Source: Chapters 15 and 16
Tags: alternating current, AC circuit, RMS, reactance, impedance, resonance, transformer, Maxwell's equations, electromagnetic waves, Poynting vector, radiation pressure, EM spectrum
Difficulty: Advanced
Prerequisites: Chapters 9-14 notes (DC circuits, Ohm's law, capacitors, inductors, Faraday's law, LC oscillations).
AC circuits extend everything you learnt about DC circuits to the case where current and voltage oscillate sinusoidally. Capacitors and inductors no longer simply store energy; they also introduce phase shifts and frequency-dependent behaviour (reactance). Chapter 16 then brings the entire electromagnetism sequence to its climax: Maxwell's equations predict the existence of electromagnetic waves, which travel at the speed of light and carry energy and momentum. This is the unification of electricity, magnetism, and optics.
In AC circuits, voltage and current are sinusoidal, and capacitors and inductors each have a frequency-dependent "resistance" called reactance. The total opposition to current is impedance. At a particular resonant frequency, the circuit lets maximum current through. Electromagnetic waves are self-propagating oscillations of electric and magnetic fields, predicted by Maxwell's equations, and they carry energy described by the Poynting vector.
Alternating current (AC)
Current that periodically reverses direction, typically sinusoidal: i = I₀ sin(ωt).
Direct current (DC)
Current that flows in only one direction.
RMS (root-mean-square) values
The effective values of oscillating quantities. I_rms = I₀/√2 and V_rms = V₀/√2. These are the values that produce the same average power as a DC circuit.
Capacitive reactance (X_C)
The opposition of a capacitor to AC current: X_C = 1/(ωC). Decreases with frequency: at high frequencies, a capacitor passes current easily.
Inductive reactance (X_L)
The opposition of an inductor to AC current: X_L = ωL. Increases with frequency: at high frequencies, an inductor strongly opposes current changes.
Impedance (Z)
The total opposition to current in an AC circuit, combining resistance and reactance: Z = √(R² + (X_L - X_C)²). Measured in ohms. This is the AC generalisation of resistance.
Phase angle (φ)
The angle by which the current lags or leads the voltage: tan φ = (X_L - X_C)/R.
Resonance
The frequency at which X_L = X_C, so impedance is minimised (Z = R) and current is maximised. Resonant frequency: ω₀ = 1/√(LC).
Quality factor (Q)
A measure of the sharpness of the resonance peak. Higher Q means a narrower, taller peak.
Transformer
A device that changes AC voltage from one level to another using mutual induction: V_s/V_p = N_s/N_p. A step-up transformer increases voltage; a step-down decreases it.
Displacement current (I_d)
An additional term in Ampere's law, equal to ε₀ dΦ_E/dt, that accounts for changing electric fields producing magnetic fields even without actual charge flow. This was Maxwell's key addition.
Maxwell's equations
The four equations that unify all of electricity and magnetism. Together they predict electromagnetic waves.
Electromagnetic wave
A self-propagating oscillation of perpendicular electric and magnetic fields, travelling at c = 1/√(μ₀ε₀) ≈ 3 x 10⁸ m/s in free space.
Poynting vector (S)
The rate of energy transport per unit area in an EM wave: S = (1/μ₀) E x B. Its magnitude is the intensity.
Intensity (I)
The average power per unit area carried by an EM wave: I = S_avg = ½cε₀E₀² = ½E₀B₀/μ₀.
Radiation pressure
The pressure exerted by EM waves on a surface. For a perfect absorber: P = I/c. For a perfect reflector: P = 2I/c.
AC voltage: v = V₀ sin(ωt). AC current: i = I₀ sin(ωt).
RMS values: I_rms = I₀/√2, V_rms = V₀/√2.
For a resistor: V = IR still holds instantaneously. Average power: P_avg = I_rms V_rms = I_rms² R.
The current leads the voltage by 90° (π/2 radians).
Capacitive reactance: X_C = 1/(ωC). Acts like frequency-dependent resistance.
At low frequencies, X_C is large (capacitor blocks DC). At high frequencies, X_C is small.
The current lags the voltage by 90° (π/2 radians).
Inductive reactance: X_L = ωL.
At low frequencies, X_L is small (inductor passes DC freely). At high frequencies, X_L is large.
Current: i(t) = I₀ sin(ωt - φ).
Current amplitude: I₀ = V₀/Z, where Z = √(R² + (X_L - X_C)²).
Phase angle: tan φ = (X_L - X_C)/R.
If X_L > X_C, the circuit is inductive (current lags voltage).
If X_C > X_L, the circuit is capacitive (current leads voltage).
If X_L = X_C, the circuit is purely resistive (current in phase with voltage).
Instantaneous power varies over each cycle.
Average power: P_avg = I_rms V_rms cos φ, where cos φ is the power factor.
Power is dissipated only in the resistor. Ideal capacitors and inductors absorb and return energy without net dissipation.
P_avg = I_rms² R.
At resonance: ω₀ = 1/√(LC), so X_L = X_C and Z = R.
Current is maximum, power delivered to the resistor is maximum.
The bandwidth is the range of frequencies where P_avg exceeds half of its peak value.
Quality factor: Q = ω₀L/R. Higher Q means sharper resonance.
Based on mutual induction between two coils wound on a shared core.
Voltage ratio: V_s/V_p = N_s/N_p.
Current ratio (ideal): I_s = I_p (N_p/N_s). Power is conserved: V_p I_p = V_s I_s.
Step-up: N_s > N_p (voltage increases, current decreases).
Step-down: N_s < N_p (voltage decreases, current increases).
Maxwell added the displacement current term I_d = ε₀ dΦ_E/dt to Ampere's law.
