Electromagnetic Waves

Maxwell's Modification: Ampere's law $\oint \vec{B}\cdot d\vec{l} = \mu_0 I$ is inconsistent during capacitor charging — no conduction current flows in the gap. Maxwell introduced displacement current to fix this.

Displacement Current: Current associated with changing electric flux: $$I_d = \epsilon_0\frac{d\Phi_E}{dt}$$ For parallel plate capacitor: $\Phi_E = EA = (Q/\epsilon_0 A)A = Q/\epsilon_0$, so $I_d = dQ/dt$ = conduction current entering capacitor.

Modified Ampere-Maxwell Law: $$\oint \vec{B}\cdot d\vec{l} = \mu_0(I_c + I_d) = \mu_0\left(I_c + \epsilon_0\frac{d\Phi_E}{dt}\right)$$

Maxwell's Four Equations (in integral form):

These predict the existence of EM waves (Maxwell 1864; verified by Hertz 1887).

Speed of EM Waves in Vacuum: $$c = \frac{1}{\sqrt{\mu_0\epsilon_0}} = 3 \times 10^8 \text{ m/s}$$

In medium with permittivity $\epsilon = \epsilon_0\epsilon_r$ and permeability $\mu = \mu_0\mu_r$: $v = c/\sqrt{\epsilon_r\mu_r} = c/n$ where $n$ = refractive index.

Electromagnetic Wave: Transverse wave consisting of mutually perpendicular oscillating $\vec{E}$ and $\vec{B}$ fields, both perpendicular to direction of propagation. Carries energy and momentum.

Wave equations (in vacuum): $$\nabla^2 E = \mu_0\epsilon_0\frac{\partial^2 E}{\partial t^2}, \quad \nabla^2 B = \mu_0\epsilon_0\frac{\partial^2 B}{\partial t^2}$$

Plane Wave Solution (propagating along $+x$): $$E_y = E_0\sin(\omega t - kx), \quad B_z = B_0\sin(\omega t - kx)$$ $\vec{E}, \vec{B}, \vec{c}$ form right-handed triad: $\hat{c} = \hat{E}\times\hat{B}$.

Key Relations:

Properties of EM Waves:

1. Transverse (not longitudinal)
2. Travel through vacuum (don't need medium)
3. Speed in vacuum = $c$ (universal constant)
4. Energy is shared equally between $\vec{E}$ and $\vec{B}$
5. Carry momentum: $p = U/c$
6. Exert radiation pressure on absorbing/reflecting surface
7. Can be polarized (since transverse)
8. Obey reflection, refraction, interference, diffraction

Energy Density: $u = \dfrac{1}{2}\epsilon_0 E^2 + \dfrac{B^2}{2\mu_0} = \epsilon_0 E^2$ (since electric and magnetic components contribute equally).

Average energy density: $\langle u\rangle = \dfrac{1}{2}\epsilon_0 E_0^2$.

Poynting Vector: $\vec{S} = \dfrac{1}{\mu_0}(\vec{E}\times\vec{B})$ gives instantaneous energy flux (W/m²). Average: $\langle S\rangle = \dfrac{E_0^2}{2\mu_0 c} = \dfrac{1}{2}c\epsilon_0 E_0^2$.

EM Spectrum spans frequencies from $\sim 10^0$ Hz to $> 10^{22}$ Hz. All regions travel at $c$ in vacuum.

Major Regions (in order of increasing frequency / decreasing wavelength):

Visible Light Spectrum (VIBGYOR):

• Violet: $380-450$ nm
• Indigo: $450-485$ nm
• Blue: $485-500$ nm
• Green: $500-565$ nm
• Yellow: $565-590$ nm
• Orange: $590-625$ nm
• Red: $625-700$ nm

Mnemonic: "VIBGYOR" (Violet → Red).

Notable points:

• Ozone layer absorbs harmful UV-B and UV-C
• Microwave background radiation ($2.7$ K) — relic of Big Bang
• IR radiation responsible for greenhouse effect
• X-rays were discovered by Roentgen in 1895
• Gamma rays have highest penetration

Intensity $I$ = power per unit area carried by EM wave (W/m²): $$I = \langle S\rangle = \frac{1}{2}\epsilon_0 c E_0^2 = \frac{E_0^2}{2\mu_0 c} = \frac{cB_0^2}{2\mu_0}$$

Relation between $E_0, B_0$, and intensity: $$I = \frac{E_{\text{rms}}^2}{\mu_0 c} = c\epsilon_0 E_{\text{rms}}^2$$

Momentum carried by EM wave: $$p = U/c$$ where $U$ is the energy of the wave.

Radiation Pressure:

Solar Constant (intensity of sunlight at Earth): $\sim 1370$ W/m².

For point source of power $P$, intensity at distance $r$: $I = P/(4\pi r^2)$ (inverse square law).

Power radiated by accelerating charge: $P = q^2a^2/(6\pi\epsilon_0 c^3)$ (Larmor formula).