Electromagnetic Wave Properties to Know for AP Physics 2

What You Need to Know

Electromagnetic (EM) waves are self-propagating oscillating electric and magnetic fields. In AP Physics 2, you’re expected to connect wave behavior (optics) with field/energy ideas (Maxwell + photons), and to fluently move between \lambda, f, v, energy, intensity, and field amplitudes.

Core ideas (the “must-say” properties)

• EM waves are transverse: \vec E \perp \vec B \perp \text{direction of travel}.
• In vacuum, all EM waves travel at the same speed:
  c = 3.00\times 10^8\ \text{m/s}
• Speed in vacuum relates to constants:
  c = \frac{1}{\sqrt{\mu_0\epsilon_0}}
• Fields are linked (vacuum plane wave):
  \frac{E}{B} = c\quad\Rightarrow\quad E = cB
• EM waves carry energy and momentum (radiation pressure).
• Light can be treated as a wave and as photons:
  E_{\text{photon}} = hf = \frac{hc}{\lambda}

Big exam trigger: If you see intensity, think energy transport (Poynting vector) and/or field amplitudes. If you see \lambda or f, think wave relation and photon energy.

Step-by-Step Breakdown

A) Converting between \lambda, f, v (vacuum or medium)

1. Use the wave relation:
   v = f\lambda
2. If it’s vacuum/air (usually treated as vacuum), set v=c.
3. If it’s a material, use refractive index:
   n = \frac{c}{v}\quad\Rightarrow\quad v = \frac{c}{n}
4. At a boundary, remember:
   • Frequency stays the same: f_1=f_2
   • Speed and wavelength change: v_2 = \frac{c}{n_2} and \lambda_2 = \frac{v_2}{f}

Mini-check: If n increases, speed decreases, so wavelength decreases (frequency unchanged).

B) From intensity to electric/magnetic field amplitude

For a sinusoidal plane EM wave, average intensity is
I = \left\langle S\right\rangle = \frac{1}{2}c\epsilon_0 E_0^2 = \frac{1}{2}\frac{c}{\mu_0}B_0^2

1. Choose which form matches what you need (usually solve for E_0 first).
2. Solve for amplitude:
   E_0 = \sqrt{\frac{2I}{c\epsilon_0}}
3. Then get the other field using
   B_0 = \frac{E_0}{c}

Decision point: Use E_0 and B_0 (amplitudes), not instantaneous values, unless the question explicitly asks for instantaneous fields.

C) Photon energy/momentum (when it’s “quantum light”)

1. From wavelength or frequency:
   E = hf = \frac{hc}{\lambda}
2. Photon momentum:
   p = \frac{E}{c} = \frac{h}{\lambda}
3. If you have power and want photons per second:
   \text{photons/s} = \frac{P}{E_{\text{photon}}}

D) Radiation pressure (force from light)

1. If light with intensity I hits area A, power on it is
   P = IA
2. Radiation pressure:
   • Absorbed: p = \frac{I}{c}
   • Reflected: p = \frac{2I}{c}
3. Force:
   F = pA

Key Formulas, Rules & Facts

Constants you should recognize

EM wave structure (vacuum plane wave)

Wave relations & media

Critical reminder: At refraction, students often (wrongly) change frequency. Don’t. Only v and \lambda change.

Intensity, energy, photons

Radiation pressure

Polarization (transverse wave property)

“Wave optics” conditions (often tested as EM-wave behavior)

Reflection phase shifts (thin film / reflection logic)

• Reflecting from lower n to higher n gives a phase shift of \pi.
• Reflecting from higher n to lower n gives no phase shift.

This shows up when deciding whether two reflected rays are in-phase or out-of-phase.

Examples & Applications

1) Photon energy from wavelength (spectrum + quantum)

Problem: What is the energy of a photon with \lambda = 500\ \text{nm}?

Setup:

• Convert: 500\ \text{nm} = 5.00\times 10^{-7}\ \text{m}
• Use E=\frac{hc}{\lambda}

Key insight:
E = \frac{\left(6.63\times 10^{-34}\right)\left(3.00\times 10^8\right)}{5.00\times 10^{-7}} \approx 3.98\times 10^{-19}\ \text{J}
Higher frequency (shorter wavelength) means higher photon energy.

2) Field amplitude from intensity (link wave energy to E-field)

Problem: Sunlight intensity is about I = 1000\ \text{W/m}^2. Find E_0 and B_0.

