Source: UIUC Electricity, Magnetism, and Thermodynamics Final Examination
Tags: AC circuits, capacitive reactance, inductive reactance, impedance, resonance, RLC series circuit, transformer, electromagnetic waves, Poynting vector, polarization, Malus's law, first law of thermodynamics, specific heat capacity, mechanical equivalent of heat, RMS voltage
Difficulty: Intermediate | Prerequisites: Parts 1 and 2 of these study notes (electrostatics, capacitors, magnetism, induction, RL/RC/LC circuits).
AC circuits bring together everything from the first two parts of the course: resistors dissipate energy, capacitors and inductors store it, and the interplay between them at different frequencies produces resonance, impedance, and phase shifts. Electromagnetic waves are the payoff of the entire electricity and magnetism sequence, showing that oscillating electric and magnetic fields can propagate through empty space at the speed of light. Thermodynamics, while a distinct topic, rounds out the final exam by connecting energy conservation (the first law) and heat transfer to the mechanical and electrical systems studied earlier.
In AC circuits, capacitors and inductors have frequency-dependent opposition to current (reactance), and the total opposition (impedance) is minimised at resonance when inductive and capacitive reactances cancel. Electromagnetic waves consist of perpendicular, oscillating E and B fields that travel at c, with the Poynting vector indicating energy flow. The first law of thermodynamics is energy conservation applied to heat and work, and specific heat capacity tells you how much energy is needed to raise a substance's temperature.
Capacitive Reactance (X_C)
The opposition a capacitor offers to alternating current: X_C = 1/(ωC) = 1/(2πfC). Measured in ohms. In simple terms, a capacitor blocks low frequencies and passes high frequencies. Doubling the frequency halves the reactance.
Inductive Reactance (X_L)
The opposition an inductor offers to alternating current: X_L = ωL = 2πfL. Measured in ohms. Think of it as the opposite of capacitive reactance: an inductor passes low frequencies easily but resists high frequencies.
Impedance (Z)
The total opposition to current in an AC circuit: Z = √(R² + (X_L − X_C)²). Measured in ohms. In simple terms, impedance is the AC equivalent of resistance, but it includes the effects of capacitors and inductors as well.
Resonance (in a Series RLC Circuit)
The condition where X_L = X_C, so the impedance reduces to Z = R (its minimum value). The resonant angular frequency is ω₀ = 1/√(LC). Think of it as the "sweet spot" frequency where the circuit allows the maximum current to flow.
Transformer
A device that uses electromagnetic induction to change the voltage of an AC signal. The voltage ratio equals the turns ratio: V_s/V_p = N_s/N_p. In simple terms, more turns on the secondary coil means higher voltage out (step-up), fewer turns means lower voltage out (step-down).
RMS Voltage (V_rms)
The root-mean-square voltage of a sinusoidal AC source: V_rms = V_peak / √2. Think of it as the "effective" voltage: a DC source at V_rms delivers the same average power as the AC source at V_peak.
Electromagnetic Wave
A self-propagating wave of oscillating electric and magnetic fields, perpendicular to each other and to the direction of travel. Travels at c = 3 × 10⁸ m/s in vacuum. In simple terms, light, radio, microwaves, and X-rays are all electromagnetic waves at different frequencies.
Poynting Vector (S)
S = (1/μ₀) E × B. It gives both the direction of energy flow and the intensity (power per unit area) of an electromagnetic wave. Units: W/m². Think of it as an arrow showing where the wave's energy is going and how much of it there is.
Malus's Law
When polarised light of intensity I₀ passes through a polarising filter at angle θ to the polarisation direction, the transmitted intensity is I = I₀ cos² θ. In simple terms, the closer the filter is aligned to the light's polarisation, the more light gets through.
First Law of Thermodynamics
ΔU = Q − W, where ΔU is the change in internal energy, Q is heat added to the system, and W is work done by the system. In simple terms, the energy of a system changes by the amount of heat you put in, minus the work it does on its surroundings.
Specific Heat Capacity (c)
The amount of heat required to raise 1 kg of a substance by 1 K (or 1°C): Q = mcΔT. Units: J/(kg·K). Think of it as a material's thermal stubbornness: water's high specific heat (4190 J/(kg·K)) is why it heats up slowly and cools down slowly.
X_C = 1/(2πfC). Reactance is inversely proportional to frequency.
If the frequency of an AC source connected to a pure capacitor is doubled, X_C is halved.
At very high frequencies the capacitor behaves almost like a short circuit (X_C approaches 0). At DC (f = 0) it behaves like an open circuit (X_C is infinite).
Z = √(R² + (X_L − X_C)²).
