04 // Interactive Formula Compendium // PHYSICSc = 1 / √(μ₀ · ε₀) = f · λ, E_photon = h · f = h · c / λ
Maxwell's Equations, Electromagnetic Wave Speed & Photon Energy Sheet
Maxwell's Equations unify electric and magnetic fields and predict transverse electromagnetic waves propagating in vacuum at the speed of light c = 1/√(μ₀ε₀) = 299,792,458 m/s with wavelength λ = c / f.
Formula Reference
Main Formula
c = 1 / √(μ₀ · ε₀) = f · λ, E_photon = h · f = h · c / λ
Rearranged Forms
∇ · E = ρ / ε₀ (Gauss's Law), ∇ · B = 0 (No Magnetic Monopoles)
∇ × E = −∂B / ∂t (Faraday), ∇ × B = μ₀ J + μ₀ ε₀ ∂E / ∂t (Ampère-Maxwell)
S = (1 / μ₀) (E × B), I_avg = ½ c ε₀ E₀² (Poynting Vector & Wave Intensity)
Variables & Units
| Symbol | Name | Unit |
|---|---|---|
| lambda_m | Electromagnetic Wavelength (λ = c / f) | m |
| f_MHz | Wave Frequency f (MHz) | MHz |
| n | Medium Refractive Index (n ≥ 1) | 1 |
Inputs & Parameters
[MHz]
[1]
Calculated Result (Electromagnetic Wavelength (λ = c / f))
lambda_m =2.997925m
Formula Notes & Background
- In charge-free vacuum (ρ = 0, J = 0), take the curl of Faraday's law: ∇ × (∇ × E) = −∂/∂t (∇ × B).
- Use the vector identity ∇ × (∇ × E) = −∇²E (since ∇ · E = 0) and substitute Ampère-Maxwell's displacement current ∇ × B = μ₀ε₀ ∂E/∂t.
- This yields the 3D wave equation ∇²E = μ₀ε₀ (∂²E/∂t²) with phase velocity c = 1/√(μ₀ε₀) ≈ 2.99792 × 10⁸ m/s.
Step-by-Step Examples
#1: Find the vacuum wavelength of a 100 MHz FM radio broadcast and a 2.4 GHz Wi-Fi signal.
c = 2.9979 × 10⁸ m/sf₁ = 100 × 10⁶ Hzf₂ = 2.4 × 10⁹ Hz
λ₁ = 2.9979×10⁸ / 10⁸ = 2.998 m; λ₂ = 2.9979×10⁸ / 2.4×10⁹ = 0.1249 m
→ λ_FM = 3.00 m, λ_WiFi = 12.5 cm
#2: Compute the photon energy in eV of green laser light with λ = 532 nm.
h c ≈ 1239.84 eV·nmλ = 532 nm
E = 1239.84 eV·nm / 532 nm
→ E = 2.33 eV (3.73 × 10⁻¹⁹ J)
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Frequently Asked Questions
What was Maxwell's crucial addition to Ampère's Law?+
Maxwell added the displacement current term μ₀ε₀(∂E/∂t), proving that a changing electric field creates a magnetic field just as a changing magnetic field creates an electric field.
When light enters glass of refractive index n = 1.5, does its frequency or wavelength change?+
Frequency f remains strictly constant across boundary interfaces; phase speed drops to v = c/n and wavelength shortens to λ = λ₀/n.
How are the electric field E₀ and magnetic field B₀ magnitudes related in a vacuum EM wave?+
At every point in a plane electromagnetic wave, E₀ = c · B₀, with E and B perpendicular to each other and to the direction of travel.