Setup:
E_0 = \sqrt{\frac{2I}{c\epsilon_0}}

Key insight (numbers rough OK on AP):
E_0 \approx \sqrt{\frac{2(1000)}{(3.00\times 10^8)(8.85\times 10^{-12})}} \approx 870\ \text{N/C}
Then
B_0 = \frac{E_0}{c} \approx \frac{870}{3.00\times 10^8} \approx 2.9\times 10^{-6}\ \text{T}

3) Radiation pressure and force (momentum transfer)

Problem: A perfectly reflecting sail of area A=2.0\ \text{m}^2 receives I=600\ \text{W/m}^2 normally. Find force.

Setup:

• Reflecting: p=\frac{2I}{c}
• Force: F=pA

Key insight:
F = \left(\frac{2I}{c}\right)A = \frac{2(600)(2.0)}{3.00\times 10^8} \approx 8.0\times 10^{-6}\ \text{N}
Tiny force, but real.

4) Refraction: what changes and what doesn’t

Problem: Light enters glass with n=1.50 from air. If \lambda_{\text{air}}=600\ \text{nm}, find \lambda_{\text{glass}}.

Setup:

• Frequency unchanged.
• Speed reduces: v=\frac{c}{n}
• So wavelength scales the same way: \lambda_{\text{glass}} = \frac{\lambda_{\text{air}}}{n}

Key insight:
\lambda_{\text{glass}} = \frac{600\ \text{nm}}{1.50} = 400\ \text{nm}

Common Mistakes & Traps

1. Changing frequency when light enters a medium

   • Wrong move: Setting f_2=\frac{c}{\lambda_2} using vacuum speed after refraction.
   • Why wrong: At boundaries, f stays fixed; the source sets it.
   • Fix: Use f_1=f_2, then update v and \lambda with n.

2. Forgetting the factor \frac{1}{2} in average intensity

   • Wrong move: Using I=c\epsilon_0E_0^2.
   • Why wrong: Intensity uses time-averaged Poynting vector for sinusoidal waves.
   • Fix: Memorize I=\frac{1}{2}c\epsilon_0E_0^2.

3. Mixing amplitude with instantaneous fields

   • Wrong move: Treating E_0 as the field at all times.
   • Why wrong: The fields oscillate: E(t) varies from -E_0 to +E_0.
   • Fix: If intensity is given, you’re almost always solving for amplitudes E_0,B_0.

4. Using E=cB in a material without thinking

   • Wrong move: Applying E/B=c inside a medium automatically.
   • Why wrong: That exact ratio is for a plane wave in vacuum; AP problems usually keep it vacuum/air unless stated, but be cautious.
   • Fix: Unless the problem explicitly moves into a medium and asks about fields, use the vacuum relationship in vacuum/air contexts.

5. Confusing electric field units with energy units

   • Wrong move: Thinking bigger E field means bigger photon energy.
   • Why wrong: Photon energy depends on f: E_{\text{photon}}=hf. Field amplitude affects intensity, not per-photon energy.
   • Fix: Separate: frequency => photon energy, amplitude => intensity.

6. Radiation pressure factor of 2 mistake

   • Wrong move: Using p=\frac{I}{c} for a reflecting surface.
   • Why wrong: Reflection reverses momentum, doubling change in momentum.
   • Fix: Absorb: I/c. Reflect: 2I/c.

7. Interference sign errors (path vs phase shift on reflection)

   • Wrong move: Using \Delta L=m\lambda without accounting for a \pi phase flip.
   • Why wrong: A reflection from higher n can add an extra half-wavelength phase shift.
   • Fix: Track whether one (and only one) ray gets a \pi shift; that swaps constructive/destructive conditions.

8. Spectrum ordering backwards (energy vs wavelength)

   • Wrong move: Saying radio has higher energy than gamma because “longer is bigger.”
   • Why wrong: E\propto f\propto 1/\lambda.
   • Fix: Shorter wavelength => higher frequency => higher energy.

Memory Aids & Quick Tricks

Quick Review Checklist

• You can state: EM waves are transverse with \vec E \perp \vec B \perp \text{travel}.
• You know in vacuum: c=3.00\times 10^8\ \text{m/s} and c=\frac{1}{\sqrt{\mu_0\epsilon_0}}.
• You can use v=f\lambda and in media v=c/n.
• You remember refraction rule: f stays the same; v and \lambda change.
• You can convert photon energy: E=hf=\frac{hc}{\lambda} and momentum: p=\frac{h}{\lambda}.
• You can get field amplitudes from intensity: I=\frac{1}{2}c\epsilon_0E_0^2 and B_0=E_0/c.
• You can compute radiation pressure: absorb I/c, reflect 2I/c, then F=pA.
• You can apply polarization rules: I=\frac{1}{2}I_0 then I=I_0\cos^2\theta.
• You can rank spectrum correctly: radio => gamma increases f and energy, decreases \lambda.

You’ve got this—if you keep frequency vs amplitude straight, most EM-wave questions collapse into a one-line equation.