EM Waves

EM Waves

🟢 Lite — Quick Review (1h–1d)

Rapid summary for last-minute revision before your exam.

Electromagnetic waves are transverse waves generated by mutually perpendicular, in-phase, time-varying electric (E) and magnetic (B) fields that propagate in vacuum at speed c = 1/√(μ₀ε₀) ≈ 3 × 10⁸ m/s — the value that unifies electricity, magnetism, and optics. The fields and the propagation direction k form a right-handed orthogonal set (k = E × B direction). The ratio E/B in vacuum equals c, so B = E/c. Energy is carried by the Poynting vector S = (1/μ₀) E × B, whose magnitude gives instantaneous energy flux in W/m². High-yield JEE pointers: (1) the displacement current ε₀ dΦ_E/dt completes Ampère’s law and is the reason waves exist in the gap of a charging capacitor; (2) average intensity I = ½ ε₀ c E₀² scales as 1/r² for an isotropic source; (3) for a perfect reflector radiation pressure is 2I/c, while an absorber sees I/c — a frequent single-mark trap.

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🟡 Standard — Regular Study (2d–2mo)

Standard content for students with a few days to months.

Maxwell’s Prediction and Wave Speed

Maxwell’s four equations predict self-sustaining oscillations of E and B that propagate through empty space. Combining Faraday’s law and the Ampère–Maxwell law (with the displacement current I_d = ε₀ dΦ_E/dt) yields the wave equation ∇²E = μ₀ε₀ ∂²E/∂t², giving c = 1/√(μ₀ε₀) ≈ 3 × 10⁸ m/s. The displacement-current term is the conceptual key: without it, Ampère’s law would forbid changing magnetic flux between capacitor plates, and no radiation could cross a capacitor gap.

Field Geometry and Energy Transport

E and B are perpendicular to each other and to k, oscillating in phase with E/B = c. Energy density is shared equally: u = ½ε₀E² + B²/(2μ₀) = ε₀E² (using B = E/c). The Poynting vector S = (1/μ₀) E × B points along k with magnitude EB/μ₀; its time average gives intensity I = ½ε₀c E₀² = P/(4πr²) for an isotropic point source, so doubling the distance quarters the intensity.

Spectrum and Boundary Behaviour

The EM spectrum orders by frequency: radio, microwave, infrared, visible (~400–700 nm), UV, X-ray, gamma — each band characterised by distinct production (e.g., antenna oscillation for radio, bremsstrahlung for X-rays) and detection mechanisms. At a dielectric interface, Brewster’s angle θ_B = tan⁻¹(n₂/n₁) polarises the reflected ray completely; for n₁ > n₂, total internal reflection occurs beyond the critical angle θ_c = sin⁻¹(n₂/n₁).

Quantity	Formula	SI unit
Wave speed in vacuum	c = 1/√(μ₀ε₀)	m/s
Field ratio	E/B = c	m/s
Average intensity	I = ½ ε₀ c E₀²	W/m²
Radiation pressure (absorber)	P = I/c	N/m²
Brewster’s angle	tan θ_B = n₂/n₁	rad

Typical JEE Patterns

Questions pair the Poynting vector with intensity/radiation-pressure calculations, ask for the displacement current in a parallel-plate capacitor being charged at rate dV/dt (I_d = ε₀ A dV/dt), or test polarisation at a Brewster’s-angle setup with one numerical twist.

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🔴 Extended — Deep Study (3mo+)

Comprehensive coverage for students on a longer study timeline.

Dispersive Media: Phase vs Group Velocity

In a non-conducting medium of refractive index n, the phase velocity is v_p = c/n while the group velocity v_g = dω/dk differs from v_p whenever n = n(ω) (normal dispersion: v_g < v_p; anomalous: v_g > v_p, even exceeding c — but information and energy still travel at v_g ≤ c, never violating relativity). JEE rarely quantifies this, but a multiple-choice may assert that “phase velocity equals signal velocity in vacuum” — true — or “in glass” — false. Conductor behaviour adds the skin depth δ = √(2/(μσω)): fields attenuate as e^(−x/δ), and the wave is no longer a true transverse plane wave because E and B fall out of phase.

Radiation Pressure and Momentum

Each photon carries momentum p = h/λ = U/c, giving radiation pressure P = I/c (absorber) and P = 2I/c (perfect reflector). A common trap: for an isotropic source, intensity falls as 1/r², so radiation pressure also falls as 1/r² — not 1/r. For a perfectly reflecting flat surface, only the normal component of momentum reverses, so the tangential Poynting component contributes nothing to force; the full factor 2 appears only for normal incidence.

Worked Micro-Example

A 100 W isotropic bulb sits at the centre of a spherical room of radius r = 2 m. (a) Intensity at the wall: I = P/(4πr²) = 100/(16π) ≈ 1.99 W/m². (b) Peak E-field: E₀ = √(2I/(ε₀c)) = √(2·1.99/(8.85×10⁻¹²·3×10⁸)) ≈ 1.22 × 10² V/m. (c) Peak B-field: B₀ = E₀/c ≈ 4.07 × 10⁻⁷ T. (d) Radiation pressure if the wall is a perfect absorber: P_rad = I/c ≈ 6.6 nPa — vanishingly small, which is why light pressure needs focused lasers or large-area sails to be measurable.

Common Mistakes to Eliminate

• Writing S = E × B and forgetting the 1/μ₀ factor; units immediately expose the error.
• Using I ∝ E (linear) instead of I ∝ E² when comparing fields at two distances.
• Stating “EM waves need a medium” — Maxwell’s equations show vacuum propagation is intrinsic.
• Mixing Brewster’s angle (polarisation) with the critical angle (total internal reflection); they use different formulae and different physical setups.

Practice Prompts

1. A parallel-plate capacitor of capacitance 10 pF is charged from 0 to 100 V in 1 µs. Compute the displacement current between the plates and verify Ampère–Maxwell law across a circular Amperian loop of radius 5 cm centred on the axis.
2. A linearly polarised beam of intensity I₀ is incident on a glass surface (n = 1.5) at Brewster’s angle from air. Find the polarisation state and relative intensity of the reflected ray, then the angle between reflected and refracted rays.

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