Electromagnetic Waves, Antennas, and RF: From a 1D Wave to a Field
AntennasElectromagneticWaves
2026-09-15 00:53:09
2026-09-15 00:53:09
Topic
shorten title
electromagnetic waves
radio waves
scalar field
vector field
one-dimensional wave
spatial field
time-dependent field
electric field
plane wave
transverse wave
position vector
field snapshot
time history
GPS
RF
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\section*{Electromagnetic Waves, Antennas, and RF: From a 1D Wave to a Field}
The Wave Mechanics series developed waves gradually from an oscillation at one point to quantities that depend on both position and time. A typical one-dimensional wave was written as
\begin{equation}
u=u(x,t).
\end{equation}
That notation already contains the central idea needed to begin electromagnetism: a physical quantity can be assigned to every point in space and can change with time. Such a quantity is called a \emph{field}.
Electromagnetism uses fields as its basic language. Electric and magnetic fields are not merely single numbers attached to one object. They are quantities defined throughout space, and they can vary from place to place and from instant to instant \cite{Griffiths2017,OpenStaxV2,FeynmanV2,MIT802}.
EM01 makes the transition in four small steps:
\begin{equation}
\boxed{
u(t)
\longrightarrow
u(x,t)
\longrightarrow
\psi(x,y,z,t)
\longrightarrow
\mathbf E(x,y,z,t).}
\end{equation}
The first three expressions are scalar-valued. The last is vector-valued. Maxwell's equations are intentionally deferred. The goal here is only to become comfortable with what a field is and how wave language extends from one spatial dimension to three.
\section{A one-dimensional wave is already a field}
Consider a one-dimensional wave
\begin{equation}
u=u(x,t).
\end{equation}
At a fixed time $t=t_0$, the function becomes
\begin{equation}
u=u(x,t_0).
\end{equation}
Every position $x$ is assigned one number $u$. That is already a scalar field on a one-dimensional space.
For example, the snapshot
\begin{equation}
u(x,t_0)=A\cos(kx)
\end{equation}
assigns a displacement to every point along the $x$ axis.
\begin{center}
\includegraphics{EM01_fig01_1d_wave_field.png}
\vspace{0.45em}
\textbf{Figure.}
At one instant, a one-dimensional wave assigns one scalar value to every position on the line. The plotted curve is therefore a picture of a scalar field in one spatial dimension.
\end{center}
This viewpoint is more general than a vibrating string. The quantity $u$ could represent pressure perturbation, voltage, density perturbation, temperature variation, or another scalar-valued physical quantity.
The essential structure is
\begin{equation}
\boxed{\text{position and time} \longmapsto \text{physical value}.}
\end{equation}
\section{Function notation as a map}
It is useful to read
\begin{equation}
u(x,t)
\end{equation}
as an instruction:
\begin{quote}
Give the function a position $x$ and a time $t$; the function returns the value of the field there and then.
\end{quote}
For example, if
\begin{equation}
u(x,t)=A\cos(kx-\omega t),
\end{equation}
then the number
\begin{equation}
u(2.0\,\text{m},0.10\,\text{s})
\end{equation}
is the value of the field at one particular event in space and time.
An \emph{event} in this elementary sense is simply a specified place and time.
This language will be important later because electromagnetic fields such as
\begin{equation}
\mathbf E(\mathbf r,t)
\end{equation}
are evaluated in exactly the same way.
\section{From one spatial coordinate to three}
The simplest wave model uses one spatial coordinate:
\begin{equation}
u=u(x,t).
\end{equation}
A field in ordinary three-dimensional space can depend on
\begin{equation}
x,\qquad y,\qquad z,
\end{equation}
as well as time. A scalar field may therefore be written
\begin{equation}
\boxed{\psi=\psi(x,y,z,t).}
\end{equation}
At every point $(x,y,z)$ and time $t$, the field returns one scalar number.
Examples of scalar fields include temperature,
\begin{equation}
T=T(x,y,z,t),
\end{equation}
and pressure,
\begin{equation}
p=p(x,y,z,t).
\end{equation}
The physical meaning differs, but the mathematics has the same structure.
