Electromagnetic Waves – MIT‑Level Study Notes
(Prepared for the INSPIRE Scholarship Physics Programme)
Goal – Build a deep, rigorous understanding of electromagnetic (EM) wave propagation from first principles, including advanced topics such as relativistic covariance, anisotropic media, and non‑linear optics.
1. Maxwell’s Equations in Vacuum
| Equation | Differential form | Integral form |
|---|---|---|
| Gauss’s law for E | (\nabla!\cdot!\mathbf{E}= \frac{\rho}{\varepsilon_0}) | (\oint_{\partial V}\mathbf{E}!\cdot! d\mathbf{S}= \frac{Q_{\text{enc}}}{\varepsilon_0}) |
| Gauss’s law for B | (\nabla!\cdot!\mathbf{B}=0) | (\oint_{\partial V}\mathbf{B}!\cdot! d\mathbf{S}=0) |
| Faraday’s law | (\nabla\times!\mathbf{E}= -\frac{\partial\mathbf{B}}{\partial t}) | (\oint_{\partial S}\mathbf{E}!\cdot! d\mathbf{l}= -\frac{d}{dt}\int_{S}\mathbf{B}!\cdot! d\mathbf{S}) |
| Ampère–Maxwell law | (\nabla\times!\mathbf{B}= \mu_0\mathbf{J}+ \mu_0\varepsilon_0\frac{\partial\mathbf{E}}{\partial t}) | (\oint_{\partial S}\mathbf{B}!\cdot! d\mathbf{l}= \mu_0 I_{\text{enc}}+\mu_0\varepsilon_0\frac{d}{dt}\int_{S}\mathbf{E}!\cdot! d\mathbf{S}) |
Assumptions:
- Linear, isotropic, homogeneous medium (vacuum).
- No free charges or currents: (\rho=0,\ \mathbf{J}=0).
2. Derivation of the Wave Equation
2.1 From Faraday & Ampère–Maxwell
Take the curl of Faraday’s law: [ \nabla\times(\nabla\times\mathbf{E})= -\frac{\partial}{\partial t}(\nabla\times\mathbf{B}). ]
Use vector identity (\nabla\times(\nabla\times\mathbf{E}) = \nabla(\nabla!\cdot!\mathbf{E})-\nabla^2\mathbf{E}).
With (\nabla!\cdot!\mathbf{E}=0) (vacuum, no charges), we get
[
-\nabla^2\mathbf{E}= -\frac{\partial}{\partial t}\bigl(\mu_0\varepsilon_0\frac{\partial\mathbf{E}}{\partial t}\bigr).
]
Thus, [ \nabla^2\mathbf{E} - \mu_0\varepsilon_0,\frac{\partial^2\mathbf{E}}{\partial t^2}=0. ]
Similarly for (\mathbf{B}): [ \nabla^2\mathbf{B} - \mu_0\varepsilon_0,\frac{\partial^2\mathbf{B}}{\partial t^2}=0. ]
2.2 Wave Speed and Speed of Light
Define wave speed (c) by [ c^2 \equiv \frac{1}{\mu_0\varepsilon_0}. ] Hence, [ \nabla^2\mathbf{E} - \frac{1}{c^2}\frac{\partial^2\mathbf{E}}{\partial t^2}=0,\quad \nabla^2\mathbf{B} - \frac{1}{c^2}\frac{\partial^2\mathbf{B}}{\partial t^2}=0. ]
Result: Electromagnetic fields in vacuum propagate as waves at speed (c\approx 3\times10^8) m/s.
3. Plane Wave Solutions
Assume monochromatic plane wave: [ \mathbf{E}(\mathbf{r},t)=\Re{\mathbf{E}_0 e^{i(\mathbf{k}!\cdot!\mathbf{r}-\omega t)}}, ] [ \mathbf{B}(\mathbf{r},t)=\Re{\mathbf{B}_0 e^{i(\mathbf{k}!\cdot!\mathbf{r}-\omega t)}}. ]
Insert into wave equation → dispersion relation: [ |\mathbf{k}|^2 = \frac{\omega^2}{c^2}\quad\Rightarrow\quad \omega = ck. ]
3.1 Transversality
From (\nabla!\cdot!\mathbf{E}=0):
[
\mathbf{k}!\cdot!\mathbf{E}_0 = 0.
]
Same for (\mathbf{B}_0).
Thus electric and magnetic fields are transverse to the propagation direction (\hat{\mathbf{k}}).
3.2 Relation between (\mathbf{E}) and (\mathbf{B})
Faraday’s law in phasor form: [ i\mathbf{k}\times\mathbf{E}_0 = i\omega \mathbf{B}_0 ;;\Rightarrow;;\mathbf{B}_0 = \frac{1}{\omega}\mathbf{k}\times\mathbf{E}_0. ]
Hence (\mathbf{E},\mathbf{B},\hat{\mathbf{k}}) form a right‑handed set. Magnitudes satisfy [ |\mathbf{B}_0| = \frac{|\mathbf{E}_0|}{c}. ]
3.3 Polarization
- Linear: (\mathbf{E}_