1. Electrostatics, Properly This Time

We met electrostatics in Chapter 0, but we sketched it. To do EM waves with confidence we need the toolkit a little sharper. None of this is busywork. Each equation here is a piece you will use when we derive the wave equation, when we compute coax capacitance, when we design a microstrip impedance, or when we reason about why a shielded cable does not leak.

1.1 Coordinate systems: matching geometry to symmetry

The single most common reason students get tangled in EM is using Cartesian coordinates for problems that scream for cylindrical or spherical ones. Three coordinate systems are used in EM, and the right choice cuts the work down by an order of magnitude.

• Cartesian $(x, y, z)$: flat geometries. Parallel-plate capacitors, infinite charged sheets, microstrip on a uniform PCB. Anything where the natural directions are perpendicular flat planes.
• Cylindrical $(\rho, \phi, z)$: cylindrically symmetric systems. Coaxial cable is the textbook example: an inner conductor along the $z$-axis, an outer shield at radius $b$, with everything depending only on $\rho$. Long straight wires, solenoids, optical fibers all live in cylindrical coordinates.
• Spherical $(r, \theta, \phi)$: anything radiating from a point. Antenna far fields are spherical: at large distance, the radiated wave looks like a sphere expanding outward from a point source, the field amplitude falling as $1/r$ and the intensity as $1/r^2$ (the inverse-square law you may have heard for sound).

Geometry analogy. Imagine you are wrapping a present. A book wraps cleanly with rectangular paper (Cartesian). A bottle of wine wants a cylindrical sleeve. A globe wants a sphere of cloth. Forcing one wrapping onto the wrong shape produces ugly creases. Forcing the wrong coordinates onto an EM problem produces ugly integrals.

1.2 Coulomb's law revisited

Two point charges in vacuum, separated by distance $r$, exert a force on each other:

$\vec{F}_{12} = \frac{1}{4\pi\varepsilon_0}\frac{q_1 q_2}{r^2}\hat{r}$

The constant $1/(4\pi\varepsilon_0) \approx 9\times 10^9$ N·m²/C². The permittivity of free space $\varepsilon_0 = 8.854\times 10^{-12}$ F/m sets how strongly empty space resists letting an electric field exist. Bigger $\varepsilon_0$ would mean a weaker force.

Trampoline analogy revisited. A heavy ball deformed the trampoline; nearby balls rolled into the dip. The deformation is the field; the rolling is the force. Two balls feel each other's dips with a strength that falls off as $1/r^2$ because the dip's slope spreads its energy over the surface of an ever-larger imaginary sphere as you go outward.

1.3 Electric field intensity for canonical configurations

Field per unit positive test charge, $\vec{E} = \vec{F}/q_t$, with units V/m. Three configurations recur enough that you should commit them to memory.

• Point charge at origin: $\displaystyle\vec{E} = \frac{Q}{4\pi\varepsilon_0 r^2}\hat{r}$. Falls as $1/r^2$.
• Infinite line of charge with linear density $\rho_L$ along the $z$-axis: $\displaystyle\vec{E} = \frac{\rho_L}{2\pi\varepsilon_0\rho}\hat{\rho}$. Falls only as $1/\rho$, slower than a point charge.
• Infinite charged plane with surface density $\rho_S$: $\displaystyle\vec{E} = \frac{\rho_S}{2\varepsilon_0}\hat{n}$. Constant with distance. The field of a flat sheet does not fall off.

The reason for the different falloffs is geometric. Around a point charge, field lines fan into a sphere whose surface area grows as $r^2$. Around a line, they fan into a cylinder whose surface grows as $r$. Around a plane, they march straight out, the cross-sectional area constant. The strength of the field is the line density, so it scales inversely with that area.

1.4 Electric flux density and Gauss's law

When fields enter materials they polarize the bound charges, and tracking everything by $\vec{E}$ alone gets clumsy. Define electric flux density

$\vec{D} = \varepsilon\vec{E}, \qquad \varepsilon = \varepsilon_0\varepsilon_r$

where $\varepsilon_r$ is the relative permittivity of the material. For air, $\varepsilon_r \approx 1$. For FR-4 PCB substrate, $\varepsilon_r \approx 4.5$. For Teflon, $\approx 2.1$. For Rogers RF substrates, $\approx 2.2$ to $\approx 10$. For ceramic capacitor dielectrics, $\varepsilon_r$ can reach thousands.

The advantage of $\vec{D}$ is that Gauss's law in terms of $\vec{D}$ does not depend on the medium:

$\oint_S \vec{D}\cdot d\vec{A} = Q_{enclosed}$

In words: the total flux of $\vec{D}$ through a closed surface equals the charge it encloses. Choose the surface to match the symmetry of the problem and the integral evaluates by inspection.

Coax cable field, by Gauss. Pick a Gaussian cylinder of radius $\rho$ inside a coax cable, between inner and outer conductors. By symmetry, $\vec{D}$ points radially and has the same magnitude on the cylinder's curved surface. Multiply by the surface area $2\pi\rho L$, and you get $D \cdot 2\pi\rho L = \rho_L L$, where $\rho_L$ is the charge per unit length on the inner conductor. So $D = \rho_L/(2\pi\rho)$. From there, $E$, voltage, and capacitance fall out by integration. Now pick a Gaussian cylinder outside the shield. The enclosed charge is zero (inner conductor and shield carry equal and opposite charge). So $\vec{D} = 0$ outside the cable. Coax confines its field perfectly. This single fact is why coax does not leak signals into adjacent cables, and why it is the workhorse of cable TV, RF test equipment, base-station antenna feedlines, and TEMPEST-resistant data links.

