Difficulty: Intermediate | Prerequisites: Maxwell's equations (conceptual), sinusoidal wave notation, basic AC circuits.
Tags: electromagnetic wave, EM wave, plane wave, Poynting vector, magnetic field, electric field, wave equation, frequency, wavenumber, transformer, ideal transformer, turns ratio, RMS voltage, average power, PHYS 212, University Physics
Electromagnetic waves are the payoff of the entire electricity and magnetism sequence. Maxwell showed that changing electric fields create magnetic fields and vice versa, and that these coupled oscillations propagate through space at the speed of light. This section asks you to read the physics out of a wave equation: frequency, wavelength, direction of travel, field magnitudes, and energy flow. Transformers, meanwhile, are a practical application of electromagnetic induction, and they show up on the exam because they test your understanding of turns ratios, RMS vs. peak values, and power conservation.
An EM wave's equation tells you everything: the wavenumber gives wavelength and frequency, the coefficient gives field amplitude, and the sign in the argument tells you the direction of travel. The Poynting vector (E × B) gives the direction of energy flow. For transformers, the voltage ratio equals the turns ratio, and power in equals power out (for an ideal transformer).
Plane electromagnetic wave
A wave in which the electric and magnetic fields are uniform over any plane perpendicular to the direction of propagation. The standard form is E = E₀ sin(ωt ± ky) with a direction unit vector. In simple terms, it is a flat sheet of oscillating electric and magnetic fields moving through space.
Wavenumber (k)
The spatial frequency of the wave, measured in radians per metre. Related to wavelength by k = 2π/λ. Think of it as "how many radians of oscillation fit in one metre."
Angular frequency (ω)
The temporal frequency in radians per second. Related to ordinary frequency by ω = 2πf. In simple terms, it tells you how fast the fields oscillate in time.
Dispersion relation (in vacuum)
ω = ck, where c = 3 × 10⁸ m/s. This links the temporal and spatial parts of the wave. In simple terms, it says EM waves in vacuum always travel at the speed of light, no matter their frequency.
Poynting vector (S)
S = (1/μ₀)(E × B). It points in the direction the wave carries energy and its magnitude is the power per unit area (intensity). Think of it as an arrow showing which way the light is going and how bright it is.
E/B ratio in vacuum
For an EM wave in vacuum, E₀ / B₀ = c. If you know the electric field amplitude, you can immediately find the magnetic field amplitude. In simple terms, the electric field is always c times larger than the magnetic field (in SI units).
Direction of propagation from the wave equation
If the argument is (ωt + ky), the wave travels in the −y direction. If the argument is (ωt − ky), the wave travels in the +y direction. Think of it this way: a "+" between ωt and ky means the wave moves in the negative coordinate direction, because the spatial phase must decrease over time to keep the argument constant.
Ideal transformer
A device with two coils (primary and secondary) wound on a shared magnetic core. For an ideal transformer, V_s/V_p = N_s/N_p and I_s/I_p = N_p/N_s. Power is conserved: V_p I_p = V_s I_s. In simple terms, it trades voltage for current (or vice versa) without losing energy.
Turns ratio
The ratio N_s/N_p of secondary to primary coil turns. A step-down transformer has N_s < N_p (lower voltage, higher current on the secondary side). A step-up transformer has N_s > N_p.
RMS voltage
The root-mean-square voltage of an AC signal. For a sinusoidal source, V_rms = V_peak / √2. RMS values are what you use to compute average power. In simple terms, it is the "effective" DC-equivalent voltage for power calculations.
Given: E = 2100 sin(ωt + 0.8y) x̂
The electric field oscillates in the x-direction (x̂ is the polarisation direction).
The spatial part is +0.8y, so k = 0.8 rad/m and the wave travels in the −y direction (because of the "+" sign).
Wavelength: λ = 2π/k = 2π/0.8 ≈ 7.85 m.
Frequency: use ω = ck → ω = (3 × 10⁸)(0.8) = 2.4 × 10⁸ rad/s, so f = ω/(2π) ≈ 3.82 × 10⁷ Hz ≈ 38.2 MHz.
Amplitude: B₀ = E₀/c = 2100 / (3 × 10⁸) = 7 × 10⁻⁶ T = 7 μT.
Direction: the wave travels in −ŷ. E is in x̂. Since S ∝ E × B and S points in −ŷ:
x̂ × B̂ = −ŷ → B̂ = ẑ.
So B = 7 × 10⁻⁶ sin(ωt + 0.8y) ẑ.
S = (1/μ₀)(E × B).
E is in x̂, B is in ẑ → E × B = x̂ × ẑ = −ŷ... wait, let's be careful: x̂ × ẑ = −ŷ. But we need the wave to travel in −ŷ (from the +0.8y argument). That checks out: S points in −ŷ.
Answer: −ŷ.
Given: N_p = 100, N_s = 10, V_p,rms = 120 V, R_load = 20 Ω.
Voltage across the resistor (secondary side):
V_s / V_p = N_s / N_p = 10/100 = 0.1.
