Electromagnetic Waves & Spectrum

Displacement Current and Maxwell's Equations

Maxwell identified an inconsistency in Ampere's law: a charging capacitor has no conduction current between the plates, yet the magnetic field must be continuous.
He introduced displacement current: $I_{d}$ = $\epsilon_{0}$ * d($\Phi_{E}$)/dt, where $\Phi_{E}$ is the electric flux.
The modified Ampere-Maxwell law: the line integral of B around a closed loop = $\mu_{0}$*($I_{c}$ + $I_{d}$), where $I_{c}$ is conduction current.
Between capacitor plates, $I_{d}$ = $I_{c}$ (continuity of current is maintained). Displacement current exists wherever the electric field changes with time, even in vacuum.

Maxwell's four equations (qualitative): (1) Gauss's law for electricity — electric flux through a closed surface equals enclosed charge/$\epsilon_{0}$. (2) Gauss's law for magnetism — magnetic flux through any closed surface is zero (no magnetic monopoles). (3) Faraday's law — a changing magnetic flux produces an electric field (EMF). (4) Ampere-Maxwell law — a current or changing electric flux produces a magnetic field. Together, these predict that mutually sustaining oscillating electric and magnetic fields can propagate through space as electromagnetic waves.

Properties of Electromagnetic Waves

Electromagnetic Wave — E and B Field Oscillation:

EM waves are transverse waves with oscillating E and B fields perpendicular to each other and to the direction of propagation.
If the wave propagates along the z-axis: E = $E_{0}$ sin(kz - $\omega$*t) x-hat, B = $B_{0}$ sin(kz - $\omega$*t) y-hat. Key properties:

Speed in vacuum: c = 1/$\sqrt{mu_0 * epsilon_0}$ = 3 x $10^{8}$ m/s. This was a triumph of Maxwell's theory — it predicted the speed of light from purely electromagnetic constants.

The ratio $\frac{E_{0}}{B_{0}}$ = c always holds. E and B are in phase and reach their maxima and zeros simultaneously.

EM waves carry energy.
The energy density: u = ($\frac{1}{2}$)$\epsilon_{0}$$E^{2}$ + ($\frac{1}{2}$)$\frac{B^{2}}{\mu_{0}}$.
At any instant, the electric and magnetic energy densities are equal: ($\frac{1}{2}$)$\epsilon_{0}$$E^{2}$ = ($\frac{1}{2}$)$\frac{B^{2}}{\mu_{0}}$.

The Poynting vector S = (1/$\mu_{0}$)(E x B) gives the energy flow per unit area per unit time (W/$m^{2}$). Its magnitude is the intensity.
Average intensity: I = ($\frac{1}{2}$)c$\epsilon_{0}$$E_{0}^{2}$ = $E_{0}$$B_{0}$/(2$\mu_{0}$).

EM waves carry momentum. Radiation pressure on a perfectly absorbing surface: P = I/c. On a perfectly reflecting surface: P = 2I/c. Force = P x A.

EM waves are produced by accelerating charges. An oscillating charge of frequency f produces EM waves of the same frequency. Stationary charges produce no EM waves; charges in uniform motion produce no EM radiation (they produce static or uniformly moving fields).

In a medium with permittivity $\epsilon$ and permeability $\mu$: v = 1/$\sqrt{\mu*\epsilon}$ = c/n, where n = $\sqrt{mu_r * epsilon_r}$ is the refractive index.

The Electromagnetic Spectrum

EM Spectrum — From Radio to Gamma Rays:

The EM spectrum is a continuous range of frequencies/wavelengths, all traveling at speed c in vacuum. The regions, from lowest to highest frequency:

Radio waves (f < $10^{9}$ Hz, $\lambda$ > 0.3 m): Produced by oscillating circuits (LC oscillators, antennas). Used in communication, broadcasting, radar. AM radio (~500-1500 kHz), FM radio (~88-108 MHz), TV (~54-890 MHz).

Microwaves ($10^{9}$ - $10^{11}$ Hz, 0.3 m - 1 mm): Produced by klystrons, magnetrons. Used in microwave ovens (2.45 GHz — resonant frequency of water molecules), radar, satellite communication, WiFi.

Infrared ($10^{11}$ - 4 x $10^{14}$ Hz, 1 mm - 700 nm): Produced by hot bodies and molecules. Used in thermal imaging, remote controls, greenhouse effect, night vision. Earth's atmosphere is partly opaque to IR (greenhouse gases absorb and re-emit).

Visible light (4 x $10^{14}$ - 8 x $10^{14}$ Hz, 700 - 400 nm): Produced by atomic electron transitions. VIBGYOR: Violet (~400 nm) to Red (~700 nm). The only EM radiation detected by human eyes.

Ultraviolet (8 x $10^{14}$ - $10^{17}$ Hz, 400 - 1 nm): Produced by very hot objects, gas discharges. Causes sunburn, vitamin D synthesis, fluorescence. Ozone layer absorbs harmful UV-B and UV-C.

X-rays ($10^{16}$ - $10^{19}$ Hz, 10 nm - 0.01 nm): Produced by bombarding high-Z metal targets with fast electrons. Used in medical imaging, crystallography (Bragg diffraction), security scanning.

Gamma rays (f > $10^{19}$ Hz, $\lambda$ < 0.01 nm): Produced by nuclear transitions and particle annihilation. Highest frequency, most energetic, most penetrating. Used in cancer treatment, sterilization.

The key problem-solving concept is that all EM waves share the same fundamental properties (speed c, transverse nature, E/B = c) but differ in wavelength/frequency, which determines their source, interaction with matter, and applications.

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