At resonance, X_L = X_C, so Z = R. This is the minimum possible impedance for the circuit.
The resonant frequency is f₀ = 1/(2π√(LC)).
For R = 40 Ω, L = 80 mH, C = 50 μF: f₀ = 1/(2π√(0.08 × 50 × 10⁻⁶)) = 1/(2π√(4 × 10⁻⁶)) = 1/(2π × 0.002) = 1/(0.01257) ≈ 79.6 Hz.
At resonance the current is in phase with the voltage (phase angle φ = 0).
The phase angle between voltage and current in a series RLC circuit is φ = arctan((X_L − X_C)/R).
At resonance, X_L = X_C, so φ = 0. Voltage and current are in phase.
The statement "at resonance the phase angle is zero" is true.
For a sinusoidal AC source, V_rms = V_peak / √2.
This relationship is exact for sinusoidal waveforms only. It does not hold for square waves or other shapes.
The statement "RMS voltage equals peak voltage divided by √2" is true.
V_s / V_p = N_s / N_p.
A step-up transformer has more turns on the secondary than the primary (N_s > N_p), so V_s > V_p.
For N_p = 100, N_s = 500, V_p = 120 V: V_s = 120 × (500/100) = 600 V.
An ideal transformer conserves power: V_p I_p = V_s I_s. Stepping up voltage steps down current by the same factor.
E and B oscillate perpendicular to each other and perpendicular to the direction of propagation.
The relationship between amplitudes is B₀ = E₀ / c.
EM waves travel at c = 1/√(ε₀μ₀) ≈ 3 × 10⁸ m/s in vacuum.
They do not require a medium for propagation. This is what distinguishes them from mechanical waves like sound.
The Poynting vector S = (1/μ₀) E × B points in the direction of energy flow. The statement "the Poynting vector represents the direction of energy flow" is true.
Unpolarised light has electric field oscillations in all directions perpendicular to propagation.
After passing through one ideal polariser, the intensity drops to I₀/2 and the light becomes linearly polarised.
After a second polariser at angle θ to the first, Malus's law applies: I = (I₀/2) cos² θ.
For I₀ = 1000 W/m² and θ = 30°: I = (1000/2) cos²(30°) = 500 × (√3/2)² = 500 × 0.75 = 375 W/m².
ΔU = Q − W (using the convention that W is work done by the system).
If 500 J of heat is added (Q = +500 J) and the system does 200 J of work on its surroundings (W = +200 J): ΔU = 500 − 200 = 300 J.
The internal energy increases by 300 J.
Q = mcΔT, so ΔT = Q/(mc).
When gravitational potential energy is converted entirely into heat: Q = Mgh.
For a 2.0 kg block dropped 100 m, absorbed by 1.0 kg of water: Q = 2.0 × 9.8 × 100 = 1960 J. ΔT = 1960 / (1.0 × 4190) ≈ 0.47 K.
For a 5.0 kg block dropped 100 m, heating 2.5 kg of water: Q = 5.0 × 9.8 × 100 = 4900 J. ΔT = 4900 / (2.5 × 4190) ≈ 0.47 K.
Specific heat capacity is an intrinsic property of the material. It does not depend on the mass of the sample.
Quantity | Formula |
|---|---|
Capacitive reactance | X_C = 1/(ωC) = 1/(2πfC) |
Inductive reactance | X_L = ωL = 2πfL |
Impedance (series RLC) | Z = √(R² + (X_L − X_C)²) |
Resonant frequency | f₀ = 1/(2π√(LC)) |
Phase angle | φ = arctan((X_L − X_C)/R) |
Transformer voltage ratio | V_s/V_p = N_s/N_p |
RMS voltage | V_rms = V_peak / √2 |
EM wave speed | c = 1/√(ε₀μ₀) ≈ 3 × 10⁸ m/s |
E and B amplitude relation | B₀ = E₀ / c |
Poynting vector | S = (1/μ₀) E × B |
Malus's law | I = I₀ cos² θ |
Unpolarised through first polariser | I = I₀ / 2 |
First law of thermodynamics | ΔU = Q − W |
Heat and temperature change | Q = mcΔT |
Transformers are the reason electricity can be transmitted efficiently over long distances: power stations step voltage up to reduce current (and therefore resistive losses in the wires), and local substations step it back down for safe household use. Polarising filters are used in sunglasses to reduce glare from horizontal surfaces, in LCD screens to control pixel brightness, and in photography to cut reflections from glass and water. The first law of thermodynamics governs everything from car engines to refrigerators to your body's metabolism.
Students often think that doubling the frequency of an AC source doubles the capacitive reactance. It halves it, because X_C = 1/(2πfC) is inversely proportional to frequency.
