Introduction to Electromagnetism
Electromagnetism is the branch of physics that describes how electric charges and electric currents interact through fields. It explains why a capacitor stores energy, why a transformer works, why a coaxial cable has a characteristic impedance, and why a piece of wire half a wavelength long radiates power into space. For an RF engineer, electromagnetism is not background theory — it is the rule book that every component on the bench obeys.
The central idea is the field. Instead of saying that one charge "reaches across" empty space to push another, we say that every charge fills the space around it with an electric field, and every moving charge also creates a magnetic field. Any other charge placed in that space feels a force determined by the local value of the field. The field exists whether or not a test charge is there to feel it, it carries energy, and — as we will see in later parts of this series — it can detach from its source and travel as a wave.
This series builds the theory one step at a time. This first article introduces the two fields and their basic properties. Parts 2 to 6 develop the four laws that govern them — Gauss's law, Gauss's law for magnetism, Faraday's law and the Ampère–Maxwell law — and Parts 7 to 10 combine them into Maxwell's equations, derive electromagnetic waves, and apply the results to transmission lines, waveguides, microstrip circuits and antennas.
Vector quantities are written in bold: 𝐄 (electric field), 𝐃 (electric flux density), 𝐁 (magnetic flux density), 𝐇 (magnetic field intensity), 𝐉 (current density). A hat marks a unit vector, e.g. r̂. Constants: ε₀ = 8.854 × 10⁻¹² F/m (permittivity of free space) and μ₀ ≈ 4π × 10⁻⁷ H/m (permeability of free space).
What Is an Electric Field?
Definition of an Electric Field
An electric field is the region around an electric charge in which another charge experiences a force. It is defined operationally: place a very small positive test charge q₀ at a point, measure the force 𝐅 on it, and divide by the charge. The test charge must be small so that it does not disturb the charges that create the field.
Once the field is known, the force on any charge q at that point follows immediately as 𝐅 = q𝐄. A positive charge is pushed along the field; a negative charge such as an electron is pushed against it. This is exactly what happens to the conduction electrons in a wire or a PCB trace when a voltage is applied: the electric field inside the conductor drives them, and their motion is the current.
Electric Field Intensity
The magnitude of 𝐄 at a point is called the electric field intensity (or field strength). For a single point charge q in free space, Coulomb's law gives the field at distance r:
Field intensity in practice spans a huge range. The fair-weather field near the ground is about 100 V/m; the field 1 m from a 1 kW broadcast transmitter antenna can be tens of V/m; dry air breaks down (sparks) at about 3 MV/m, which sets the peak-power limit of air-filled coaxial cables and waveguides. Receiver sensitivities, by contrast, correspond to fields of microvolts per metre.
A charge of q = 1 nC produces, at r = 1 cm: E = 8.988 × 10⁹ × 10⁻⁹ / (0.01)² = 89.9 kV/m. At r = 10 cm the field is 100 times smaller: 899 V/m. This rapid fall-off is why electrostatic discharge (ESD) damage is concentrated where a charged object touches a pin.
Electric Field Lines and Their Properties
Field lines (lines of force), introduced by Michael Faraday, are a way to draw a vector field. A field line is a curve whose tangent at every point is in the direction of 𝐄. They obey a small set of rules:
- They start on positive charges and end on negative charges (or extend to infinity). In electrostatics they never form closed loops.
- The tangent gives the field direction at every point along the line.
- Line density shows field strength. Where lines crowd together the field is strong; where they spread apart it is weak.
- Field lines never cross. The field has a single direction at each point, so two lines cannot pass through the same point.
- The number of lines leaving a charge is proportional to the charge. This idea becomes exact in Gauss's law (Part 2).
- Lines meet a conductor at right angles. In electrostatic equilibrium the surface of a conductor is an equipotential, so the field has no component along the surface. This is one of the boundary conditions studied in Part 9.