1.5 Electric potential and the gradient relation

Voltage between two points equals the work per unit charge to move a test charge from one to the other against the field:

$V_{AB} = -\int_A^B \vec{E}\cdot d\vec{l}$

Equivalently, the field is the negative gradient of the potential:

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

Equipotential surfaces are perpendicular to field lines everywhere. This is why a charged conductor's surface is equipotential; field lines must enter or leave it perpendicularly.

1.6 Conductors, dielectrics, and capacitance

Inside a perfect conductor at electrostatic equilibrium, $\vec{E} = 0$. Charges have rearranged themselves on the surface until any internal field would cause them to move, so they did move, until none was left. This produces three properties used constantly:

• The interior of a conductor is field-free.
• All free charge sits on the outer surface, never in the bulk.
• The conductor surface is an equipotential.

In a dielectric (insulator), the free charge cannot wander, but the bound charge can polarize. An external field stretches each atom or molecule a little: the electron cloud shifts opposite to the field, the nucleus shifts with it, creating tiny induced dipoles. Their net effect is to partially cancel the external field inside the dielectric, with the residual field given by $\vec{E} = \vec{D}/\varepsilon$.

Capacitance is the ratio of charge stored to voltage across a conductor pair: $C = Q/V$. Three classic results, each derivable by Gauss + integration:

• Parallel plate (area $A$, separation $d$): $C = \varepsilon A/d$.
• Coaxial cable (inner radius $a$, outer $b$, length $L$): $\displaystyle C = \frac{2\pi\varepsilon L}{\ln(b/a)}$.
• Spherical capacitor (inner radius $a$, outer $b$): $\displaystyle C = 4\pi\varepsilon\,\frac{ab}{b - a}$.

For RG-58 50 Ω coax in air-equivalent dielectric: about 100 pF/m. For a typical 50 Ω microstrip on FR-4: a similar order of magnitude. We will use this when we derive characteristic impedance.

1.7 Energy stored in an electric field

The energy density of an electrostatic field is

$w_E = \frac{1}{2}\varepsilon E^2 = \frac{1}{2}\vec{D}\cdot\vec{E}$

Total energy in any volume is the integral of $w_E$. For a charged capacitor, this volume integral evaluates to $\frac{1}{2}CV^2$, the same expression you saw in Chapter 2 from a different starting point. The two pictures (energy in plates, energy in field) are equivalent, and the field picture is the one that generalizes to radiation.

1.8 Continuity equation and the relaxation time

Charge cannot be created or destroyed, only redistributed. The mathematical statement is the continuity equation:

$\nabla\cdot\vec{J} + \frac{\partial\rho}{\partial t} = 0$

Inside a conductor with conductivity $\sigma$, Ohm's law in field form gives $\vec{J} = \sigma\vec{E}$, and combining with Gauss's law leads to a first-order ODE for any temporary excess charge density $\rho$:

$\frac{\partial\rho}{\partial t} + \frac{\sigma}{\varepsilon}\rho = 0$

The solution is $\rho(t) = \rho_0\,e^{-t/\tau}$ with relaxation time $\tau = \varepsilon/\sigma$.

For copper, $\sigma \approx 5.8\times 10^7$ S/m and $\varepsilon \approx \varepsilon_0$, so $\tau \approx 10^{-19}$ seconds, far shorter than any timescale you will encounter in practice. Free charge inside a conductor disperses essentially instantaneously to the surface. That is why the interior of a metal box is field-free even when the outside is held at a high voltage: any charge in the bulk would self-disperse in tens of attoseconds.

For a poor conductor like distilled water ($\sigma \sim 10^{-4}$ S/m) the relaxation time is microseconds, slow enough to matter in some specialized devices.

1.9 Poisson and Laplace equations

Combine Gauss's law with $\vec{E} = -\nabla V$:

$\nabla\cdot(-\varepsilon\nabla V) = \rho \quad\Rightarrow\quad \nabla^2 V = -\rho/\varepsilon \qquad (\text{Poisson})$

In a charge-free region, $\rho = 0$ and we get

$\nabla^2 V = 0 \qquad (\text{Laplace})$

Capacitance, microstrip impedance, MEMS plate capacitance, and PCB cross-talk are all in the end Poisson or Laplace problems. Modern EM solvers (HFSS, CST, Ansys SIwave, OpenEMS) do nothing more sophisticated than discretize $\nabla^2 V$ and solve the resulting matrix. Knowing this means that when a solver tells you a number, you understand exactly what it computed and why.

1.10 Where electrostatics shows up in real hardware

• Capacitive touchscreens. Your finger forms a capacitor with each grid intersection.
• MEMS pressure sensors. A diaphragm changes plate spacing, hence capacitance.
• Electrostatic chucks in semiconductor processing, holding wafers with high-voltage fields.
• High-voltage transmission lines designed to keep field below the corona discharge threshold of about 3 MV/m.
• Lightning rods, where a sharp tip concentrates field, ionizes air, and steers strikes to ground.
• Smartcards and security tokens, where electrostatic discharge can fault a chip mid-cryptographic-operation, leaking key material in the resulting glitch. Defending against this is one of the simplest reasons your bank card has carefully placed ESD diodes and conformal coating.