Difficulty: Intermediate to Advanced | Prerequisites: RL and LC circuit notes, basic trigonometry and complex numbers helpful
Tags: RLC circuit, AC circuit, resonance, impedance, reactance, phase angle, power factor, electromagnetic waves, Poynting vector, polarization, Malus's Law, relativistic Doppler effect, EM spectrum
This is the capstone material for a typical E&M course. AC circuits extend the DC analysis you already know by introducing frequency-dependent behaviour: inductors and capacitors no longer behave as short or open circuits, but instead have reactances that vary with the driving frequency. At one special frequency (resonance), the impedance is minimised and the circuit draws maximum current. Electromagnetic waves tie the whole course together: oscillating electric and magnetic fields propagate through space at the speed of light, carrying energy described by the Poynting vector. Polarisation and the Doppler effect round out the optics portion.
You should be comfortable with RL and LC circuits, phasor diagrams (or at least the idea that voltage and current can be out of phase), and the relationship E = cB for electromagnetic waves.
A series RLC circuit driven by an AC source has impedance Z = sqrt(R² + (X_L - X_C)²), where X_L and X_C are the inductive and capacitive reactances. At resonance (X_L = X_C), the impedance drops to R alone, current is maximised, and the power factor is 1. Electromagnetic waves carry energy described by the Poynting vector S = (1/mu_0)(E x B). Unpolarised light passing through a polariser loses half its intensity; subsequent polarisers follow Malus's Law. The relativistic Doppler effect shifts wavelengths for sources moving relative to an observer.
Inductive reactance (X_L)
The opposition an inductor presents to alternating current: X_L = omega L = 2 pi f L, measured in ohms. At higher frequencies, the inductor opposes current more strongly. Think of it as the inductor's "resistance" to AC, which grows with frequency.
Capacitive reactance (X_C)
The opposition a capacitor presents to alternating current: X_C = 1/(omega C) = 1/(2 pi f C), measured in ohms. At higher frequencies, the capacitor passes current more freely. In simple terms, it is the capacitor's "resistance" to AC, which shrinks with frequency.
Impedance (Z)
The total opposition to current in an AC circuit, combining resistance and reactance: Z = sqrt(R² + (X_L - X_C)²). Measured in ohms. It plays the role of resistance in Ohm's Law for AC: V_rms = I_rms times Z.
Resonance (in a series RLC circuit)
The driving frequency at which X_L = X_C, so the impedance is minimised (Z = R). At resonance, the current is at its maximum and the voltage and current are in phase. The resonant angular frequency is omega_0 = 1/sqrt(LC), and the resonant frequency in Hz is f_0 = 1/(2 pi sqrt(LC)).
Phase angle (delta or phi)
The angle by which the voltage leads or lags the current in an AC circuit: tan(delta) = (X_L - X_C)/R. When X_L > X_C, the voltage leads (inductive behaviour). When X_C > X_L, the voltage lags (capacitive behaviour). At resonance, delta = 0.
Power factor (cos delta)
The fraction of the apparent power that is dissipated as real power: P_avg = V_rms I_rms cos(delta). At resonance, cos(delta) = 1 and all the power drawn from the source is dissipated in the resistor.
Poynting vector (S)
S = (1/mu_0)(E x B). It gives the direction and magnitude of energy flow (power per unit area) in an electromagnetic wave. The direction of S is the direction the wave travels. In simple terms, it tells you where the energy is going and how fast.
Malus's Law
When polarised light of intensity I passes through a polariser whose axis is at angle theta to the polarisation direction, the transmitted intensity is I cos²(theta). The first polariser applied to unpolarised light always transmits exactly half the original intensity, regardless of its orientation.
Relativistic Doppler effect (for light)
For a source moving away from an observer at speed v, the observed wavelength is lambda_obs = lambda_source sqrt((1 + v/c)/(1 - v/c)). Moving away produces a redshift (longer wavelength); moving toward produces a blueshift (shorter wavelength).
Given: V_rms = 120 V, R = 30 Omega, L = 200 mH, C = 100 muF, f = 60 Hz.
Step 1: calculate the angular frequency.
omega = 2 pi f = 2 pi (60) = 120 pi ≈ 377 rad/s.
Step 2: calculate the reactances.
X_L = omega L = 377 times 0.200 = 75.4 Omega.
X_C = 1/(omega C) = 1/(377 times 100 x 10⁻⁶) = 1/0.0377 ≈ 26.5 Omega.
Step 3: calculate the impedance.
Z = sqrt(R² + (X_L - X_C)²) = sqrt(30² + (75.4 - 26.5)²) = sqrt(900 + 2390) = sqrt(3290) ≈ 57.4 Omega.
Step 4: calculate the RMS current.
I_rms = V_rms / Z = 120 / 57.4 ≈ 2.09 A.