Electric Fields Produced by Electric Charges
Electric fields obey the principle of superposition: the total field from several charges is the vector sum of the fields each would produce alone. For N point charges, and for a continuous charge distribution with volume charge density ρ_v, the field is:
Three standard results are worth memorising because they appear throughout RF engineering:
| Source | Field magnitude | Fall-off | Where you meet it |
|---|---|---|---|
| Point charge q | q / (4πε₀r²) | 1/r² | ESD, charged particles |
| Infinite line charge λ (C/m) | λ / (2πε₀r) | 1/r | Inner conductor of a coaxial cable, wires |
| Infinite sheet σ (C/m²) | σ / (2ε₀) | Constant | Parallel-plate capacitor (σ/ε₀ between plates) |
| Electric dipole p = qd (far field) | p / (4πε₀r³) × (angular factor) | 1/r³ | Molecules, short dipole antenna (static part) |
What Is a Magnetic Field?
Definition of a Magnetic Field
A magnetic field is the region around a moving charge, an electric current or a magnetised material in which a moving charge experiences a force. A stationary charge feels no magnetic force at all. The magnetic flux density 𝐁 is defined through the force on a charge q moving with velocity 𝐯:
Magnetic Field Intensity
Two related vectors describe the magnetic field, and keeping them apart avoids a great deal of confusion:
- Magnetic flux density 𝐁, measured in tesla (T = Wb/m²). This is the quantity that appears in the force law and in Faraday's law. The Earth's field is about 50 μT; a fridge magnet is about 5 mT; an MRI scanner is 1.5–3 T.
- Magnetic field intensity 𝐇, measured in amperes per metre (A/m). This is the quantity set directly by currents (Ampère's law, Part 5), independent of the material.
Magnetic Field Lines and Their Properties
Magnetic field lines are drawn with the same convention as electric ones — tangent along 𝐁, density proportional to field strength — but their topology is fundamentally different:
- They always form closed loops. They have no starting or ending point, because there are no isolated magnetic charges (monopoles). Part 3 shows that this is precisely what Gauss's law for magnetism states.
- Outside a magnet they run from the north pole to the south pole; inside the magnet they continue from south back to north, closing the loop.
- Around a straight current they are concentric circles, with a direction given by the right-hand rule: thumb along the current, fingers curl in the direction of 𝐁.
- They never cross, and their density indicates the magnitude of 𝐁.
Magnetic Fields Produced by Electric Currents
In 1820 Hans Christian Ørsted noticed that a current-carrying wire deflects a nearby compass needle — the first proof that electricity and magnetism are connected. Shortly afterwards, Biot and Savart quantified the field produced by a small current element I d𝐥:
A wire carrying I = 1 A produces, at r = 1 cm: B = μ₀I / (2πr) = 2 × 10⁻⁷ × 1 / 0.01 = 20 μT — comparable to the Earth's field. This is why high-current DC traces and switching-regulator loops can disturb magnetometers and why compasses are kept away from power cables.
Electric Fields vs. Magnetic Fields
Although electric and magnetic fields are two aspects of one electromagnetic field, in static situations they behave very differently. The table below summarises the comparison.
| Property | Electric field | Magnetic field |
|---|---|---|
| Source | Electric charges (stationary or moving) | Moving charges, currents, magnetic materials |
| Field vectors | 𝐄 (V/m), 𝐃 = ε𝐄 (C/m²) | 𝐇 (A/m), 𝐁 = μ𝐇 (T) |
| Acts on | Any charge, moving or not | Moving charges only |
| Force | 𝐅 = q𝐄 (along the field) | 𝐅 = q𝐯 × 𝐁 (perpendicular) |
| Does work on charges? | Yes — changes their kinetic energy | No — only changes their direction |
| Field lines | Start on + charges, end on − charges | Always closed loops |
| Material constant | Permittivity ε = ε₀ε_r | Permeability μ = μ₀μ_r |
| Energy density | w_e = ½ εE² | w_m = ½ μH² |
| Circuit element that stores it | Capacitor | Inductor |
| Governing laws (this series) | Gauss's law — Part 2 | Gauss's law for magnetism — Part 3, Ampère — Part 5 |
The last rows matter most for circuit designers. A capacitor is a device that stores energy in an electric field; an inductor stores energy in a magnetic field. At RF, every trace has both: the electric field between the trace and the ground plane gives it capacitance per unit length, and the magnetic field around its current gives it inductance per unit length. Together they determine its characteristic impedance — a result derived from field theory in Parts 2, 5 and 10.
