The Ampère–Maxwell Law
The fourth and final equation:
$$\nabla \times \mathbf{B} = \mu_0 \mathbf{J} + \mu_0 \varepsilon_0 \frac{\partial \mathbf{E}}{\partial t}$$
In words: the curl of the magnetic field is created by electric currents AND by changing electric fields.
The first term was Ampère’s discovery. The second was Maxwell’s — and it changed everything.
Ampère’s law (the original)
André-Marie Ampère discovered in the 1820s that electric currents create magnetic fields:
$$\nabla \times \mathbf{B} = \mu_0 \mathbf{J}$$
where $\mathbf{J}$ is the current density (amps per square metre) and $\mu_0$ is the permeability of free space.
This is the mathematical form of what you see in the magnetic fields node: current through a wire creates circular field lines around it.
Maxwell’s correction
In the 1860s, James Clerk Maxwell noticed that Ampère’s original law had a problem. Consider charging a capacitor: current flows into one plate and out of the other, but between the plates there’s a gap — no current flows through the gap, yet the magnetic field doesn’t suddenly vanish there.
Maxwell realized that the changing electric field between the plates acts like a current. He called it the displacement current:
$$\mathbf{J}_d = \varepsilon_0 \frac{\partial \mathbf{E}}{\partial t}$$
Adding this to Ampère’s law fixed the inconsistency:
$$\nabla \times \mathbf{B} = \mu_0 \mathbf{J} + \mu_0 \varepsilon_0 \frac{\partial \mathbf{E}}{\partial t}$$
A magnetic field curls around both real currents (moving charges) and displacement currents (changing E fields). Adjust each source to see its contribution.
The symmetry with Faraday
Compare the two curl equations:
| Faraday’s law | Ampère-Maxwell law | |
|---|---|---|
| Equation | $\nabla \times \mathbf{E} = -\partial\mathbf{B}/\partial t$ | $\nabla \times \mathbf{B} = \mu_0 \mathbf{J} + \mu_0 \varepsilon_0\, \partial\mathbf{E}/\partial t$ |
| Says | Changing B creates curling E | Changing E creates curling B |
| Source | Time-varying magnetic field | Current + time-varying electric field |
The two fields chase each other: changing $\mathbf{B}$ creates $\mathbf{E}$, changing $\mathbf{E}$ creates $\mathbf{B}$. This mutual feedback is the engine of electromagnetic waves.
Equation #4 of 4
With all four equations in hand, we can see the complete picture of how electric and magnetic fields behave — and predict something Ampère never imagined: light.