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Maxwell's Equations

Source lecture(s): SC134 Lec 13–14

Intuition

Four equations that contain the whole of classical electromagnetism — and, once assembled, predict light without anyone having asked about light.

\[\oint\mathbf{E}\cdot d\mathbf{A} = \frac{Q_{\rm enc}}{\varepsilon_0} \qquad\qquad \nabla\cdot\mathbf{E} = \frac{\rho}{\varepsilon_0}\]
\[\oint\mathbf{B}\cdot d\mathbf{A} = 0 \qquad\qquad \nabla\cdot\mathbf{B} = 0\]
\[\oint\mathbf{E}\cdot d\boldsymbol{\ell} = -\frac{d\Phi_B}{dt} \qquad\qquad \nabla\times\mathbf{E} = -\frac{\partial\mathbf{B}}{\partial t}\]
\[\oint\mathbf{B}\cdot d\boldsymbol{\ell} = \mu_0I_{\rm enc} + \mu_0\varepsilon_0\frac{d\Phi_E}{dt} \qquad \nabla\times\mathbf{B} = \mu_0\mathbf{J} + \mu_0\varepsilon_0\frac{\partial\mathbf{E}}{\partial t}\]

Read in order they say: charge makes diverging E; there are no magnetic charges; changing B makes circulating E; current and changing E make circulating B.

The displacement current

The last term is Maxwell's own, and the reason his name is on the set. Consider charging a capacitor and applying Ampère's law to a loop around the wire. Choose a flat surface cutting the wire: current passes through, so \(\oint\mathbf{B}\cdot d\boldsymbol{\ell} = \mu_0I\). Now choose a bulging surface that passes between the plates instead: the same boundary loop, but no current crosses it at all.

Same loop, two answers. Ampère's law as it stood was inconsistent.

Maxwell's fix: between the plates \(\mathbf{E}\) is growing, and \(\varepsilon_0\,d\Phi_E/dt\) equals exactly the current in the wire. Adding that term makes the answer independent of the surface — as it must be, since the loop is the same.

This is not a patch on a technicality. Symmetry now holds: a changing \(\mathbf{B}\) makes \(\mathbf{E}\), and a changing \(\mathbf{E}\) makes \(\mathbf{B}\). Each can sustain the other with no charges present at all — and that is a wave.

Light falls out

In vacuum (\(\rho = 0\), \(\mathbf{J} = 0\)), take the curl of Faraday's law, substitute Ampère's, and use \(\nabla\cdot\mathbf{E} = 0\):

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

A wave equation, with speed

\[v = \frac{1}{\sqrt{\mu_0\varepsilon_0}} = \frac{1}{\sqrt{(4\pi\times10^{-7})(8.854\times10^{-12})}} = 3.00\times10^{8}\ \text{m/s}\]

Both constants were measured in the laboratory — \(\varepsilon_0\) from the force between charges, \(\mu_0\) from the force between currents. Neither had anything to do with optics. Their combination came out at the measured speed of light, and Maxwell concluded that light is an electromagnetic wave.

It is worth sitting with how strange that is: two benchtop electrical measurements, combined, predict the speed of light. Nothing else in the syllabus makes this kind of leap.

What each equation forbids

Sometimes clearer than what they assert:

  • No magnetic monopoles (\(\nabla\cdot\mathbf{B} = 0\)): magnetic field lines never begin or end. Every field in the Biot–Savart lab obeys this identically.
  • No perpetual motion (the minus sign in Faraday): induced effects oppose their cause.
  • No surface ambiguity (displacement current): Ampère's law is now well posed.
  • No static charge without a field (Gauss): charge always announces itself.

Common mistakes

  • Thinking displacement current is a current. No charge flows between the plates. It is a changing electric field with the same units, producing the same magnetic effect.
  • Applying the vacuum wave speed inside matter. In a medium \(v = 1/\sqrt{\mu\varepsilon} = c/n\); see dielectrics.
  • Forgetting Maxwell's equations need the Lorentz force to be complete. They tell you the fields from the sources; \(\mathbf{F} = q(\mathbf{E}+\mathbf{v}\times\mathbf{B})\) tells you what the fields do to charges. Neither alone is the theory.
  • Reading the integral and differential forms as different physics. They are equivalent by the divergence and Stokes' theorems.

Knowledge graph position

Prerequisites: Gauss's law, Faraday's law, Ampère's law, magnetic flux. Leads to: electromagnetic waves, optics, special relativity, plasma physics, computational EM.

Quiz

Q1 (conceptual). Why was the displacement current necessary?

Answer

Without it Ampère's law gives different answers for different surfaces sharing the same boundary loop — a flat surface cutting the wire of a charging capacitor encloses current, a bulging one passing between the plates does not. The term \(\varepsilon_0\,d\Phi_E/dt\) between the plates exactly equals the wire current, restoring consistency. Its by-product is electromagnetic waves.

Q2 (computational). Compute \(1/\sqrt{\mu_0\varepsilon_0}\) and say why it is remarkable.

Answer

\(\mu_0\varepsilon_0 = (1.2566\times10^{-6})(8.854\times10^{-12}) = 1.1127\times10^{-17}\), so \(1/\sqrt{\cdot} = 2.998\times10^{8}\) m/s. Remarkable because both constants come from purely electrical and magnetic benchtop measurements with no reference to light, yet their combination is exactly the speed of light.

Q3 (MCQ). \(\nabla\cdot\mathbf{B} = 0\) expresses the fact that:

  • (a) magnetic fields are always zero
  • (b) there are no magnetic monopoles, so field lines never begin or end
  • (c) magnetic fields do no work
  • (d) magnetism is weaker than electricity
Answer

(b). It is the magnetic counterpart of Gauss's law with zero on the right — no magnetic charge has ever been observed, so magnetic field lines always close on themselves.