Deriving the Electromagnetic Wave Equation in a Vacuum

Hello! Welcome to the first lesson in our new module, Electromagnetic Waves and Optics.

In our previous lesson, we reached a major milestone by assembling the complete set of Maxwell's equations. We saw how these four equations, especially the two curl equations, describe a beautiful and symmetric interplay between electric and magnetic fields changing in time. This dynamic coupling is not just an elegant mathematical feature; it's the very mechanism that allows energy and information to travel through empty space.

Today, we will use this framework to make one of the most profound predictions in the history of physics. Our learning outcome is to derive the electromagnetic wave equation in a vacuum from Maxwell's equations. We will demonstrate mathematically that these equations inherently predict the existence of self-propagating waves of electric and magnetic fields, and in the process, we will uncover their speed.

1. Maxwell's Equations in Free Space

Our journey begins by considering the simplest possible environment: a perfect vacuum, or "free space." In this region, there are no electric charges ( $ρ = 0$ ) and no conduction currents ( $J = 0$ ). Under these conditions, the four Maxwell's equations in their differential form simplify considerably:

Gauss's Law for $E$ : $\nabla \cdot E = 0$
Gauss's Law for $B$ : $\nabla \cdot B = 0$
Faraday's Law: $\nabla \times E = - \frac{\partial B}{\partial t}$
Ampère-Maxwell Law: $\nabla \times B = μ_{0} ϵ_{0} \frac{\partial E}{\partial t}$

Notice the enhanced symmetry here. The two divergence equations are identical in form, telling us that in a vacuum, both $E$ and $B$ field lines are continuous—they don't start or stop. The two curl equations now show a perfectly reciprocal relationship: a changing $B$ field creates a curling $E$ field, and a changing $E$ field creates a curling $B$ field. This is the engine of an electromagnetic wave.

2. A First Pass: Deriving the 1D Wave Equation

Before tackling the full three-dimensional derivation using vector calculus, it is highly instructive to analyze a simplified one-dimensional case. This approach builds strong physical intuition by using the integral forms of Faraday's and the Ampère-Maxwell laws on infinitesimal loops of space. It makes the causal link between the changing fields and the wave propagation very concrete.

The following video provides an excellent step-by-step walkthrough of this derivation.

Maxwell's Equations and Electromagnetic Waves

To build our intuition, we will first derive the wave equation for a simplified one-dimensional case. This approach uses the integral forms of Faraday's and the Ampère-Maxwell laws on infinitesimal loops, making the connection between the changing fields and the resulting wave very clear.

Please watch from 03:09 to 21:44. Follow the logic carefully. The video first applies Faraday's Law to an infinitesimal rectangular loop to derive one relation between the spatial derivative of E and the time derivative of B. It then does the same with the Ampère-Maxwell Law for a differently oriented loop. Finally, these two first-order equations are combined via differentiation to arrive at a second-order wave equation for E and B separately.

For your reference, a similar derivation can be found in the MIT course notes, "Chapter 13 Maxwell's Equations and Electromagnetic Waves," in Section 13.4, "Plane Electromagnetic Waves."

After this process, we arrive at two separate, uncoupled equations—one for the electric field component $E_{y}$ and one for the magnetic field component $B_{z}$ :

\frac{\partial ^{2} E _{y}}{\partial x ^{2}} = μ_{0} ϵ_{0} \frac{\partial ^{2} E _{y}}{\partial t ^{2}}

\frac{\partial ^{2} B _{z}}{\partial x ^{2}} = μ_{0} ϵ_{0} \frac{\partial ^{2} B _{z}}{\partial t ^{2}}

3. The Speed of Light

These equations have a very specific and famous mathematical form. The standard one-dimensional wave equation for a wave function $ψ (x, t)$ propagating with speed $v$ is:

\frac{\partial ^{2} ψ}{\partial x ^{2}} = \frac{1}{v ^{2}} \frac{\partial ^{2} ψ}{\partial t ^{2}}

By comparing this general form to the equations we just derived for $E_{y}$ and $B_{z}$ , we can immediately identify the speed of these electromagnetic waves.

