Difficulty: Intermediate to Advanced | Prerequisites: Capacitance, DC Circuits & Magnetism study notes, trigonometric functions, complex number basics helpful but not essential
This final set of notes covers AC circuit analysis (including the full RLC series circuit, which is a favourite long-answer exam problem), electromagnetic wave propagation, transformers, and the thermodynamics lab material that appeared on this exam. If you can work through the RLC impedance calculation and understand what resonance means physically, you are well-positioned for the final.
AC circuits behave differently from DC because inductors and capacitors create frequency-dependent opposition (reactance) to current flow. At resonance, the inductive and capacitive reactances cancel, impedance is minimised, and current is maximised. Electromagnetic waves carry energy in the direction of the Poynting vector, and transformers exploit Faraday's Law to step voltage up or down.
Impedance (Z)
The total opposition to current in an AC circuit, measured in ohms. It combines resistance and reactance: Z = √[R² + (X_L – X_C)²]. Think of it as: the AC equivalent of resistance, but frequency-dependent.
Inductive reactance (X_L)
The opposition to AC current provided by an inductor: X_L = 2πfL. In simple terms, the higher the frequency, the more an inductor resists current changes.
Capacitive reactance (X_C)
The opposition to AC current provided by a capacitor: X_C = 1/(2πfC). Think of it as: at low frequencies, a capacitor blocks current; at high frequencies, it passes it easily.
Resonance
The condition in an RLC circuit when X_L = X_C. At resonance, impedance is at its minimum (Z = R), current is at its maximum, and the phase angle is zero.
RMS voltage / current
The root-mean-square value of an AC quantity: V_rms = V_max / √2. In simple terms, RMS values let you calculate average power as though the AC signal were a steady DC signal.
Lenz's Law
The direction of an induced current is such that it produces a magnetic field opposing the change in flux that caused it. Think of it as: nature resists change in magnetic flux.
Poynting vector (S)
The vector S = (1/μ₀)(E x B) that gives the direction and magnitude of energy flow in an electromagnetic wave. In simple terms, it points in the direction the wave is carrying energy.
Transformer
A device that uses electromagnetic induction to change AC voltage levels. A step-up transformer increases voltage and decreases current; a step-down does the reverse.
Specific heat (c)
The amount of heat energy required to raise the temperature of 1 kg of a substance by 1°C. Measured in J/(kg·°C). Think of it as: how much energy a material "soaks up" per degree of warming.
An induced current flows in a direction that creates a magnetic field opposing the change in flux that caused the induction.
If the external flux through a loop is increasing, the induced current flows in a direction to create flux opposing the increase (i.e. in the opposite direction to the external field).
This is a direct consequence of conservation of energy.
In a series RLC circuit driven by an AC source, the current is the same through all components.
The voltage across each component is different in both magnitude and phase.
The inductor voltage leads the current by 90°. The capacitor voltage lags the current by 90°. The resistor voltage is in phase with the current.
Total impedance: Z = √[R² + (X_L – X_C)²].
The RMS current: I_rms = V_rms / Z.
Average power dissipated: P_avg = I_rms² R (only the resistor dissipates power).
Resonance occurs when X_L = X_C, i.e. 2πfL = 1/(2πfC).
At resonance, Z = R (impedance is at its minimum), current is at its maximum, and the phase angle between voltage and current is zero.
The resonant frequency is f₀ = 1/(2π√(LC)).
In a purely capacitive AC circuit, the current leads the voltage by 90°. The capacitor charges before the voltage across it builds up.
In a purely inductive AC circuit, the current lags the voltage by 90°.
Mnemonic: "ELI the ICE man" (voltage E leads current I in an inductor L; current I leads voltage E in a capacitor C).
A step-up transformer increases voltage and decreases current. It has more turns on the secondary coil than the primary.
A step-down transformer decreases voltage and increases current.
The voltage ratio equals the turns ratio: V_s / V_p = N_s / N_p.
Power is (ideally) conserved: V_p I_p = V_s I_s.
V_rms = V_max / √2 ≈ 0.707 V_max.
For V_max = 170 V: V_rms = 170 / √2 ≈ 120 V.
RMS values are what you use in power calculations: P = V_rms I_rms cosφ.
