Electromagnetic Waves

Maxwell noticed an inconsistency in Ampere's law when applied to a charging capacitor: conduction current flows in the connecting wire but no charge flows between the capacitor plates.
He introduced the displacement current Id = $\epsilon_{0}$ (dPhi_E/dt), where $\epsilon_{0}$ is the permittivity of free space, and $\Phi_{E}$ is the electric flux between the plates; [Id] = [A], SI unit: ampere (A). This is not a real current of moving charges — it arises from the changing electric field in the gap. The modified Ampere-Maxwell law becomes: $\oint B .<br/>dl$ = $\mu_{0}$($I_{c}$ + $I_{d}$) = $\mu_{0}$($I_{c}$ + $\epsilon_{0}$ dPhi_E/dt).

Maxwell's four equations (qualitative): (1) Gauss's law for electricity — electric charges produce diverging E fields (source of E = charges). (2) Gauss's law for magnetism — no magnetic monopoles; B field lines are always closed loops. (3) Faraday's law — a changing B field produces an E field (electromagnetic induction). (4) Ampere-Maxwell law — electric currents and changing E fields produce B fields. These equations show a beautiful symmetry: changing B produces E, and changing E produces B, enabling self-sustaining electromagnetic waves.

EM wave production: An accelerating charge produces oscillating E and B fields that propagate outward as a transverse wave.
Properties: E is perpendicular to B, both perpendicular to the direction of propagation; E and B oscillate in phase (peaks occur simultaneously); speed in vacuum c = 1/$\sqrt{mu_0 epsilon_0}$ = 3 x $10^{8}$ m/s; [c] = [L $T^{-1}$], SI unit: m/s; the ratio $\frac{E_{0}}{B_{0}}$ = c.

EM waves carry energy: intensity I = ($\frac{1}{2}$) $\epsilon_{0}$ c $E_{0}^{2}$; [I] = [M $T^{-3}$], SI unit: W/$m^{2}$.
They also carry momentum: p = U/c for complete absorption, p = 2U/c for total reflection (radiation pressure). This means EM waves exert pressure on surfaces.

The EM spectrum in order of increasing frequency (decreasing wavelength): Radio waves ($10^{3}$-$10^{9}$ Hz; source: oscillating circuits; use: communication), Microwaves ($10^{9}$-$10^{12}$ Hz; source: klystron/magnetron; use: radar, microwave ovens), Infrared ($10^{12}$-$4x10^{14}$ Hz; source: hot bodies; use: night vision, physiotherapy), Visible light ($4x10^{14}$-7.$5x10^{14}$ Hz, 400-700 nm VIBGYOR; source: sun, bulbs; use: vision), Ultraviolet (7.$5x10^{14}$-$10^{17}$ Hz; source: sun, mercury lamp; use: sterilization, LASIK), X-rays ($10^{16}$-$10^{21}$ Hz; source: X-ray tubes, fast electrons hitting metal target; use: medical imaging), Gamma rays ($10^{18}$-$10^{24}$ Hz; source: radioactive nuclear decay; use: cancer treatment).

The distinction between X-rays and $\gamma$ rays is source-based, not frequency-based, as their ranges overlap: X-rays come from electron deceleration or inner-shell transitions, while $\gamma$ rays originate from nuclear transitions. All EM waves travel at the same speed c in vacuum — they differ only in frequency and wavelength (c = f $\lambda$). In a medium, different wavelengths travel at different speeds (dispersion).

The key testable concept is the EM spectrum ordering (frequency, wavelength, source, and application for each type) and the properties of EM waves (transverse, E perpendicular to B, in phase, same speed in vacuum).

Solved Numericals

N₁. The electric field in an EM wave is E = 50 sin(wt - kx) V/m. Find the magnitude of the magnetic field $B_{0}$ and verify using c = $\frac{E_{0}}{B_{0}}$. Find the intensity.

Given: $E_{0}$ = 50 V/m, c = 3 x $10^{8}$ m/s.

Magnetic field amplitude: $B_{0}$ = $E_{0}$/c = 50 V/m / (3 x $10^{8}$ m/s) = 1.667 x $10^{-7}$ T = 166.7 nT.

Verification: c = $\frac{E_{0}}{B_{0}}$ = 50 / (1.667 x $10^{-7}$) = 3.0 x $10^{8}$ m/s. Verified.

Intensity: I = ($\frac{1}{2}$) $\epsilon_{0}$ c $E_{0}^{2}$ = ($\frac{1}{2}$) x 8.85 x $10^{-12} \; C^{2}$/(N $m^{2}$) x 3 x $10^{8}$ m/s x (50)² (V/m)² = ($\frac{1}{2}$) x 8.85 x $10^{-12}$ x 3 x $10^{8}$ x 2500 W/$m^{2}$ = ($\frac{1}{2}$) x 8.85 x 3 x 2500 x $10^{-4}$ W/$m^{2}$ = ($\frac{1}{2}$) x 6637.5 x $10^{-4}$ W/$m^{2}$ = 3318.75 x $10^{-4}$ W/$m^{2}$ = 0.3319 W/$m^{2}$ ~ 0.33 W/$m^{2}$.

Dimensional check: [$C^{2}$/(N $m^{2}$)] x [m/s] x [$\frac{V^{2}}{m^{2}}$] = [$C^{2} \; V^{2}$/(N $m^{3}$ s)] = [W/$m^{2}$].
(Since 1 V = 1 J/C, 1 W = 1 J/s, this simplifies correctly.)

N₂. A parallel plate capacitor with plate area 0.1 $m^{2}$ is being charged. If the electric field between plates changes at rate dE/dt = 5 x $10^{12}$ V/(m s), find the displacement current.

Given: A = 0.1 $m^{2}$, dE/dt = 5 x $10^{12}$ V/(m s), $\epsilon_{0}$ = 8.85 x $10^{-12} \; C^{2}$/(N $m^{2}$).

Electric flux: $\Phi_{E}$ = EA (for uniform field perpendicular to plates). dPhi_E/dt = A x dE/dt = 0.1 $m^{2}$ x 5 x $10^{12}$ V/(m s) = 5 x $10^{11}$ V m/s.

Displacement current: $I_{d}$ = $\epsilon_{0}$ x dPhi_E/dt = 8.85 x $10^{-12} \; C^{2}$/(N $m^{2}$) x 5 x $10^{11}$ V m/s = 8.85 x 5 x $10^{-1}$ A = 44.25 x $10^{-1}$ A = 4.425 A.

This displacement current in the gap is exactly equal to the conduction current in the connecting wire, ensuring continuity of current through the complete circuit (Ampere-Maxwell law).