\section{A scalar-field snapshot}
At a fixed time $t=t_0$, a three-dimensional scalar field becomes
\begin{equation}
\psi=\psi(x,y,z,t_0).
\end{equation}
If we further examine only a plane such as $z=0$, then
\begin{equation}
\psi=\psi(x,y,0,t_0).
\end{equation}
This can be drawn using contours. Each contour joins points with equal field value.
\begin{center}
\includegraphics{EM01_fig02_scalar_field_snapshot.png}
\vspace{0.45em}
\textbf{Figure.}
A two-dimensional slice through a scalar field at one instant. Each spatial point has one scalar value, and a contour connects locations with the same value.
\end{center}
A contour plot is therefore not a trajectory of a moving particle. It is a map of field values across space.
That distinction will matter later when electric-field lines and wavefronts are introduced. A line drawn in a field diagram does not automatically mean that matter or energy physically travels along that line.
\section{The position vector}
Writing all three coordinates repeatedly becomes cumbersome. We therefore introduce the position vector
\begin{equation}
\boxed{
\mathbf r
=x\hat{\mathbf x}
+y\hat{\mathbf y}
+z\hat{\mathbf z}.}
\end{equation}
The notation
\begin{equation}
\psi(\mathbf r,t)
\end{equation}
is shorthand for
\begin{equation}
\psi(x,y,z,t).
\end{equation}
Likewise,
\begin{equation}
\mathbf E(\mathbf r,t)
\end{equation}
means a vector field evaluated at position $\mathbf r$ and time $t$.
No new physics has been introduced by this notation. It is only a compact way to refer to a location in three-dimensional space.
\section{Scalar fields and vector fields}
A scalar field assigns one number to every point. A vector field assigns a vector to every point.
A two-dimensional vector field can be written
\begin{equation}
\boxed{
\mathbf A(x,y,t)
=A_x(x,y,t)\hat{\mathbf x}
+A_y(x,y,t)\hat{\mathbf y}.}
\end{equation}
A three-dimensional vector field has three components:
\begin{equation}
\boxed{
\mathbf A(x,y,z,t)
=A_x\hat{\mathbf x}
+A_y\hat{\mathbf y}
+A_z\hat{\mathbf z}.}
\end{equation}
Each component is itself a scalar field.
Thus a vector field can be understood as several scalar component fields combined with basis directions.
\section{The electric field is vector-valued}
The electric field will eventually be defined physically through the force experienced by charge. For now, we need only its mathematical type:
\begin{equation}
\boxed{
\mathbf E=\mathbf E(x,y,z,t).}
\end{equation}
At each position and time, the electric field has both magnitude and direction.
In Cartesian coordinates,
\begin{equation}
\boxed{
\mathbf E
=E_x\hat{\mathbf x}
+E_y\hat{\mathbf y}
+E_z\hat{\mathbf z}.}
\end{equation}
\begin{center}
\includegraphics{EM01_fig03_vector_field_components.png}
\vspace{0.45em}
\textbf{Figure.}
A vector field assigns a vector to each point. The vector can be decomposed into Cartesian components such as $E_x\hat{\mathbf x}$ and $E_y\hat{\mathbf y}$.
\end{center}
EM02 will develop vector components, dot products, cross products, and transverse directions more systematically.
\section{A vector field is not a moving arrow}
A common conceptual mistake is to imagine that the arrow representing a vector field is itself an object moving through space.
Instead, at one instant the field specifies a vector at every position.
For example, suppose
\begin{equation}
\mathbf E(x,y,t_0)
=(2x)\hat{\mathbf x}
+(3y)\hat{\mathbf y}.
\end{equation}
At the point
\begin{equation}
x=1,\qquad y=2,
\end{equation}
we obtain
\begin{equation}
\mathbf E(1,2,t_0)
=2\hat{\mathbf x}+6\hat{\mathbf y}.
\end{equation}
At another location, the vector can be different.
The collection of all these vectors is the field.
\section{Two complementary ways to inspect a time-dependent field}
A field that depends on space and time can be inspected in two especially useful ways.