Maxwell's Equations and Electromagnetic Waves

Now that we have the wave equation, let's identify the speed of propagation and confirm that a traveling wave is indeed a solution.

Continue watching the Physics Ninja video from 21:44 to 27:05. Pay attention to how the wave speed is extracted by comparing the derived equation to the standard wave equation. Note the monumental result when the numerical values for \mu_0 and \epsilon_0 are substituted.

By comparing the equations, we see that $1/ v^{2} = μ_{0} ϵ_{0}$ , which gives the wave speed:

v = \frac{1}{μ _{0} ϵ _{0}}

This is one of the most important results in all of physics. The speed of these predicted waves depends only on two fundamental constants of nature: the permeability of free space ( $μ_{0}$ ) and the permittivity of free space ( $ϵ_{0}$ ). When 19th-century physicists first plugged in the experimentally measured values for these constants:

$μ_{0} = 4 π \times 1 0^{- 7} T \cdot m/A$
$ϵ_{0} \approx 8.85 \times 1 0^{- 12} C^{2} / N \cdot m^{2}$

They found:

v = \frac{1}{( 4 π \times 1 0 ^{- 7} ) \times ( 8.85 \times 1 0 ^{- 12} )} \approx 2.998 \times 1 0^{8} m/s

This value was astonishingly close to the measured speed of light, denoted by $c$ . This theoretical triumph led Maxwell to the inescapable conclusion that light is an electromagnetic wave.

4. The Rigorous 3D Derivation

The 1D derivation was insightful, but a full proof requires us to work with the vector equations in three dimensions. The strategy is to decouple the curl equations for $E$ and $B$ to obtain a second-order partial differential equation for each field separately. The key mathematical tool is the "curl of the curl" vector identity.

For any vector field $A$ , the following identity holds:

\nabla \times (\nabla \times A) = \nabla (\nabla \cdot A) - \nabla^{2} A

where $\nabla^{2} A$ is the vector Laplacian.

Because you value mathematical rigor, let's carefully follow the derivation for the electric field $E$ . The following video presents this derivation clearly and concisely.

Maxwell's Equations and Electromagnetic Waves

We will now derive the full three-dimensional wave equation. This derivation is more abstract but also more powerful, as it holds for any direction of propagation. We'll use the differential forms of Maxwell's equations and the curl of the curl vector identity.

Watch from 29:01 to 37:14. The video first shows how to convert the integral forms of Faraday's and Ampère-Maxwell's laws into differential form using Stokes' theorem (which we covered previously). The core derivation starts at 32:01. Follow these key steps: Take the curl of Faraday's Law. Apply the curl of the curl identity to the left side and simplify it using Gauss's Law (( abla \cdot \mathbf{E} = 0)). Substitute the Ampère-Maxwell Law into the right side to eliminate \mathbf{B}.

Let's summarize the steps from the video to derive the wave equation for $E$ :

Start with Faraday's Law:
$\nabla \times E = - \frac{\partial B}{\partial t}$
Take the curl of both sides:
$\nabla \times (\nabla \times E) = \nabla \times (- \frac{\partial B}{\partial t})$
Use the vector identity on the left-hand side (LHS):
$\nabla (\nabla \cdot E) - \nabla^{2} E$
Since $\nabla \cdot E = 0$ in a vacuum, the LHS simplifies to $- \nabla^{2} E$ .
Manipulate the right-hand side (RHS):
Swap the order of the curl and time derivative: $- \frac{\partial}{\partial t} (\nabla \times B)$ .
Now, substitute the Ampère-Maxwell law ( $\nabla \times B = μ_{0} ϵ_{0} \frac{\partial E}{\partial t}$ ):
$- \frac{\partial}{\partial t} (μ_{0} ϵ_{0} \frac{\partial E}{\partial t}) = - μ_{0} ϵ_{0} \frac{\partial ^{2} E}{\partial t ^{2}}$
Equate the simplified LHS and RHS:
$- \nabla^{2} E = - μ_{0} ϵ_{0} \frac{\partial ^{2} E}{\partial t ^{2}}$

Canceling the negative signs gives us the three-dimensional electromagnetic wave equation for the electric field:

\nabla^{2} E - μ_{0} ϵ_{0} \frac{\partial ^{2} E}{\partial t ^{2}} = 0

Test your understanding!

