Why Waves? The Necessity of Propagation

To an early physicist, electricity and magnetism were static, separate forces. The leap to Electromagnetic Waves required two fundamental shifts: one mathematical and one physical.

1. The "Leap-Frog" Effect

Imagine a region of empty space. If you somehow create a changing electric field $\vec{E}$, what happens?

Ampère-Maxwell A time-varying $\vec{E}$ is linked to a circulating $\vec{B}$.

Faraday A time-varying $\vec{B}$ is linked to a circulating $\vec{E}$.

These fields don't just "stay" where they are. They kick-start each other. A time-varying $\vec{E}$ appears in the Ampère-Maxwell law as circulation of $\vec{B}$ in the surrounding space. A time-varying $\vec{B}$ appears in Faraday's law as circulation of $\vec{E}$. In a wave, the coupled curl equations pass the disturbance onward, forming a "leap-frog" through space.

2. Maxwell's Missing Link

Before Maxwell, Ampère's Law was incomplete. It lacked the term $\mu_0 \epsilon_0 \frac{\partial \vec{E}}{\partial t}$ (the Displacement Current).

                Without this term, information couldn't travel in a vacuum. The equations would effectively say that
                fields
                at different points are independent. By adding the displacement current, Maxwell linked the "curl"
                of one field to the "change in time" of another. This link is what makes the Wave
                    Equation possible.
            

3. Mathematical Emergence

In a source-free vacuum ($\rho=0, \vec{J}=0$), Maxwell's equations take their most symmetric form:

(1) \nabla \cdot \vec{E} = 0, \quad (2) \nabla \cdot \vec{B} = 0 $$ $$ (3) \nabla \times \vec{E} = -\pd{\vec{B}}{t}, \quad (4) \nabla \times \vec{B} = \mu_0 \epsilon_0 \pd{\vec{E}}{t}

To decouple them, we take the curl of equation (3): $$ \nabla \times (\nabla \times \vec{E}) = \nabla \times \left(-\pd{\vec{B}}{t}\right) = -\pd{}{t} (\nabla \times \vec{B}) $$

Next, we use a standard Vector Identity for the left side: $$ \nabla \times (\nabla \times \vec{E}) = \nabla(\nabla \cdot \vec{E}) - \nabla^2 \vec{E} $$

Substituting the identity into our curl equation, and replacing $\nabla \times \vec{B}$ with equation (4): $$ \nabla(\nabla \cdot \vec{E}) - \nabla^2 \vec{E} = -\pd{}{t} \left( \mu_0 \epsilon_0 \pd{\vec{E}}{t} \right) $$

Since $\nabla \cdot \vec{E} = 0$ in vacuum, the first term vanishes: $$ - \nabla^2 \vec{E} = -\mu_0 \epsilon_0 \pd{^2 \vec{E}}{t^2} $$

\nabla^2 \vec{E} = \mu_0 \epsilon_0 \frac{\partial^2 \vec{E}}{\partial t^2}

Comparing this to the General Wave Equation ($\nabla^2 f = \frac{1}{v^2} \frac{\partial^2 f}{\partial t^2}$), we identify the speed of propagation: $$ \frac{1}{v^2} = \mu_0 \epsilon_0 \implies v = \frac{1}{\sqrt{\mu_0 \epsilon_0}} $$

This predicted speed $v$ is approximately $c \approx 3 \times 10^8$ m/s—the speed of light—leading to the revolutionary realization that light is an electromagnetic wave.

            In Summary
            Waves are not "added" to Maxwell's equations. They are built-in. Any system that
            obeys these four laws must support self-propagating electromagnetic disturbances.
        