Difficulty: Intermediate to Advanced | Prerequisites: Parts 1 and 2 notes, trigonometry, complex numbers helpful but not essential
This is the capstone material for PHYS 212. LC and RLC circuits describe oscillations and resonance, which underpin everything from radio tuning to signal filtering. AC circuit analysis (impedance, phase, transformers) is how real power systems work. Electromagnetic waves tie the entire course together: changing electric fields create magnetic fields and vice versa, producing self-sustaining waves that travel at the speed of light. Polarisation is the final piece, connecting wave optics to everyday technology like sunglasses and LCD screens.
LC circuits oscillate at a natural frequency determined by L and C. In a series RLC circuit, impedance depends on resistance and the difference between inductive and capacitive reactance; at resonance these cancel, leaving only R. Transformers step voltage up or down using turn ratios. EM waves are transverse, travel at c, and carry both energy and momentum. Polarisers reduce light intensity according to Malus's law.
LC circuit
A circuit with only an inductor and a capacitor. Energy oscillates back and forth between the electric field of the capacitor and the magnetic field of the inductor. Think of it as the electrical equivalent of a mass on a spring.
Natural angular frequency (omega_0)
The frequency at which an LC circuit oscillates: omega_0 = 1/sqrt(LC). In simple terms, smaller L or C means faster oscillations.
Impedance (Z)
The AC equivalent of resistance, measured in ohms. It combines resistance R with the effects of inductors and capacitors: Z = sqrt(R^2 + (X_L - X_C)^2).
Inductive reactance (X_L)
The opposition to current flow from an inductor in an AC circuit: X_L = omega L. It increases with frequency.
Capacitive reactance (X_C)
The opposition to current flow from a capacitor in an AC circuit: X_C = 1/(omega C). It decreases with frequency.
Resonance
The condition where X_L = X_C, so impedance is minimised (Z = R) and current is maximised. At resonance, current and voltage are in phase.
Quality factor (Q)
A dimensionless number measuring how "sharp" a resonance is: Q = omega_0 L / R, or equivalently Q = (1/R) sqrt(L/C). A higher Q means a narrower resonance peak and less energy dissipation per cycle.
Transformer
A device that changes AC voltage using two coils (primary and secondary) wound around a shared core. The voltage ratio equals the turns ratio: V_s/V_p = N_s/N_p.
Self-inductance of a solenoid
L = mu_0 N^2 A / l, where N is total turns, A is cross-sectional area, and l is length. Inserting a ferromagnetic core multiplies L by the relative permeability.
Electromagnetic wave
A self-propagating wave of oscillating electric and magnetic fields, travelling at c = 3 x 10^8 m/s in vacuum. The E and B fields are perpendicular to each other and to the direction of travel.
Malus's law
When polarised light passes through a polariser, the transmitted intensity is I = I_in cos^2(theta), where theta is the angle between the polarisation direction and the transmission axis.
Unpolarised light
Light whose electric field oscillates in all directions perpendicular to propagation. After passing through an ideal polariser, its intensity drops to I_0/2.
An LC circuit with inductance L and capacitance C oscillates at angular frequency omega_0 = 1/sqrt(LC).
Energy swings between the capacitor (U_E = Q^2/2C) and the inductor (U_B = LI^2/2). Total energy is conserved.
Example: L = 20 mH, C = 5 microfarads. omega_0 = 1/sqrt(0.020 x 5 x 10^-6) = 1/sqrt(10^-7) = 1/(3.162 x 10^-4) = 3162 rad/s.
The impedance of a series RLC circuit is Z = sqrt(R^2 + (X_L - X_C)^2).
X_L = omega L (inductive reactance), X_C = 1/(omega C) (capacitive reactance).
Example: R = 30 ohms, X_L = 80 ohms, X_C = 40 ohms. Z = sqrt(30^2 + (80 - 40)^2) = sqrt(900 + 1600) = sqrt(2500) = 50 ohms.
The maximum current is I_max = V_max / Z.
Resonance occurs when X_L = X_C, which means omega = 1/sqrt(LC).
At resonance, Z = R (its minimum value), so current is maximised.
The current is in phase with the generator voltage (phase angle phi = 0).
The average power dissipated is at its maximum, not zero.