\subsection{Snapshot in space}
Fix the time:
\begin{equation}
t=t_0.
\end{equation}
Then examine
\begin{equation}
\psi(\mathbf r,t_0).
\end{equation}
This answers:
\begin{quote}
What does the field look like everywhere in space at this instant?
\end{quote}
This is analogous to taking a photograph of a wave.
\subsection{Time history at one point}
Instead fix the position:
\begin{equation}
\mathbf r=\mathbf r_0.
\end{equation}
Then examine
\begin{equation}
\psi(\mathbf r_0,t).
\end{equation}
This answers:
\begin{quote}
What does one detector at one location observe as time passes?
\end{quote}
A radio receiver and a GPS antenna are physically much closer to this second viewpoint: the hardware occupies a limited region of space and measures time-varying electromagnetic quantities there.
\section{The one-dimensional traveling-wave pattern survives}
The Wave Mechanics series used expressions such as
\begin{equation}
u(x,t)=A\cos(kx-\omega t).
\end{equation}
The same mathematical idea survives in electromagnetism, but the field itself becomes vector-valued.
A simple illustrative field is
\begin{equation}
\boxed{
\mathbf E(z,t)
=\hat{\mathbf x}E_0\cos(kz-\omega t).}
\end{equation}
Read this equation carefully:
\begin{itemize}
\item the field varies with $z$ and $t$;
\item the field vector points in the $\hat{\mathbf x}$ direction;
\item the pattern propagates in the $+z$ direction because the phase is $kz-\omega t$;
\item the field direction and propagation direction are different.
\end{itemize}
\begin{center}
\includegraphics{EM01_fig04_plane_wave_vector_field.png}
\vspace{0.45em}
\textbf{Figure.}
An illustrative transverse vector wave. The electric-field vector points along $\hat{\mathbf x}$ while the sinusoidal pattern propagates along $+z$.
\end{center}
The fact that electromagnetic waves are transverse will later follow from Maxwell's equations. EM01 uses the form only to show how a familiar one-dimensional sinusoid can become a vector field.
\section{Field direction versus direction of variation}
The equation
\begin{equation}
\mathbf E(z,t)
=\hat{\mathbf x}E_0\cos(kz-\omega t)
\end{equation}
contains two different directions.
The basis vector
\begin{equation}
\hat{\mathbf x}
\end{equation}
tells us the direction in which the field vector points.
The coordinate
\begin{equation}
z
\end{equation}
tells us the direction along which the phase changes in this example.
Confusing these two directions is a common early mistake.
A field can point in one direction while changing from place to place in another direction.
\section{Field magnitude}
For a vector
\begin{equation}
\mathbf E
=E_x\hat{\mathbf x}
+E_y\hat{\mathbf y}
+E_z\hat{\mathbf z},
\end{equation}
its magnitude is
\begin{equation}
\boxed{
|\mathbf E|
=\sqrt{E_x^2+E_y^2+E_z^2}.}
\end{equation}
This is simply the three-dimensional Pythagorean theorem.
For the illustrative plane-wave field
\begin{equation}
\mathbf E(z,t)
=\hat{\mathbf x}E_0\cos(kz-\omega t),
\end{equation}
we have
\begin{equation}
E_x=E_0\cos(kz-\omega t),
\end{equation}
and
\begin{equation}
E_y=E_z=0.
\end{equation}
Therefore
\begin{equation}
|\mathbf E|
=|E_0\cos(kz-\omega t)|.
\end{equation}
The vector itself can point in either the positive or negative $x$ direction as the sinusoid changes sign.
\section{A field may be uniform or nonuniform}
A uniform vector field has the same vector at every spatial point. For example,
\begin{equation}
\mathbf E(\mathbf r,t_0)=E_0\hat{\mathbf x}
\end{equation}
is spatially uniform at time $t_0$.
By contrast,
\begin{equation}
\mathbf E(z,t_0)
=\hat{\mathbf x}E_0\cos(kz)
\end{equation}
is nonuniform because its value changes with $z$.
The words uniform and constant therefore need context. A field may be spatially uniform yet change with time, or spatially varying yet frozen at one selected instant.