V_s,rms = 0.1 × 120 = 12 V.
This is a step-down transformer.
Average power in the resistor:
P_avg = V_s,rms² / R = (12)² / 20 = 144/20 = 7.2 W.
Equivalently, I_s,rms = V_s,rms / R = 12/20 = 0.6 A, and P = I² R = (0.6)²(20) = 7.2 W.
Quantity | Formula |
|---|---|
Wavenumber | k = 2π/λ |
Angular frequency | ω = 2πf |
Dispersion relation (vacuum) | ω = ck, or equivalently c = fλ |
Speed of light | c = 3 × 10⁸ m/s |
E/B ratio | E₀ = cB₀ |
Poynting vector | S = (1/μ₀)(E × B) |
Average intensity | ⟨S⟩ = E₀ B₀ / (2μ₀) = E₀² / (2μ₀c) |
Transformer voltage ratio | V_s/V_p = N_s/N_p |
Transformer current ratio | I_s/I_p = N_p/N_s |
Power conservation | V_p I_p = V_s I_s |
Average power (resistor) | P = V_rms² / R = I_rms² R |
The EM wave equations describe everything from radio transmissions to the light reaching your eyes. The Poynting vector is how engineers calculate the power density of a signal arriving at an antenna, which determines whether your phone can pick it up. Transformers are the reason the electrical grid works: power stations step voltage up to hundreds of kilovolts for efficient long-distance transmission, then step it back down to 120 V or 240 V at your house.
Students often confuse the sign convention in the wave equation. A "+" between ωt and ky means the wave travels in the −y direction, not +y. Think of it as: to keep the phase constant, as t increases, y must decrease.
The E/B amplitude ratio is c ≈ 3 × 10⁸, which is enormous. Students sometimes invert this and write B₀ = cE₀. It is B₀ = E₀/c.
For the Poynting vector direction, students sometimes guess rather than computing E × B. Always use the cross product, or at minimum the right-hand rule: fingers curl from E to B, and the thumb points in the direction of propagation.
In transformer problems, students frequently use peak voltage where RMS is required (or vice versa). The problem says "RMS voltage of 120 V," so 120 V is already the RMS value. Do not divide by √2 again.
⚠️ Be ready to extract f, λ, k, ω, and the direction of travel from a single wave equation. This is almost certainly one full exam question.
⚠️ Know the E × B cross product for all six possible combinations of coordinate axes. The exam will give you a specific polarisation and expect you to find the Poynting vector direction quickly.
⚠️ Transformer problems are quick marks if you remember V_s/V_p = N_s/N_p and P_in = P_out. They are easy to lose if you mix up which is primary and which is secondary.
⚠️ "Average voltage" across a resistor in a transformer problem means V_rms (the average of a pure sinusoid is zero, so the question is always asking for RMS).
True or false: If E = E₀ sin(ωt − kz), the wave travels in the +z direction.
Fill in the blank: B₀ = E₀ / ___.
True or false: A step-down transformer has more turns on the secondary coil than the primary.
Fill in the blank: For a plane EM wave, the Poynting vector points in the direction of ___ × ___.
True or false: V_rms = V_peak / 2.
Answers: 1. True. 2. c (speed of light). 3. False (fewer turns on secondary). 4. E × B. 5. False, V_rms = V_peak / √2.
Q: An EM wave has E = 2100 sin(ωt + 0.8y) x̂ in vacuum. What is the frequency?
A: k = 0.8 rad/m. ω = ck = (3 × 10⁸)(0.8) = 2.4 × 10⁸ rad/s. f = ω/(2π) ≈ 3.82 × 10⁷ Hz.
Q: What is the magnitude of the magnetic field for the wave above?
A: B₀ = E₀/c = 2100/(3 × 10⁸) = 7 × 10⁻⁶ T = 7 μT.
Q: What is the direction of the Poynting vector for the wave above?
A: The wave travels in −ŷ (positive sign in the argument with +ky). The Poynting vector points in −ŷ.
Q: A transformer with 100 primary turns and 10 secondary turns is connected to a 120 V RMS source. What is the secondary voltage?
A: V_s = (N_s/N_p) × V_p = (10/100) × 120 = 12 V RMS.
Q: If the secondary feeds a 20 Ω resistor, what is the average power dissipated?
A: P = V²/R = (12)²/20 = 7.2 W.
EM wave properties connect back to Maxwell's equations (the theoretical foundation) and forward to optics: polarisation, reflection, and refraction all describe what happens when these waves encounter matter. Transformer physics connects to Faraday's law of induction and mutual inductance, which you covered earlier in the course. The concept of RMS values reappears in every AC circuit problem.
Related Terms / Search Tags: electromagnetic wave, EM wave, plane wave, Poynting vector, energy flux, wave propagation direction, wavenumber, angular frequency, E/B ratio, speed of light, transformer, ideal transformer, turns ratio, step-up, step-down, RMS voltage, average power, AC power, PHYS 212, University Physics, Exam 3