UY1: Electromagnetic Spectrum & Sinusoidal EM Plane Waves

This page gives the UY1 working model/result for Electromagnetic Spectrum & Sinusoidal EM Plane Waves. You reuse it when you build fields/potentials by symmetry or superposition, and when you connect fields to forces, energy, and circuits.

Module path: Electromagnetism (UY1)
Practice: UY1 Electromagnetism Quiz
Full routing: UY1 Assessment Map
Math toolkit: Mathematics for Undergraduate Physics

1) At a glance

Electromagnetic waves are coupled oscillations of vec E and vec B fields.
In vacuum: no medium required and speed is constant c.
Key relations:

λ f = c, E₀ = cB₀.

Wave parameters:

k = 2π/λ, ω = 2π f, ω/k = c (vacuum)

Direction of propagation is vec E × vec B.
Phase convention: cos(kx-ω t) propagates in + x; cos(kx + ω t) propagates in -x.

Prerequisites: Displacement Current
Next uses: Energy & Momentum In Electromagnetic Waves, Standing Electromagnetic Waves

2) Setup

Take a linearly polarized plane wave traveling in + x direction.

vec E along hat y,
vec B along hat z,
both perpendicular to propagation direction.

This right-handed triad makes sign checks straightforward.

If you choose vec E and vec B, the propagation direction must satisfy hatkparallel vec E × vec B.
If your phase is cos(kx-ω t), increasing t shifts the pattern toward + x (so it moves in + x).
If you flip one field direction (say vec B → -vec B) but keep the same phase, the wave would propagate the other way because vec E × vec B flips.

3) Core derivation/explanation

A standard sinusoidal vacuum plane wave is:

vec E(x,t) = E₀cos(kx-ω t) hat y, vec B(x,t) = B₀cos(kx-ω t) hat z.

Parameters:

k = 2π/λ, ω = 2π f, ω/k = c.

Field relations in vacuum:

E₀ = cB₀, vec Eperpvec Bperphatk.

If direction reverses to -x, the phase becomes (kx + ω t) form and orientation signs must still satisfy propagation by vec E × vec B.

Spectrum idea: all EM bands follow same physics; they differ mainly by f (or λ), from radio to gamma.

4) Worked example(s)

A wave has frequency f = 100 MHz in vacuum and electric-field amplitude E₀ = 30 V m⁻¹.

Wavelength:

λ = c/f = 3.00 × 10⁸/1.00 × 10⁸ = 3.0 m.

Magnetic amplitude:

B₀ = E₀/c = 30/3.00 × 10⁸ = 1.0 × 10⁻⁷ T.

Band: 100 MHz lies in the radio region.

5) Practice set (with hints + answers)

If wavelength decreases by factor 10 in vacuum, what happens to frequency? Hint: λ f = c. Answer: frequency increases by factor 10.
A wave has B₀ = 2.0 × 10⁻⁶ T. Find E₀ in vacuum. Hint: E₀ = cB₀. Answer: 600 V m⁻¹.
How do you quickly check if chosen vec E and vec B directions are consistent with propagation direction? Hint: cross product. Answer: verify vec E × vec B points along propagation.

6) Summary + next steps

EM waves are transverse and self-propagating field disturbances.
Vacuum relations λ f = c and E₀ = cB₀ solve many quick problems.
Next we quantify energy flow, momentum, and radiation pressure.

Quick checks to run on any plane-wave answer:

Units: λ in m, f in Hz, ω in rad s⁻¹, k in rad m⁻¹.
Direction: verify vec Eperp vec B and vec E × vec B points along the stated propagation direction.

Next: Energy & Momentum In Electromagnetic Waves Previous: Resonance & Power In A.C. Circuits Back To Electromagnetism (UY1)