The impedance is at its minimum, not maximum.
Capacitor: current leads voltage by 90 degrees. The mnemonic is "ICE" (I leads C in E, where E stands for voltage/EMF).
Inductor: voltage leads current by 90 degrees. The mnemonic is "ELI" (E leads I in L).
Resistor: voltage and current are in phase.
At resonance in an RLC circuit, the inductive and capacitive phase shifts cancel, so the net current is in phase with the source voltage.
Q = omega_0 L / R = (1/R) sqrt(L/C).
A high Q means sharp resonance (the circuit responds strongly only near omega_0) and low energy loss per cycle.
Example: R = 10 ohms, L = 100 mH, C = 10 microfarads. omega_0 = 1/sqrt(0.1 x 10^-5) = 1/sqrt(10^-6) = 1000 rad/s. Q = (1000)(0.1)/10 = 10.
An ideal transformer conserves power: V_p I_p = V_s I_s.
The voltage ratio equals the turns ratio: V_s / V_p = N_s / N_p.
Example: N_p = 100, N_s = 500, V_p = 120 V (rms). V_s = 120 x (500/100) = 600 V.
Step-up transformers increase voltage but decrease current. Step-down transformers do the opposite.
L = mu_0 N^2 A / l.
Increasing N, increasing A, or inserting a ferromagnetic core all increase L.
Increasing the length l while keeping N constant decreases L, because the turns are more spread out (n = N/l decreases).
EM waves are transverse: E and B oscillate perpendicular to the direction of propagation and perpendicular to each other.
They travel at c = 3 x 10^8 m/s in vacuum. No medium is required.
They carry both energy and momentum. The energy is described by the Poynting vector S = (1/mu_0) E x B.
The ratio of field amplitudes is E_0 / B_0 = c. Example: E_0 = 300 V/m gives B_0 = 300 / (3 x 10^8) = 1.0 x 10^-6 T.
Unpolarised light through a first ideal polariser: transmitted intensity = I_0 / 2.
Polarised light through a second polariser at angle theta: I = I_in cos^2(theta).
Two-polariser problem: unpolarised I_0 through first polariser gives I_0/2. That polarised light then passes through a second polariser at 60 degrees: I = (I_0/2) cos^2(60) = (I_0/2)(1/4) = I_0/8.
For already-polarised light at angle theta to a single polariser: I_out = I_in cos^2(theta). Example: theta = 30 degrees gives I_out = I_in (sqrt(3)/2)^2 = I_in (3/4) = 75% of I_in.
\omega_0 = \frac{1}{\sqrt{LC}} \quad \text{(LC natural frequency)}Z = \sqrt{R^2 + (X_L - X_C)^2} \quad \text{(RLC impedance)}X_L = \omega L \qquad X_C = \frac{1}{\omega C}Q = \frac{\omega_0 L}{R} = \frac{1}{R}\sqrt{\frac{L}{C}} \quad \text{(quality factor)}\frac{V_s}{V_p} = \frac{N_s}{N_p} \quad \text{(transformer turns ratio)}L = \frac{\mu_0 N^2 A}{l} \quad \text{(solenoid inductance)}\frac{E_0}{B_0} = c = 3 \times 10^8 \text{ m/s} \quad \text{(EM wave field ratio)}I = I_{\text{in}} \cos^2\theta \quad \text{(Malus's law)}For unpolarised light through a first polariser: I = I_0 / 2.
LC and RLC resonance is how radio receivers select a single station from the spectrum: the circuit is tuned so that omega_0 matches the broadcast frequency. Transformers are the reason mains power is transmitted at high voltage and stepped down for household use, reducing resistive losses in the cables. Polarising filters in sunglasses block horizontally polarised glare from reflective surfaces.
Students often think that at resonance the power dissipated is zero. The opposite is true: power dissipation is maximised at resonance because the current is maximised.
Students confuse which component leads and which lags. Remember ELI the ICE man: voltage (E) leads current (I) in an inductor (L); current (I) leads voltage (E) in a capacitor (C).
Students sometimes add reactances as X_L + X_C when computing impedance. The correct combination is (X_L - X_C), because they act in opposite directions.
