How Maxwell Made the Electromagnetic Wave Equation Using Calculus

It is one of the most beautiful "Aha!" moments in the history of science. To see exactly how Maxwell did it, we have to look under the hood at the calculus he used.

We are going to look at the actual vector calculus that changed the world.

Here is the step-by-step mathematical synthesis of how James Clerk Maxwell united electricity, magnetism, and light.

1. The Raw Materials (Before Maxwell)

In the mid-1800s, physicists had compiled several laws describing electricity and magnetism, but they were disjointed. In the language of vector calculus, the two most important for this story were:

Faraday’s Law: A changing magnetic field (B) induces an electric field (E).

\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t}

(In English: The "curl" or swirling of an electric field is created by the rate of change of a magnetic field over time).

Ampère’s Law (Original): An electric current (J) creates a swirling magnetic field.

\nabla \times \mathbf{B} = \mu_0 \mathbf{J}

(Here, \mu_0 is just a constant of nature related to magnetic strength).

2. The Mathematical Contradiction

Maxwell took Ampère’s Law and applied a basic rule of calculus. There is a mathematical identity which states that the divergence of a curl is always zero. So, if you take the divergence (\nabla \cdot) of both sides of Ampère's Law, you get:

\nabla \cdot (\nabla \times \mathbf{B}) = \mu_0 (\nabla \cdot \mathbf{J})

0 = \mu_0 (\nabla \cdot \mathbf{J})

This implies that \nabla \cdot \mathbf{J} must always equal zero. In physics terms, this means electric current can never accumulate in one place; whatever flows in must perfectly flow out.

But wait! That isn't true in reality. If you charge a capacitor (two metal plates separated by a gap), current flows into the plate and piles up as electrical charge (\rho). By the law of conservation of charge, if charge is piling up, \nabla \cdot \mathbf{J} is not zero:

\nabla \cdot \mathbf{J} = -\frac{\partial \rho}{\partial t}

Maxwell realized that Ampère's Law was mathematically flawed when dealing with changing currents.

3. Maxwell’s Ingenious Fix: The Displacement Current

Maxwell needed to fix Ampère's Law so that it wouldn't violate the conservation of charge. He looked at another established law, Gauss's Law, which links charge (\rho) to the electric field (E):

\nabla \cdot \mathbf{E} = \frac{\rho}{\varepsilon_0}

(Here, \varepsilon_0 is the constant of nature for electrical strength).

He rearranged this to solve for the charge: \rho = \varepsilon_0 (\nabla \cdot \mathbf{E}). Then, he plugged this into the conservation of charge equation:

\nabla \cdot \mathbf{J} = -\frac{\partial (\varepsilon_0 \nabla \cdot \mathbf{E})}{\partial t}

Moving everything to one side, he got:

\nabla \cdot (\mathbf{J} + \varepsilon_0 \frac{\partial \mathbf{E}}{\partial t}) = 0

To make the math perfectly consistent, Maxwell argued that Ampère's Law was missing a piece. The current \mathbf{J} needed a partner. He added the term \varepsilon_0 \frac{\partial \mathbf{E}}{\partial t} (which he called the "displacement current") to Ampère's original law.

Maxwell’s Updated Ampère's Law:

\nabla \times \mathbf{B} = \mu_0 \mathbf{J} + \mu_0 \varepsilon_0 \frac{\partial \mathbf{E}}{\partial t}

The Physical Meaning: This addition was revolutionary. It meant that not only does a physical electric current \mathbf{J} create a magnetic field, but a changing electric field \frac{\partial \mathbf{E}}{\partial t} creates a magnetic field too.

4. The Churning: Deriving the Wave

Now the trap was set. Maxwell looked at what happens in a perfect vacuum—empty space, where there are no wires, no charges (\rho = 0), and no physical currents (\mathbf{J} = 0).

Without charges or currents, his two star equations simplified beautifully:

1. \nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t}

   (Faraday)

2. \nabla \times \mathbf{B} = \mu_0 \varepsilon_0 \frac{\partial \mathbf{E}}{\partial t}

   (Maxwell-Ampère)

Notice the beautiful symmetry? A changing magnetic field creates an electric field, and a changing electric field creates a magnetic field. They feed into each other.

To see what this loop creates, Maxwell took the curl (\nabla \times) of Faraday's Law:

\nabla \times (\nabla \times \mathbf{E}) = -\frac{\partial (\nabla \times \mathbf{B})}{\partial t}

Using a standard calculus identity on the left side, and substituting his updated Ampère's Law into the right side, the equation perfectly unwinds into:

\nabla^2 \mathbf{E} = \mu_0 \varepsilon_0 \frac{\partial^2 \mathbf{E}}{\partial t^2}

5. The Revelation: The Speed of Light

When Maxwell looked at that final equation, he recognized it immediately. In physics, the standard equation for any wave (like a sound wave or a ripple on a pond) traveling at a velocity v looks like this:

\nabla^2 \mathbf{E} = \frac{1}{v^2} \frac{\partial^2 \mathbf{E}}{\partial t^2}

By comparing his derived equation to the standard wave equation, Maxwell saw that these electric and magnetic fields were creating a self-sustaining wave traveling through empty space at a very specific velocity:

v = \frac{1}{\sqrt{\mu_0 \varepsilon_0}}

He knew the values of \mu_0 (from magnetic experiments with wires) and \varepsilon_0 (from electrostatic experiments with capacitors). When he multiplied them together and took the square root, the math spat out a number: 299,792,458 meters per second.

That number was already known to scientists of the time. It was the measured speed of light (c).

Through pure mathematical deduction—by fixing a calculus error in Ampère's Law—Maxwell proved that light is nothing more than an invisible, self-propagating wave of electricity and magnetism.