Maxwell's Equations in Differential Form
Maxwell's Equations in Differential Form
Maxwell unified electromagnetism with four elegant equations. His critical addition — the displacement current — predicted electromagnetic waves and established that light is an electromagnetic phenomenon, one of physics' greatest triumphs.
Definition
Maxwell's equations (SI): \(\nabla\cdot\mathbf{E}=\rho/\epsilon_0\), \(\nabla\cdot\mathbf{B}=0\), \(\nabla\times\mathbf{E}=-\partial_t\mathbf{B}\), \(\nabla\times\mathbf{B}=\mu_0\mathbf{J}+\mu_0\epsilon_0\partial_t\mathbf{E}\).
Key Result
Taking the curl of Faraday's law and using Ampère's law yields the wave equation \(\nabla^2\mathbf{E} = \mu_0\epsilon_0\partial_{tt}\mathbf{E}\) with \(c = 1/\sqrt{\mu_0\epsilon_0}\approx 3\times10^8\) m/s.
Example 1
In a charge-free medium, plane wave solutions are \(\mathbf{E} = E_0\hat{x}\cos(kz-\omega t)\), \(\mathbf{B} = (E_0/c)\hat{y}\cos(kz-\omega t)\), with \(k=\omega/c\).
Example 2
The Poynting vector \(\mathbf{S} = \mathbf{E}\times\mathbf{B}/\mu_0\) gives the energy flux density. For a plane wave, \(\langle S\rangle = E_0^2/(2\mu_0 c)\).
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Practice
- Show that Maxwell's equations predict charge conservation.
- What is the displacement current, and why did Maxwell introduce it?
- Derive the boundary conditions for E and B at a dielectric interface.
- In what sense are Maxwell's equations relativistically covariant?
Show Answer Key
1. Take the divergence of Ampère-Maxwell: $\nabla\cdot(\nabla\times\mathbf{B}) = 0 = \mu_0\nabla\cdot\mathbf{J}+\mu_0\epsilon_0\frac{\partial}{\partial t}(\nabla\cdot\mathbf{E}) = \mu_0(\nabla\cdot\mathbf{J}+\frac{\partial\rho}{\partial t})$. So $\frac{\partial\rho}{\partial t}+\nabla\cdot\mathbf{J}=0$ (continuity equation = charge conservation).
2. Displacement current: $\mathbf{J}_d = \epsilon_0\frac{\partial\mathbf{E}}{\partial t}$. Maxwell added it to Ampère's law to ensure consistency with charge conservation (without it, $\nabla\cdot(\nabla\times\mathbf{B}) \neq 0$ for time-varying fields). It accounts for the changing electric field between capacitor plates acting as a source of magnetic field.
3. At a dielectric interface: tangential $\mathbf{E}$ is continuous ($E_{1t}=E_{2t}$), normal $\mathbf{D}$ jumps by surface charge ($D_{2n}-D_{1n}=\sigma_f$). For $\mathbf{B}$: normal $\mathbf{B}$ is continuous ($B_{1n}=B_{2n}$), tangential $\mathbf{H}$ jumps by surface current ($H_{2t}-H_{1t}=K_f$). Derived from integrating Maxwell's equations over pillbox/loop straddling the interface.
4. The field-strength tensor $F^{\mu\nu}$ (antisymmetric 4×4 matrix containing $\mathbf{E}$ and $\mathbf{B}$) transforms as a rank-2 tensor under Lorentz transformations. Maxwell's equations become $\partial_\mu F^{\mu\nu}=\mu_0 J^\nu$ and $\partial_{[\mu}F_{\nu\lambda]}=0$ — manifestly covariant (same form in all inertial frames). This ensures $\mathbf{E}$ and $\mathbf{B}$ mix under boosts, as required by special relativity.