Electromagnetic Waves & Maxwell Review – Quick Notes

Classical Foundations

• Remaining classical topic: electromagnetic waves (light)

• Core laws summarizing classical physics

  • Newton’s 3 laws, universal gravitation

  • 1st & 2nd laws of thermodynamics

  • Maxwell’s equations + Lorentz force

• Light exhibits dual nature

  • Wave: transverse $\mathbf{E},\;\mathbf{B}\perp$ to direction of travel, needs no medium

  • Particle (photon) treatment postponed to modern physics

Faraday’s Law (General Form)

• Changing magnetic flux creates a circulating, non-conservative $\mathbf{E}$-field even in empty space

• Integral form

  • $\displaystyle \oint \mathbf{E}\cdot d\mathbf{l} = -\frac{d\Phi_B}{dt}$ (Lenz sign shows opposition to flux change)

• Distinguish fields

  • Electrostatic $\mathbf{E}$ from stationary charges (conservative)

  • Induced $\mathbf{E}$ from $d\Phi_B/dt$ (non-conservative)

Maxwell’s Five Equations

• Gauss (Electric): $\displaystyle \oint \mathbf{E}\cdot d\mathbf{A}=\frac{q<em>{\text{in}}}{\varepsilon</em>0}$

• Gauss (Magnetic): $\displaystyle \oint \mathbf{B}\cdot d\mathbf{A}=0$ (no monopoles observed)

• Faraday (Induction): $\displaystyle \oint \mathbf{E}\cdot d\mathbf{l}= -\frac{d\Phi_B}{dt}$

• Ampère–Maxwell: $\displaystyle \oint \mathbf{B}\cdot d\mathbf{l}=\mu<em>0 I</em>{\text{thru}}+\mu<em>0\varepsilon</em>0\frac{d\Phi_E}{dt}$

• Lorentz force: $\displaystyle \mathbf{F}=q(\mathbf{E}+\mathbf{v}\times\mathbf{B})$

Electromagnetic Waves

• Combine Faraday & Ampère–Maxwell → wave equations (vacuum)

  • $\displaystyle \nabla^2 \mathbf{E}=\mu<em>0\varepsilon</em>0\frac{\partial^2 \mathbf{E}}{\partial t^2}$

  • $\displaystyle \nabla^2 \mathbf{B}=\mu<em>0\varepsilon</em>0\frac{\partial^2 \mathbf{B}}{\partial t^2}$

• Plane wave (linearly polarized, propagating +x)

  • $\mathbf{E}(x,t)=E_{\max}\sin(kx-\omega t)\,\hat{j}$

  • $\mathbf{B}(x,t)=B_{\max}\sin(kx-\omega t)\,\hat{k}$

  • Speed: $c=\frac{1}{\sqrt{\mu<em>0\varepsilon</em>0}}\approx3.00\times10^8\,\text{m/s}$

  • Field amplitudes related: $E<em>{\max}=c\,B</em>{\max}$; holds instantaneously $\mathbf{E}=c\mathbf{B}$

  • Wave parameters: $c=\lambda f=\omega/k$ (same for $\mathbf{E},\mathbf{B}$)

Electromagnetic Spectrum

• Continuous range classified by $f$ or $\lambda$; all satisfy $c=\lambda f$

• Order (long $\lambda$ → short): Radio, Microwave, Infrared, Visible (ROYGBIV), Ultraviolet, X-ray, Gamma

Energy & Intensity of EM Waves

• Poynting vector (instantaneous power flux)

  • $\displaystyle \mathbf{S}=\frac{1}{\mu_0}\,\mathbf{E}\times\mathbf{B}$ (points along propagation)

• Intensity (time-average power/area)

  • $\displaystyle I=\langle S\rangle=\frac{E<em>{\max}B</em>{\max}}{2\mu<em>0}=\frac{1}{2}\varepsilon</em>0 c E<em>{\max}^2=\frac{c}{2\mu</em>0}B_{\max}^2$

• Energy density

  • Equal electric & magnetic contributions: $u<em>{\text{av}}=\frac{1}{2}\varepsilon</em>0E<em>{\max}^2=\frac{B</em>{\max}^2}{2\mu_0}$

  • Relation to intensity: $I=cu_{\text{av}}$

• Point source: $I=P_{\text{av}}/(4\pi r^2)$

Momentum & Radiation Pressure

• Wave carries momentum $p=Q/c$ (energy $Q$)

• Pressure on surface (normal incidence)

  • Totally absorbed: $P=I/c$

  • Totally reflected: $P=2I/c$

  • Partial: between these limits

Polarization

• Linearly (plane) polarized: $\mathbf{E}$ oscillates in single plane; $\mathbf{B}\perp \mathbf{E}$

• Unpolarized: random superposition of orthogonal polarizations (each holds $I/2$)

• Polarizer (transmission axis)

  • Passes component parallel to axis, absorbs perpendicular component → output intensity halved for unpolarized input

• Malus’s law for polarized incident light

  • $\displaystyle I=I_0\cos^2\theta$ ( $\theta$ = angle between incident $\mathbf{E}$ plane and transmission axis)

• Successive polarizers

  • Perpendicular axes (90°) → $I=0$

  • Inserting intermediate axis (e.g., 45°) allows partial transmission by successive projection