla V$$

This single equation is one of the most practical tools in electromagnetics. Instead of solving for a vector field directly, you can often solve for a scalar potential and then take its gradient.

Potential differences are what you measure with a voltmeter, and they are what drive currents once you allow charges to move (electrostatics is the limit before conduction and time variation matter). The potential due to a point charge in a uniform medium takes the familiar $1/r$ form, and for continuous distributions the potential is found by integrating the contributions of all elements.

Equipotentials and Conductors

A conductor in electrostatic equilibrium is an equipotential body: $V$ is constant throughout the conductor and on its surface. Since $\mathbf{E}=-\nabla V$, a constant potential implies zero field inside. At the surface, the electric field is normal to the surface; any tangential component would drive surface charges to move, contradicting equilibrium.

These statements are not just theory. They justify why shielding works (Faraday cages), why sharp tips concentrate electric field (leading to corona and discharge), and why high-voltage hardware avoids small radii of curvature.

Boundary Value Problems: Solving Real Geometries

Most practical electrostatics problems are boundary value problems: you know something about the potential or charge on boundaries, and you want the field and potential in the region.

The governing equation is Poisson’s equation:

$${\nabla }^{2}V=-\frac{\rho }{\epsilon }$$

In charge-free regions where $\rho =0$, it reduces to Laplace’s equation:

$${\nabla }^{2}V=0$$

What makes boundary value methods powerful is that solutions to Laplace’s equation are strongly constrained by boundary conditions. In many cases, specifying the potential on all boundaries (Dirichlet conditions) or specifying the normal derivative of potential, equivalently the normal component of $\mathbf{E}$ (Neumann conditions), determines a unique solution.

Common boundary conditions in electrostatics include:

• Perfect conductor surface: $V$ is constant (often set to 0 V for a grounded conductor).
• Dielectric interface: tangential $\mathbf{E}$ is continuous; normal electric flux density $\mathbf{D}=\epsilon \mathbf{E}$ changes based on free surface charge.
• Infinity boundary: fields vanish far away for localized charge distributions.

Practical solution strategies range from separation of variables in standard coordinates (useful for parallel plates, cylinders, spheres) to method of images (useful near infinite conducting planes and spheres) to numerical methods such as finite element analysis for complex geometries.

Capacitance: Storing Energy in Electric Fields

Capacitance quantifies how much charge is stored per unit potential difference:

$$C=\frac{Q}{\Delta V}$$

A capacitor is not a container of charge so much as a structure that stores energy in the electric field between conductors. For linear dielectrics and fixed geometry, capacitance depends only on shape and permittivity, not on $Q$ or $\Delta V$ individually.

The energy stored in a capacitor is

$$U=\frac{1}{2}C(\Delta V{)}^2$$

This energy resides in the electric field in the dielectric region. In many engineering contexts, thinking in terms of field energy density is useful because it connects directly to insulation stress, breakdown, and force.

A Practical Example: Parallel-Plate Capacitor

A classic configuration is two large, closely spaced plates with area $A$ and separation $d$, filled with a dielectric of permittivity $\epsilon$. Neglecting fringing, the field is approximately uniform, and capacitance scales as $C\propto \epsilon A/d$. This simple scaling explains common design moves: increase plate area to increase capacitance, decrease separation to increase capacitance (limited by dielectric strength), or use a higher-permittivity dielectric to increase capacitance without changing geometry.

Bringing It Together: From Charges to Solutions

Electrostatics ties together a small set of laws into a coherent toolkit:

• Coulomb’s law describes forces between charges.
• Electric field generalizes forces into a spatial vector field.
• Gauss’s law links field flux to enclosed charge and enables symmetry-based solutions.
• Potential converts a vector field problem into a scalar one via $\mathbf{E}=-\nabla V$.
• Poisson and Laplace equations turn electrostatics into boundary value problems suitable for real devices.
• Capacitance packages geometry and materials into a directly usable circuit parameter, grounded in field behavior.

Understanding these ideas as a connected set is what turns electrostatics from memorized formulas into a practical design and analysis method for real electromagnetic systems.