Following the exact same procedure, derive the wave equation for the magnetic field $B$ . Start by taking the curl of the Ampère-Maxwell law: $\nabla \times (\nabla \times B) = \nabla \times (μ_{0} ϵ_{0} \frac{\partial E}{\partial t})$ .

Show answer

Start with the Ampère-Maxwell Law: $\nabla \times B = μ_{0} ϵ_{0} \frac{\partial E}{\partial t}$ .
Take the curl: $\nabla \times (\nabla \times B) = \nabla \times (μ_{0} ϵ_{0} \frac{\partial E}{\partial t})$ .
Simplify the LHS: Using the vector identity $\nabla \times (\nabla \times B) = \nabla (\nabla \cdot B) - \nabla^{2} B$ and Gauss's Law for magnetism ( $\nabla \cdot B = 0$ ), the LHS becomes $- \nabla^{2} B$ .
Simplify the RHS: Swap derivatives and substitute Faraday's Law:
$μ_{0} ϵ_{0} \frac{\partial}{\partial t} (\nabla \times E) = μ_{0} ϵ_{0} \frac{\partial}{\partial t} (- \frac{\partial B}{\partial t}) = - μ_{0} ϵ_{0} \frac{\partial ^{2} B}{\partial t ^{2}}$ .
Equate LHS and RHS: $- \nabla^{2} B = - μ_{0} ϵ_{0} \frac{\partial ^{2} B}{\partial t ^{2}}$ .
Final Equation: $\nabla^{2} B - μ_{0} ϵ_{0} \frac{\partial ^{2} B}{\partial t ^{2}} = 0$

This confirms that the magnetic field also propagates as a wave with speed $c = 1/ μ_{0} ϵ_{0}$ .

For a classic lecture-style presentation of this same derivation, you might also appreciate the first part of Professor Walter Lewin's famous lecture on the topic. In his inimitable style, he conveys the historical weight and excitement of this discovery. You can find this in the video "8.03 - Lect 13 - Electromagnetic Waves..." from 08:15 to 11:57.

Conclusion

Today we have seen how Maxwell's equations, when applied to empty space, naturally and inevitably lead to wave equations for both the electric and magnetic fields. This derivation is a cornerstone of theoretical physics.

Key Takeaways:

In a vacuum, the coupling between a changing $E$ field and a changing $B$ field leads to self-sustaining waves.
By decoupling the curl equations, we can derive the second-order wave equation for $E$ and $B$ individually.
The wave equation takes the form $\nabla^{2} E - \frac{1}{c ^{2}} \frac{\partial ^{2} E}{\partial t ^{2}} = 0$ , where the speed of propagation $c$ is given by the fundamental constants $μ_{0}$ and $ϵ_{0}$ .
The calculated value of this speed matches the measured speed of light, providing powerful theoretical evidence that light is an electromagnetic wave.

Preview of the Next Lesson:
Now that we have established that electromagnetic waves exist, we will move on to study their properties. In the next lesson, we will describe the properties of plane electromagnetic waves, including their transverse nature and polarization. We will see how Maxwell's equations dictate that the $E$ and $B$ fields in a plane wave are perpendicular to each other and to the direction of wave propagation.

Recommended resources

1. Maxwell's Equations in Free Space

2. A First Pass: Deriving the 1D Wave Equation

3. The Speed of Light

4. The Rigorous 3D Derivation

Conclusion