Step 5: determine the power factor.
cos(delta) = R/Z = 30/57.4 ≈ 0.523.
For R = 40 Omega, L = 50 mH, C = 20 muF:
The resonant frequency is f_0 = 1/(2 pi sqrt(LC)) = 1/(2 pi sqrt(50 x 10⁻³ times 20 x 10⁻⁶)).
LC = 10⁻⁶, so sqrt(LC) = 10⁻³ s.
f_0 = 1/(2 pi times 10⁻³) = 1000/(2 pi) ≈ 159 Hz.
Average power at resonance:
At resonance, Z = R and the power factor is 1.
P_avg = V_rms² / R = (120)² / 40 = 14400/40 = 360 W.
For the same RLC circuit (R = 40, L = 50 mH, C = 20 muF) driven at f = 200 Hz:
omega = 2 pi (200) = 400 pi ≈ 1257 rad/s.
X_L = 1257 times 0.050 = 62.8 Omega.
X_C = 1/(1257 times 20 x 10⁻⁶) = 1/0.02514 ≈ 39.8 Omega.
tan(delta) = (X_L - X_C)/R = (62.8 - 39.8)/40 = 23.0/40 = 0.575.
delta = arctan(0.575) ≈ 29.9 degrees.
Since X_L > X_C, the voltage leads the current (inductive behaviour).
An electromagnetic wave has perpendicular, oscillating E and B fields, both perpendicular to the direction of propagation.
E and B amplitudes are related by:
E_max = c times B_max, where c = 1/sqrt(epsilon_0 mu_0) ≈ 3 x 10⁸ m/s.
Equivalently, E_max = B_max / sqrt(epsilon_0 mu_0) = B_max c.
Given B(x,t) = B_max cos(kx - omega t), the corresponding electric field amplitude is E_max = B_max times c.
Energy flow:
The Poynting vector S = (1/mu_0)(E x B) gives both the direction and the instantaneous rate of energy transport per unit area. The direction of energy flow is determined by S, not by E or B individually and not by the wave vector k alone (though in vacuum, S, k, and the direction of propagation all point the same way).
Unpolarised light through the first polariser:
Unpolarised light contains all polarisation directions equally. Any single polariser transmits exactly half the incident intensity:
I_1 = I_0 / 2.
This result is independent of the polariser's orientation.
Through the second polariser:
After the first polariser, the light is linearly polarised along the first polariser's axis. The second polariser, oriented at angle theta to the first, transmits:
I_2 = I_1 cos²(theta) = (I_0 / 2) cos²(theta).
Worked example: theta = 60 degrees.
I_2 = (I_0 / 2) cos²(60) = (I_0 / 2)(1/2)² = (I_0 / 2)(1/4) = I_0 / 8.
For a source moving away from an observer at v = 0.1c, emitting light at lambda_source = 600 nm:
lambda_obs = lambda_source sqrt((1 + v/c) / (1 - v/c))
lambda_obs = 600 sqrt((1 + 0.1) / (1 - 0.1)) = 600 sqrt(1.1/0.9) = 600 sqrt(1.222) = 600 times 1.1055 ≈ 663 nm.
The observed wavelength is longer (redshifted), as expected for a receding source.
Inductive reactance: X_L = omega L = 2 pi f L
Capacitive reactance: X_C = 1/(omega C) = 1/(2 pi f C)
Impedance: Z = sqrt(R² + (X_L - X_C)²)
RMS current: I_rms = V_rms / Z
Resonant frequency: f_0 = 1/(2 pi sqrt(LC))
Resonant angular frequency: omega_0 = 1/sqrt(LC)
Phase angle: tan(delta) = (X_L - X_C) / R
Power factor: cos(delta) = R / Z
Average power: P_avg = V_rms I_rms cos(delta) = V_rms² R / Z²
Average power at resonance: P_avg = V_rms² / R
E and B relationship: E_max = c B_max
Speed of light: c = 1/sqrt(epsilon_0 mu_0)
Poynting vector: S = (1/mu_0)(E x B)
Malus's Law: I = I_0 cos²(theta)
Unpolarised light through first polariser: I = I_0 / 2
Relativistic Doppler: lambda_obs = lambda_source sqrt((1 + v/c)/(1 - v/c)) (receding source)
RLC resonance is the principle behind radio tuning circuits: adjusting the capacitance shifts the resonant frequency to match the desired station. Power factor correction in industrial electrical systems saves energy and reduces utility costs by bringing the phase angle closer to zero. The Poynting vector quantifies how much solar energy reaches the Earth's surface per square metre, which underpins solar panel design. Polarising filters on cameras and sunglasses exploit Malus's Law to reduce glare from reflected light.
Students often think resonance in an RLC circuit means maximum voltage across the resistor. It means maximum current; the voltage across R does increase, but the total source voltage is fixed.