\section{Fields and measurements}
Fields are mathematical models, but they are tied to measurement.
A sensor located at $\mathbf r_0$ samples a field locally. If the field is
\begin{equation}
\mathbf E(\mathbf r,t),
\end{equation}
then the sensor encounters
\begin{equation}
\mathbf E(\mathbf r_0,t).
\end{equation}
That is a time history at one location.
Later, an antenna will be treated more carefully because a real antenna occupies a finite region and responds to spatial structure, polarization, orientation, and phase across an aperture or array. But the local-field viewpoint is the correct first step.
\section{Why this matters for radio and GPS}
A radio wave is not merely a scalar sinusoid moving through empty space. It is an electromagnetic field distributed through space and time.
A GPS receiving antenna does not directly observe ``the satellite power'' as a single abstract number. The satellite generates electromagnetic fields that propagate across a very large distance. The receiving antenna interacts with the local electric and magnetic fields that arrive at its aperture.
The later chain will be
\begin{equation}
\boxed{
\text{fields}
\longrightarrow
\text{waves}
\longrightarrow
\text{energy flux}
\longrightarrow
\text{antenna response}
\longrightarrow
\text{received power}.}
\end{equation}
Only after that chain is understood will quantities such as
\begin{equation}
C/N_0
\end{equation}
and
\begin{equation}
J/S
\end{equation}
be physically transparent rather than merely memorized link-budget formulas.
\section{Worked Example 1: identify the type of field}
Classify each expression as scalar or vector-valued.
\begin{align}
T(x,y,z,t)&=T_0+ax,\\
p(x,t)&=p_0\cos(kx-\omega t),\\
\mathbf E(x,t)&=\hat{\mathbf y}E_0\cos(kx-\omega t).
\end{align}
The first expression returns one temperature value at each event, so it is a scalar field.
The second returns one pressure value at each event, so it is also a scalar field.
The third returns a vector because the scalar coefficient multiplies a basis direction $\hat{\mathbf y}$. It is therefore a vector field.
\section{Worked Example 2: evaluate a scalar field}
Suppose
\begin{equation}
\psi(x,y)=2x-y^2.
\end{equation}
Find the field at
\begin{equation}
x=3,\qquad y=2.
\end{equation}
Substitution gives
\begin{align}
\psi(3,2)
&=2(3)-(2)^2\\
&=6-4\\
&=2.
\end{align}
Thus
\begin{equation}
\boxed{\psi(3,2)=2.}
\end{equation}
The result is a scalar because the field itself is scalar-valued.
\section{Worked Example 3: evaluate a vector field}
Let
\begin{equation}
\mathbf E(x,y)
=(2x)\hat{\mathbf x}
+(3y)\hat{\mathbf y}.
\end{equation}
At
\begin{equation}
x=1,\qquad y=2,
\end{equation}
we obtain
\begin{equation}
\boxed{
\mathbf E(1,2)
=2\hat{\mathbf x}
+6\hat{\mathbf y}.}
\end{equation}
Its magnitude is
\begin{align}
|\mathbf E|
&=\sqrt{2^2+6^2}\\
&=\sqrt{40}\\
&\approx6.32.
\end{align}
Thus the vector field specifies both a direction and a magnitude at that point.
\section{Worked Example 4: snapshot versus time history}
Consider
\begin{equation}
u(x,t)=A\cos(kx-\omega t).
\end{equation}
A spatial snapshot at $t=0$ is
\begin{equation}
\boxed{u(x,0)=A\cos(kx).}
\end{equation}
A time history at the position $x=0$ is
\begin{equation}
u(0,t)=A\cos(-\omega t).
\end{equation}
Since cosine is even,
\begin{equation}
\boxed{u(0,t)=A\cos(\omega t).}
\end{equation}
These are two views of the same field: one shows variation across space at one time, and the other shows variation in time at one position.
\section{Worked Example 5: read a simple vector wave}
Consider
\begin{equation}
\mathbf E(z,t)
=\hat{\mathbf x}(5\,\text{V/m})\cos(kz-\omega t).
\end{equation}
Identify the field direction and propagation direction.
