Have you ever wondered how we can understand and analyze the behavior of electromagnetic waves, the fundamental force behind everything from visible light to radio waves and X-rays? The key lies in mathematical representation.

Mathematical equations and models can be used to describe and predict the behavior of electromagnetic waves, allowing us to study and utilize these waves in a variety of applications, from wireless communication to medical imaging. In this article, we’ll explore the mathematical representation of electromagnetic waves, uncovering the principles that underlie this powerful force of nature.

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Mathematical representation of electromagnetic waves using maxwell’s equations

The mathematical representation of an electromagnetic wave can be expressed using Maxwell’s equations as follows:

\begin{align*} \frac{\partial \mathbf{E}}{\partial t} &= -\nabla \times \mathbf{B}\\ \frac{\partial \mathbf{B}}{\partial t} &= \nabla \times \mathbf{E}\\ \nabla \cdot \mathbf{E} &= \frac{\rho}{\epsilon_0}\\ \nabla \cdot \mathbf{B} &= 0 \end{align*}

Here, $\mathbf{E}$ is the electric field, $\mathbf{B}$ is the magnetic field, $\rho$ is the charge density, and $\epsilon_0$ is the vacuum permittivity. The first two equations describe how the electric and magnetic fields are related and how they propagate through space as a wave. The second two equations describe how the electric and magnetic fields are related to the distribution of electric charges and how they satisfy the laws of conservation of charge and magnetic flux.

The solutions to these equations describe the behavior of electromagnetic waves, which are characterized by their frequency $\nu$, wavelength $\lambda$, and amplitude $A$. The relationship between these parameters can be expressed using the wave equation:

$$c=\nu \lambda$$

where $c$ is the speed of light in a vacuum. This equation shows that the frequency and wavelength of an electromagnetic wave are inversely proportional to each other and that the speed of light is constant.

What are plane electromagnetic waves?

A plane electromagnetic wave is a type of electromagnetic radiation that has a constant wavelength, frequency, and amplitude and is characterized by a constant direction of propagation perpendicular to the electric and magnetic fields.

In a vacuum, electromagnetic waves travel at the speed of light, and their electric and magnetic fields oscillate in a perpendicular fashion to one another and to the direction of wave propagation. This oscillation creates the characteristic wave pattern of the electromagnetic wave.

Plane electromagnetic waves equation

The mathematical representation of a plane electromagnetic wave can be expressed using the following equations:

\begin{align*} \mathbf{E}(x, t) &= \mathbf{E}_0 \cos(\omega t – k x + \phi) \\ \mathbf{B}(x, t) &= \mathbf{B}_0 \cos(\omega t – k x + \phi + \frac{\pi}{2}) \end{align*}

Here, $\mathbf{E}$ is the electric field, $\mathbf{B}$ is the magnetic field, $\mathbf{E}_0$, and $\mathbf{B}_0$ are the amplitudes of the electric and magnetic fields, respectively, $\omega$ is the angular frequency, $k$ is the wave vector, $x$ is the position along the direction of propagation, $t$ is time, and $\phi$ is the phase angle.

These equations describe a plane electromagnetic wave that propagates in the direction of the wave vector $\mathbf{k}$ with a constant speed equal to the speed of light $c$. The electric and magnetic fields oscillate sinusoidally in time and space with the same frequency and wavelength. The electric and magnetic fields are also perpendicular to each other and to the direction of propagation, and they are in phase with each other.

The wave vector $\mathbf{k}$ and the angular frequency $\omega$ are related to the wavelength $\lambda$ and the speed of light $c$ by the following equations:

$$k=\frac{\lambda}{2\pi},\qquad\omega=ck$$

These equations show that the wave vector and angular frequency are directly proportional to each other and that the wavelength is inversely proportional to the wave vector.

Mathematical representation of plane electromagnetic waves class 12

The below figure shows a plane electromagnetic waves traveling along the X-axis. The electric field $\overrightarrow{E}$ oscillates along Y-axis while the magnetic field $\overrightarrow{B}$ oscillates along Z-axis.

Mathematical representation of plane electromagnetic waves, source: Mini Physics

The values of electric and magnetic fields shown in the above figure depend only on x and t. The electric field vector can be represented mathematically as follows:

$$
\begin{aligned}
\vec{E} & =E_y \hat{j}=E_0 \sin (k x-\omega t) \hat{j} \\
& =E_0 \sin \left[2 \pi\left(\frac{x}{\lambda}-v t\right)\right] \hat{j} \\
& =E_0 \sin \left[2 \pi\left(\frac{x}{\lambda}-\frac{t}{T}\right)\right] \hat{j}\qquad …. (1)
\end{aligned}
$$
where $k=2 \pi / \lambda$ is the propagation constant of the wave and angular frequency, $\omega=2 \pi v$.
Clearly, $E_x=E_z=0$

The magnetic field vector may be represented mathematically as follows:
$$
\begin{aligned}
\vec{B} & =B_z \hat{k}=B_0 \sin (k x-\omega t) \hat{k} \\
& =B_0 \sin \left[2 \pi\left(\frac{x}{\lambda}-v t\right)\right] \hat{k} \\
& =B_0 \sin \left[2 \pi\left(\frac{x}{\lambda}-\frac{t}{T}\right)\right] \hat{k}\qquad …. (2)
\end{aligned}
$$
Clearly, $B_x=B_y=0$.
Here $E_0$ and $B_0$ are the amplitudes of the electric field $\vec{E}$ and magnetic field $\vec{B}$, respectively.

Equations (1) and (2) show that the variations in electric and magnetic fields are in the same phase, i.e., both attain their maxima and minima at the same instant and at the same place $(x)$.
The magnitudes of $\vec{E}$ and $\vec{B}$ are related as
$$
\frac{E}{B}=c, \text { or } \frac{E_0}{B_0}=c
$$
Maxwell also showed that the speed of an e.m. wave depends on the permeability and permittivity of the medium through which it travels. The speed of an e.m. wave in free space is given by
$$
c=\frac{1}{\sqrt{\mu_0 \varepsilon_0}}
$$
Permeability of free space,
$$
\mu_0=4 \pi \times 10^{-7} \mathrm{Ns}^2 \mathrm{C}^{-2}
$$
The permittivity of free space,
$$
\begin{aligned}
\varepsilon_0&=8.85 \times 10^{-12} \mathrm{C}^2 \mathrm{~N}^{-1} \mathrm{~m}^{-2} \\
\therefore \quad c&=\frac{1}{\sqrt{4 \pi \times 10^{-7} \times 8.85 \times 10^{-12}}} \\
&=3.0 \times 10^8 \mathrm{~ms}^{-1}\ \end{aligned}
$$
which is the speed of light in a vacuum. This fact led Maxwell to predict that light is an electromagnetic wave. The emergence of the speed of light from purely electromagnetic considerations is the crowning achievement of Maxwell’s electromagnetic theory.
The speed of an e.m. wave in any medium of permeability $\mu$ and permittivity $\varepsilon$ will be
$$
v=\frac{1}{\sqrt{\varepsilon \mu}}=\frac{1}{\sqrt{\kappa \varepsilon_0 \mu_r \mu_0}}=\frac{c}{\sqrt{\kappa \mu_r}}
$$
where $\kappa$ is the dielectric constant of the medium and $\mu_r$ is its relative permeability.

As the electric and magnetic fields in an e.m. wave are always perpendicular to each other and also perpendicular to the direction of wave propagation, electromagnetic waves are transverse in nature.

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