A common mistake is to calculate the resonant frequency using f = omega/(2 pi) but forgetting the 2 pi entirely, yielding omega in place of f. omega_0 = 1/sqrt(LC) is in rad/s; f_0 = omega_0/(2 pi) is in Hz.
Students sometimes apply Malus's Law directly to unpolarised light (writing I_0 cos²(theta) instead of I_0/2 for the first polariser). The cos² rule only applies to already-polarised light passing through a subsequent polariser.
The relativistic Doppler formula for light is different from the classical Doppler formula for sound. There is no medium for light, and the formula is symmetric: it depends only on relative velocity, not on "who is moving."
⚠️ RLC analysis is heavy on calculation. Practice computing X_L, X_C, Z, I_rms, delta, and cos(delta) until it is automatic. Exams give different component values and driving frequencies each time, so memorising numbers will not help.
⚠️ The resonant frequency formula f_0 = 1/(2 pi sqrt(LC)) is frequently tested. Know it cold, and do not confuse it with the angular frequency version.
⚠️ At resonance, the average power dissipated is P = V_rms²/R. This is a common "what is the power at resonance?" question and the answer is simpler than students expect.
⚠️ The first polariser always halves the intensity of unpolarised light. This is the single most tested fact about polarisation.
⚠️ E_max = cB_max, not B_max/c. The electric field amplitude is the larger quantity. This relationship comes directly from Maxwell's equations and is worth committing to memory.
True or false: at resonance in a series RLC circuit, the impedance equals R.
Fill in the blank: the power factor of a circuit is cos(delta) = ______ / Z.
True or false: unpolarised light passing through a single polariser emerges with intensity I_0 cos²(theta).
Fill in the blank: the Poynting vector is given by S = (1/______)(E x B).
True or false: a source moving away from an observer produces a blueshift.
Answers: 1. True. 2. R. 3. False (it emerges with I_0/2, regardless of the polariser orientation). 4. mu_0. 5. False (it produces a redshift; the wavelength gets longer).
Q: A series RLC circuit has R = 40 Omega, L = 50 mH, C = 20 muF. At what frequency does it resonate?
A: f_0 = 1/(2 pi sqrt(LC)) = 1/(2 pi times 10⁻³) ≈ 159 Hz.
Q: What is the average power dissipated at resonance if V_rms = 120 V and R = 40 Omega?
A: P = V_rms²/R = 14400/40 = 360 W.
Q: Unpolarised light of intensity I_0 passes through a vertical polariser, then through a second polariser at 60 degrees to the vertical. What is the final intensity?
A: I = (I_0/2) cos²(60) = (I_0/2)(1/4) = I_0/8.
Q: The magnetic field of an EM wave has amplitude B_max. What is the electric field amplitude?
A: E_max = cB_max, where c ≈ 3 x 10⁸ m/s.
Q: Which vector determines the direction of energy flow in an electromagnetic wave?
A: The Poynting vector, S = (1/mu_0)(E x B).
Q: A light source emitting at 600 nm moves away from an observer at 0.1c. What wavelength does the observer measure?
A: lambda_obs = 600 sqrt(1.1/0.9) ≈ 663 nm (redshifted).
Q: In a series RLC circuit driven at 200 Hz with R = 40 Omega, L = 50 mH, C = 20 muF, what is the phase angle?
A: X_L ≈ 62.8, X_C ≈ 39.8, tan(delta) = 23.0/40 = 0.575, delta ≈ 30 degrees. The voltage leads the current.
Q: For V_rms = 120 V, R = 30 Omega, L = 200 mH, C = 100 muF at 60 Hz, what is the RMS current?
A: X_L ≈ 75.4, X_C ≈ 26.5, Z ≈ 57.4 Omega, I_rms = 120/57.4 ≈ 2.09 A.
The RLC resonance condition omega_0 = 1/sqrt(LC) is the same angular frequency that governs free LC oscillations (covered in the RL/LC circuits notes), now appearing as the frequency at which a driven circuit responds most strongly. The Poynting vector and electromagnetic wave propagation tie directly back to Faraday's Law and Ampere's Law (with Maxwell's displacement current correction), completing the loop from static fields through time-varying fields to radiation. Polarisation connects to the wave nature of light and leads into topics like Brewster's angle and thin-film interference in optics courses.
RLC circuit, series RLC, AC circuit, alternating current, impedance, reactance, inductive reactance, capacitive reactance, resonance, resonant frequency, phase angle, power factor, cos phi, average power, RMS voltage, RMS current, Ohm's Law for AC, electromagnetic wave, EM wave, Poynting vector, energy flux, polarisation, polarization, Malus's Law, unpolarised light, linear polariser, transmission axis, relativistic Doppler effect, redshift, blueshift, E = cB, speed of light, Maxwell's equations