The basis vector $\hat{\mathbf x}$ shows that the electric field points along the $x$ axis.
The phase
\begin{equation}
kz-\omega t
\end{equation}
has the same right-moving form studied in the Wave Mechanics series, so the pattern propagates toward increasing $z$.
Therefore
\begin{equation}
\boxed{\text{field direction: }\pm\hat{\mathbf x}}
\end{equation}
and
\begin{equation}
\boxed{\text{propagation direction: }+\hat{\mathbf z}.}
\end{equation}
The field is transverse in this illustrative example because the field direction is perpendicular to the propagation direction.
\section{Worked Example 6: evaluate the vector wave at one event}
Using
\begin{equation}
\mathbf E(z,t)
=\hat{\mathbf x}E_0\cos(kz-\omega t),
\end{equation}
suppose
\begin{equation}
E_0=10\,\text{V/m}
\end{equation}
and the phase at a particular event is
\begin{equation}
kz-\omega t=\pi.
\end{equation}
Then
\begin{equation}
\cos\pi=-1,
\end{equation}
so
\begin{equation}
\boxed{
\mathbf E=-10\hat{\mathbf x}\,\text{V/m}.}
\end{equation}
The negative sign does not mean the field magnitude is negative. It means the vector points in the $-x$ direction at that event. Its magnitude is
\begin{equation}
\boxed{|\mathbf E|=10\,\text{V/m}.}
\end{equation}
\section{Common mistakes}
\begin{itemize}
\item \textbf{Mistake:} thinking a field is a single number for the whole system. A field assigns values throughout space and time.
\item \textbf{Mistake:} confusing a scalar field with a vector field. A scalar has magnitude only; a vector has magnitude and direction.
\item \textbf{Mistake:} interpreting a field arrow as a material object moving through space. The arrow represents the field value at a location.
\item \textbf{Mistake:} confusing the direction in which a vector points with the direction in which its value changes spatially.
\item \textbf{Mistake:} treating a contour as a path followed by the field. A contour simply joins points of equal scalar value.
\item \textbf{Mistake:} assuming that writing $\mathbf E(\mathbf r,t)$ introduces new physics. It is compact notation for a vector field depending on spatial coordinates and time.
\end{itemize}
\section{What EM01 adds to the Wave Mechanics foundation}
The Wave Mechanics series established the idea of a quantity such as
\begin{equation}
u(x,t)
\end{equation}
varying through one spatial coordinate and time.
EM01 generalizes that idea to
\begin{equation}
\boxed{\psi(x,y,z,t)}
\end{equation}
for a scalar field and
\begin{equation}
\boxed{\mathbf E(x,y,z,t)}
\end{equation}
for a vector field.
The essential conceptual bridge is
\begin{equation}
\boxed{
\text{1D scalar wave}
\longrightarrow
\text{3D scalar field}
\longrightarrow
\text{3D vector field}.}
\end{equation}
No Maxwell equation is needed yet. EM02 next develops the vector mathematics needed to describe electromagnetic fields cleanly: components, basis directions, dot products, cross products, and the geometry of transverse fields.
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\bibitem{Griffiths2017}
David J. Griffiths,
\emph{Introduction to Electrodynamics},
4th ed., Cambridge University Press, 2017.
\bibitem{OpenStaxV2}
Samuel J. Ling, Jeff Sanny, and William Moebs,
\emph{University Physics, Volume 2},
OpenStax, 2016,
chapters on electric fields, magnetic fields, and electromagnetic waves.
\bibitem{FeynmanV2}
Richard P. Feynman, Robert B. Leighton, and Matthew Sands,
\emph{The Feynman Lectures on Physics, Volume II},
Addison-Wesley, 1964,
chapters introducing electromagnetic fields.
\bibitem{MIT802}
Massachusetts Institute of Technology,
\emph{8.02 Physics II: Electricity and Magnetism},
MIT OpenCourseWare,
materials on electric and magnetic fields.
\bibitem{Crawford1968}
Frank S. Crawford, Jr.,
\emph{Waves},
Berkeley Physics Course, Volume 3,
McGraw-Hill, 1968.
\end{thebibliography}