A frequent error is believing that impedance is minimised when X_L = 0 or when R is very large. Impedance is minimised when X_L = X_C (resonance), and at that point Z = R.
Students sometimes confuse the Poynting vector with the electric field direction. The Poynting vector points in the direction of wave propagation (energy flow), which is perpendicular to both E and B.
In the first law, students mix up signs. With the convention ΔU = Q − W: heat added is positive Q, work done by the system is positive W. Forgetting which is which leads to sign errors.
⚠️ Capacitive reactance is inversely proportional to frequency. This is tested directly: "frequency doubles, what happens to X_C?" The answer is it halves.
⚠️ At resonance in a series RLC circuit, X_L = X_C, Z = R, and the phase angle is zero. All three facts are tested in different questions on the same exam.
⚠️ Unpolarised light through one ideal polariser becomes I₀/2, not I₀. Students who forget the factor of 2 at the first filter get the entire polarisation chain wrong.
⚠️ B₀ = E₀/c, not E₀ × c or E₀/c². The relationship is straightforward division by the speed of light.
⚠️ The first law sign convention matters. Know whether your course defines W as work done by the system or on the system, and apply the correct sign.
⚠️ Specific heat is a material property, independent of mass. The exam true/false section tests this directly.
Fill in the blank: Doubling the frequency of an AC source connected to a pure capacitor ______ (doubles/halves) the capacitive reactance.
True or false: At resonance in a series RLC circuit, the impedance equals R.
Fill in the blank: Unpolarised light of intensity I₀ passing through one ideal polariser has intensity ______.
True or false: A magnetic field can do work on a moving charged particle and change its kinetic energy.
Fill in the blank: According to the first law, if Q = 500 J and W = 200 J, then ΔU = ______ J.
Answers: 1. Halves. 2. True. 3. I₀/2. 4. False. 5. 300 J.
Q: An AC source is connected to a pure capacitor. If the frequency is doubled, what happens to X_C?
A: X_C is halved, because X_C = 1/(2πfC) and doubling f doubles the denominator.
Q: In a series RLC AC circuit, when is the impedance Z minimised?
A: When X_L = X_C (resonance). At that point Z = R.
Q: A step-up transformer has 100 primary turns and 500 secondary turns. If V_p = 120 V, what is V_s?
A: V_s = V_p × (N_s/N_p) = 120 × 5 = 600 V.
Q: What is the relationship between B₀ and E₀ in an electromagnetic wave?
A: B₀ = E₀/c.
Q: Unpolarised light (I₀ = 1000 W/m²) passes through two polarisers. The second is at 30° to the first. What is the final intensity?
A: After the first polariser: I₁ = 1000/2 = 500 W/m². After the second: I₂ = 500 × cos²(30°) = 500 × 0.75 = 375 W/m².
Q: A 2.0 kg block is dropped from 100 m and all its PE is absorbed by 1.0 kg of water (c = 4190 J/(kg·K)). What is ΔT?
A: Q = mgh = 2.0 × 9.8 × 100 = 1960 J. ΔT = 1960/(1.0 × 4190) ≈ 0.47 K.
Q: 500 J of heat is added to a system which then does 200 J of work. What is ΔU?
A: ΔU = Q − W = 500 − 200 = 300 J.
Q: Calculate the resonant frequency of a series RLC circuit with R = 40 Ω, L = 80 mH, C = 50 μF.
A: f₀ = 1/(2π√(LC)) = 1/(2π√(0.08 × 50 × 10⁻⁶)) = 1/(2π × 0.002) ≈ 79.6 Hz.
AC circuit analysis is the gateway to signal processing, communications engineering, and power systems. The resonance condition in RLC circuits is the same mathematics that governs tuning forks, bridges under wind loads, and atomic absorption spectra. Maxwell's equations, which predict electromagnetic waves, unify electricity and magnetism into a single theory and laid the groundwork for special relativity. The first law of thermodynamics connects to the second law (entropy), which governs the direction of spontaneous processes and the theoretical efficiency limits of heat engines.
AC circuit analysis, capacitive reactance formula, inductive reactance formula, impedance formula, series RLC circuit, resonant frequency, resonance condition, phase angle, power factor, step-up transformer, step-down transformer, turns ratio, RMS voltage, peak voltage, Vrms, electromagnetic wave propagation, speed of light, Poynting vector, energy flow, wave intensity, polarisation, Malus's law, unpolarised light, linear polarisation, polarising filter, first law of thermodynamics, internal energy, heat, work, sign convention, specific heat capacity, calorimetry, mechanical equivalent of heat, Joule's experiment, temperature change