This made the equations symmetric: a changing electric field produces a magnetic field (just as a changing magnetic field produces an electric field via Faraday's law).
The modified Ampere's law: ∮ B · dl = μ₀(I + I_d).
Maxwell's equations predict waves with speed c = 1/√(μ₀ε₀) = 3 x 10⁸ m/s.
E and B are perpendicular to each other and to the direction of propagation.
E and B are in phase and related by E = cB at all times.
Electric field: E_y(x,t) = E₀ cos(kx - ωt). Magnetic field: B_z(x,t) = B₀ cos(kx - ωt).
General wave relation: v = fλ. For EM waves: c = fλ.
Energy density: u = ½ε₀E² + B²/(2μ₀) = ε₀E² (since the electric and magnetic contributions are equal).
Energy transported: U = S × A × Δt.
Poynting vector: S = (1/μ₀) E x B. Intensity: I = S_avg = ½cε₀E₀².
EM waves carry momentum. Force on a surface: F = PA, where P is radiation pressure.
Perfect absorber: P = I/c. Perfect reflector: P = 2I/c.
All EM waves travel at c in vacuum, but differ in frequency and wavelength.
From low to high frequency: radio, microwave, infrared, visible, ultraviolet, X-ray, gamma ray.
c = fλ relates frequency and wavelength.
Quantity | Formula |
|---|---|
RMS values | I_rms = I₀/√2, V_rms = V₀/√2 |
Capacitive reactance | X_C = 1/(ωC) |
Inductive reactance | X_L = ωL |
Impedance | Z = √(R² + (X_L - X_C)²) |
Phase angle | tan φ = (X_L - X_C)/R |
Average power | P_avg = I_rms V_rms cos φ |
Resonant frequency | ω₀ = 1/√(LC) |
Transformer voltage | V_s/V_p = N_s/N_p |
Speed of light | c = 1/√(μ₀ε₀) = fλ |
Poynting vector | S = (1/μ₀) E x B |
Intensity | I = ½cε₀E₀² |
Radiation pressure (absorber) | P = I/c |
Radiation pressure (reflector) | P = 2I/c |
The resonant frequency of an RLC circuit is how radio receivers select a station: by tuning C (or L), you set ω₀ to match the broadcast frequency. Transformers are the reason long-distance power transmission works: voltage is stepped up to reduce current (and thus resistive losses in the wires), then stepped back down for household use. Radiation pressure from sunlight is used in proposals for solar sails that could propel spacecraft without fuel.
Students often forget that capacitive and inductive reactances have opposite frequency dependences. X_C decreases with ω, X_L increases. At resonance they are equal and cancel.
Impedance is not simply R + X_L + X_C. The reactances subtract from each other (because they are 90° out of phase with each other), and then you combine with R using the Pythagorean formula.
RMS values are not simply half the peak values. The factor is 1/√2 ≈ 0.707, not ½.
Students sometimes think EM waves need a medium to propagate. They do not: they travel through vacuum at the speed of light.
⚠️ Impedance and phase angle calculations are standard exam problems. Know the formula for Z and how to determine whether the circuit is inductive or capacitive from the sign of (X_L - X_C).
⚠️ At resonance, Z = R and the current is maximum. Be ready to find the resonant frequency from L and C.
⚠️ Know the relationship E₀ = cB₀ for EM waves and how to compute intensity from either E₀ or B₀.
⚠️ Transformer problems: always use the turns ratio. For an ideal transformer, power in = power out.
Fill in the blank: At resonance in an RLC circuit, the impedance equals ________.
R.
True or false: Inductive reactance increases with frequency.
True (X_L = ωL).
Fill in the blank: The speed of electromagnetic waves in vacuum is c = ________.
1/√(μ₀ε₀) ≈ 3 x 10⁸ m/s.
True or false: For a perfect reflector, the radiation pressure is I/c.
False. For a perfect reflector, it is 2I/c.
Q: An RLC series circuit has R = 100 Ω, L = 0.5 H, and C = 20 μF. What is the resonant frequency?
A: ω₀ = 1/√(LC) = 1/√(0.5 × 20 x 10⁻⁶) = 1/√(10⁻⁵) = 316 rad/s. f₀ = ω₀/(2π) ≈ 50.3 Hz.
Q: A transformer has 500 primary turns and 50 secondary turns. If the primary voltage is 120 V, what is the secondary voltage?
A: V_s = V_p × (N_s/N_p) = 120 × (50/500) = 12 V (step-down).
Q: An EM wave has an electric field amplitude of 50 V/m. What is the magnetic field amplitude?
A: B₀ = E₀/c = 50 / (3 x 10⁸) = 1.67 x 10⁻⁷ T.
Q: Sunlight has an intensity of about 1400 W/m² at Earth's orbit. What is the radiation pressure on a perfectly absorbing surface?
A: P = I/c = 1400 / (3 x 10⁸) ≈ 4.67 x 10⁻⁶ Pa.
Resonance concepts from RLC circuits carry directly into optics (interference and diffraction involve frequency selection). Maxwell's equations are the theoretical capstone of the entire PHYS 212 electromagnetism sequence. The EM wave results connect to the nature of light (Vol. 3, Ch. 1), where light is treated as an electromagnetic wave subject to reflection, refraction, and polarisation.
alternating current, AC, direct current, DC, RMS, reactance, capacitive reactance, inductive reactance, impedance, phase angle, power factor, resonance, quality factor, bandwidth, transformer, step-up, step-down, Maxwell's equations, displacement current, electromagnetic wave, Poynting vector, intensity, radiation pressure, EM spectrum, radio wave, microwave, infrared, visible light, ultraviolet, X-ray, gamma ray, speed of light, PHYS 212