Superposition and Counter-Propagating Waves
In linear media, the total electromagnetic field is the vector sum of all component solutions. For each component wave, E cross B points along that component's propagation direction; superposition does not permit amplitudes to be added without phase and direction information.
Why this shows up in the exam
Analyzing two waves travelling in opposite directions · Understanding standing electromagnetic waves · Finding resultant fields from multiple coherent sources
Learn the idea
Electromagnetic fields add vectorially, so opposing waves can reinforce or cancel locally. When two waves overlap, nature adds their electric fields and their magnetic fields at each point. Waves travelling in opposite directions have different E-B orientation rules, which can create standing-wave patterns or unequal net fields.
🧠 Memory hook: Add fields with signs and phases, not just amplitudes.
Get this one clearly and it pays off every single time it shows up in the paper. 🎯
Formulas & facts to keep ready
- E_total = sum(E_i) — vector superposition of electric fields
- B_total = sum(B_i) — vector superposition of magnetic fields
- cos(kz-omega t) + cos(kz+omega t) = 2 cos(kz) cos(omega t) — equal counter-propagating waves form a standing pattern
How to approach it
- 1Write each component with its own direction and phase
- 2Apply E cross B separately to each travelling wave
- 3Add like vector components and simplify trigonometric terms
Common slip-ups that cost marks
- •Adding amplitudes as positive scalars regardless of direction
- •Using one propagation direction for both counter-propagating waves
- •Assuming equal electric and magnetic cancellation at every point
🌟 That's the whole idea — you've got this. Try the practice set below; every question you attempt makes it stick a little harder.
Original chapter practice
Original questions for this chapter, not past-paper questions or an exact mapping to this individual concept.
An electromagnetic wave in vacuum has wavelength 1 m. Take c = 3 x 10^8 m/s. Find its frequency in units of 10^8 Hz.
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