Students assume EM waves need a medium. They do not; that is what makes them different from sound or water waves.
Students apply Malus's law directly to unpolarised light. The first polariser always halves the intensity (I_0/2); Malus's law only applies to already-polarised light hitting a subsequent polariser.
⚠️ LC frequency problems are straightforward plug-and-compute, but watch your units: convert mH to H and microfarads to F before taking the square root.
⚠️ Impedance problems test whether you subtract reactances (X_L - X_C) rather than adding them.
⚠️ Resonance questions often offer "power is zero" or "impedance is maximum" as tempting wrong answers.
⚠️ Phase-relationship questions appear frequently as multi-select. Know the ELI-ICE mnemonic cold.
⚠️ Transformer problems: multiply or divide by the turns ratio depending on step-up vs step-down.
⚠️ EM wave problems linking E_0 and B_0: divide E_0 by c. Watch the exponent.
⚠️ Two-polariser problems: first polariser halves the intensity (I_0/2), then apply Malus's law with the angle between the two polariser axes.
Fill in the blank: at resonance in a series RLC circuit, the impedance equals ___.
R (the resistance alone).
True or false: in an inductor, current leads voltage by 90 degrees.
False. Voltage leads current by 90 degrees (ELI).
Fill in the blank: when unpolarised light of intensity I_0 passes through one ideal polariser, the transmitted intensity is ___.
I_0 / 2.
True or false: electromagnetic waves require a medium to propagate.
False. They propagate through vacuum.
Fill in the blank: the quality factor of a series RLC circuit with R = 10 ohms, L = 100 mH, C = 10 microfarads is ___.
Q = 10.
Q: An LC circuit has L = 20 mH and C = 5 microfarads. What is the natural angular frequency?
A: omega_0 = 1/sqrt(LC) = 1/sqrt(0.020 x 5 x 10^-6) = 1/sqrt(10^-7) = 3162 rad/s.
Q: A series RLC circuit has R = 30 ohms, X_L = 80 ohms, X_C = 40 ohms. What is Z?
A: Z = sqrt(30^2 + (80 - 40)^2) = sqrt(900 + 1600) = 50 ohms.
Q: At resonance (X_L = X_C), which of the following is true: current in phase with voltage, power is zero, phase angle is 90 degrees, or impedance is maximum?
A: Current is in phase with the generator voltage. The phase angle is zero, impedance is at its minimum (Z = R), and power dissipation is at its maximum.
Q: A transformer has 100 primary turns and 500 secondary turns. If 120 V rms is applied to the primary, what is the secondary voltage?
A: V_s = V_p x (N_s/N_p) = 120 x (500/100) = 600 V.
Q: An EM wave in vacuum has E_0 = 300 V/m. What is B_0?
A: B_0 = E_0/c = 300 / (3 x 10^8) = 1.0 x 10^-6 T.
Q: Unpolarised light of intensity I_0 passes through two polarisers. The second is at 60 degrees to the first. What is the final intensity?
A: After the first polariser: I_0/2. After the second: (I_0/2) cos^2(60) = (I_0/2)(0.25) = I_0/8.
Q: Linearly polarised light hits a polariser at 30 degrees. What percentage of the intensity is transmitted?
A: I_out = I_in cos^2(30) = I_in (3/4) = 75%.
Q: Which changes increase the self-inductance of a solenoid?
A: Inserting a ferromagnetic core, increasing the number of turns N, and increasing the cross-sectional area A. Increasing the length l while keeping N constant decreases L.
Resonance and impedance connect to signal processing and communications: filters, amplifiers, and antenna design all rely on RLC behaviour. EM waves are the bridge from PHYS 212 into optics (PHYS 214) and modern physics. Polarisation links to materials science (birefringence, liquid crystals) and quantum mechanics (photon spin states).
LC circuit, RLC circuit, angular frequency, natural frequency, impedance, reactance, inductive reactance, capacitive reactance, resonance, resonant frequency, quality factor, Q factor, transformer, turns ratio, step-up, step-down, solenoid, self-inductance, electromagnetic wave, EM wave, speed of light, Poynting vector, transverse wave, polarisation, polarization, Malus's law, unpolarised light, polariser, analyzer, PHYS 212, electricity and magnetism, E&M final review