An electromagnetic wave is a self-sustaining ripple of electric and magnetic fields that travels through empty space at the speed of light. Before James Clerk Maxwell, the laws of electricity and magnetism were a collection of separate experimental rules. Maxwell's genius was to spot a missing piece, patch it, and in doing so reveal that light itself is an electromagnetic wave. The story begins with a subtle flaw in Ampère's law.
The problem with Ampère's circuital law. The original Ampère's law says that the line integral of the magnetic field around a closed loop equals $\mu_0$ times the current passing through any surface bounded by that loop: $\oint \vec{B}\cdot d\vec{l}=\mu_0 I$. Maxwell asked a sharp question using a charging capacitor. Imagine a wire carrying current $I$ into a parallel-plate capacitor. Choose a loop around the wire. If you stretch a flat surface across the loop, the conduction current $I$ passes through it. But if you balloon the surface so it dips between the plates, no charge actually crosses the gap — so the enclosed current is zero. The same loop gives two different answers. Something is missing.
Maxwell's fix: displacement current. Between the plates there is no conduction current, but there is a changing electric field as the capacitor charges. Maxwell proposed that a changing electric flux acts like a current and produces a magnetic field just as a real current does. He called it the displacement current:
- $I_d=\epsilon_0\frac{d\Phi_E}{dt}$, where $\Phi_E$ is the electric flux through the surface.
- It carries no moving charge — it is the changing electric field itself acting as a source of magnetic field.
- In the gap, $I_d$ exactly equals the conduction current $I$ in the wire, so the law gives one consistent answer for any surface.
The Ampère–Maxwell law. Combining the two, the corrected law is $\oint \vec{B}\cdot d\vec{l}=\mu_0(I+I_d)=\mu_0 I+\mu_0\epsilon_0\frac{d\Phi_E}{dt}$. The crucial new term tells us that a changing electric field produces a magnetic field — the partner of Faraday's discovery that a changing magnetic field produces an electric field.
Maxwell's four equations (conceptual form). The whole of classical electromagnetism is captured in four statements:
- Gauss's law for electricity: $\oint \vec{E}\cdot d\vec{A}=\frac{q}{\epsilon_0}$ — electric field lines begin and end on charges.
- Gauss's law for magnetism: $\oint \vec{B}\cdot d\vec{A}=0$ — magnetic field lines form closed loops; there are no isolated magnetic poles (monopoles).
- Faraday's law: $\oint \vec{E}\cdot d\vec{l}=-\frac{d\Phi_B}{dt}$ — a changing magnetic field creates an electric field.
- Ampère–Maxwell law: $\oint \vec{B}\cdot d\vec{l}=\mu_0 I+\mu_0\epsilon_0\frac{d\Phi_E}{dt}$ — currents and changing electric fields create a magnetic field.
How a wave sustains itself. Faraday's law and the Ampère–Maxwell law together create a feedback loop in empty space. A changing $\vec{E}$ generates a changing $\vec{B}$, which in turn generates a changing $\vec{E}$, and so on — the two fields keep regenerating each other and the disturbance marches forward as a wave. The source of an electromagnetic wave is an accelerating (or oscillating) charge: a charge moving at constant velocity gives steady fields, but an accelerating charge radiates energy as EM waves. The frequency of the wave equals the frequency of oscillation of the charge.