Relationship Between Electric Charges, Currents, and Electromagnetic Fields
The link between charges and fields can be summarised in three levels, each adding more physics:
- Stationary charges produce only an electric field (electrostatics).
- Charges moving at constant velocity — steady (DC) currents — produce an electric field and a constant magnetic field (magnetostatics). The two fields can still be analysed separately.
- Accelerating charges — time-varying currents — produce electric and magnetic fields that are coupled to each other. A changing magnetic field induces an electric field (Faraday, Part 4), and a changing electric field induces a magnetic field (Maxwell's displacement current, Part 6). This mutual coupling is what creates radiation.
The complete force on a charge in the presence of both fields is the Lorentz force:
Charge is also conserved: if current flows out of a region, the charge inside must decrease at the same rate. This is expressed by the continuity equation ∇·𝐉 = −∂ρ_v/∂t. It looks innocent, but in Part 6 it is precisely this equation that forces Maxwell to modify Ampère's law and, as a consequence, to predict electromagnetic waves.
Introduction to Electromagnetic Waves
When the current in a wire changes with time, the fields around it cannot update everywhere instantly. The change propagates outward at a finite speed. Because a changing magnetic field creates an electric field and a changing electric field creates a magnetic field, the disturbance can sustain itself without any wire at all: this self-propagating disturbance is an electromagnetic wave.
In free space an electromagnetic wave has three defining features. The electric and magnetic fields are perpendicular to each other and to the direction of travel (a transverse wave); their magnitudes are locked in a fixed ratio; and the wave travels at a speed fixed by the two constants ε₀ and μ₀:
Wavelength sets the physical scale of RF hardware. At 2.4 GHz, λ = 2.998 × 10⁸ / 2.4 × 10⁹ = 12.5 cm, so a half-wave dipole is about 6 cm long and a PCB trace longer than about λ/10 (≈ 1 cm in FR-4, where waves are slower) must be treated as a transmission line rather than a simple wire.
Importance of Electromagnetic Fields in RF Engineering
At low frequencies, circuit theory — Kirchhoff's laws, lumped resistors, capacitors and inductors — is an excellent approximation because the circuit is tiny compared with a wavelength. As frequency rises and dimensions become a noticeable fraction of λ, the approximation breaks down and the engineer must think in fields. Field theory is essential for:
Modern design tools — Ansys HFSS, CST Studio Suite, Keysight Momentum and openEMS — all work by solving Maxwell's equations numerically. Understanding the underlying field theory is what allows an engineer to set up those simulations correctly and to judge whether the results are physically sensible.
Summary
| Concept | Key relation | Unit | Remember |
|---|---|---|---|
| Electric field | 𝐄 = 𝐅 / q₀ | V/m | Force per unit charge; acts on all charges |
| Point-charge field | E = q / (4πε₀r²) | V/m | Falls off as 1/r² |
| Magnetic force | 𝐅 = q𝐯 × 𝐁 | N | Acts only on moving charges, perpendicular to motion |
| B–H relation | 𝐁 = μ₀μ_r𝐇 | T, A/m | μ_r ≈ 1 for most RF materials |
| Long straight wire | B = μ₀I / (2πr) | T | Circles the wire; right-hand rule |
| Lorentz force | 𝐅 = q(𝐄 + 𝐯 × 𝐁) | N | Complete force on a charge |
| Speed of light | c = 1/√(μ₀ε₀) | m/s | ≈ 3 × 10⁸ m/s |
| Free-space impedance | η₀ = √(μ₀/ε₀) | Ω | ≈ 377 Ω |
- Charges create electric fields; moving charges (currents) also create magnetic fields.
- Electric field lines begin and end on charges; magnetic field lines always form closed loops.
- When fields vary in time they become coupled, and the coupled fields propagate as electromagnetic waves at the speed of light.
- Above a few hundred megahertz, circuit dimensions become comparable to the wavelength and RF design has to be based on field theory.
In Part 2: Understanding Electric Flux and Gauss's Law, we make the idea of "number of field lines" precise by defining electric flux, and use it to state Gauss's law — the first of Maxwell's four equations.
Put It Into Practice
Use the NEXT ANTENNAS calculators to apply the ideas from this article to real designs.