The Poynting vector S = (1/μ₀)(E x B) gives the direction and rate of energy flow per unit area in an electromagnetic wave.
The direction of S is the direction of wave propagation.
In an EM wave, E and B are perpendicular to each other and both perpendicular to the direction of propagation.
In the falling-block experiment, a block falls and its gravitational potential energy is converted to thermal energy in water.
Increasing the block's mass (e.g. from 2.0 kg to 5.0 kg) increases the gravitational potential energy available (mgh increases), which means more heat is delivered to the water (ΔQ increases).
Specific heat is calculated from Q = mcΔT, so c = Q/(mΔT).
For a 5.0 kg water sample absorbing 1960 J with a temperature increase of 0.094°C: c = 1960 / (5.0 × 0.094) = 1960 / 0.47 ≈ 4170 J/(kg·°C). This is close to the accepted value of 4186 J/(kg·°C).
Given: R = 50 Ω, L = 0.2 H, C = 10 μF, V_rms = 120 V, f = 60 Hz.
Step 1, Reactances:
X_L = 2πfL = 2π(60)(0.2) = 75.4 Ω
X_C = 1/(2πfC) = 1/[2π(60)(10 × 10⁻⁶)] = 265.3 Ω
Step 2, Impedance:
Z = √[R² + (X_L – X_C)²] = √[50² + (75.4 – 265.3)²] = √[2500 + (–189.9)²] = √[2500 + 36062] = √38562 ≈ 196.4 Ω
Step 3, RMS current:
I_rms = V_rms / Z = 120 / 196.4 ≈ 0.611 A
Step 4, Average power:
P_avg = I_rms² R = (0.611)² × 50 ≈ 18.7 W
Note: Only the resistor dissipates power. The inductor and capacitor store and release energy but do not dissipate it on average.
X_L = 2\pi f L \qquad X_C = \frac{1}{2\pi f C}Inductive and capacitive reactance. Both depend on frequency.
Z = \sqrt{R^2 + (X_L - X_C)^2}Impedance of a series RLC circuit.
f_0 = \frac{1}{2\pi\sqrt{LC}}Resonant frequency, where X_L = X_C and impedance is minimised.
V_{\text{rms}} = \frac{V_{\text{max}}}{\sqrt{2}}RMS voltage. Similarly, I_rms = I_max / √2.
P_{\text{avg}} = I_{\text{rms}}^2 R = \frac{V_{\text{rms}}^2 R}{Z^2}Average power dissipated in an RLC circuit. Only the resistor dissipates power.
\vec{S} = \frac{1}{\mu_0} \vec{E} \times \vec{B}Poynting vector: direction and magnitude of energy flow in an EM wave.
\frac{V_s}{V_p} = \frac{N_s}{N_p}Transformer voltage ratio equals the turns ratio.
c = \frac{Q}{m \Delta T}Specific heat. Units: J/(kg·°C).
Students often think that at resonance, the current is zero. The opposite is true: at resonance, impedance is minimised and current is maximised.
Students confuse which component causes leading vs. lagging. In a capacitor, current leads voltage (ICE). In an inductor, voltage leads current (ELI).
Students sometimes assume transformers can increase both voltage and current. They cannot; power is conserved (ideally), so increasing voltage means decreasing current.
Students calculate RMS as V_max / 2 instead of V_max / √2. The factor is √2, not 2.
The RLC resonance principle is how radios tune to specific stations: the circuit's resonant frequency is adjusted to match the broadcast frequency, maximising the signal from that station. Transformers are essential in power distribution: electricity is stepped up to high voltage for long-distance transmission (reducing I²R losses) and stepped down for household use. The Poynting vector describes the energy carried by sunlight, radio waves, and Wi-Fi signals.
⚠️ The full RLC impedance calculation (reactances, impedance, current, power) is a very common 10-mark long-answer question. Practise it until you can do it without hesitation.
⚠️ Resonance condition: X_L = X_C. Know what happens to impedance, current, and phase at resonance.
⚠️ Lenz's Law: the induced current opposes the change in flux. "Opposes the change" is the key phrase, not "opposes the flux."
⚠️ V_rms = V_max / √2. For V_max = 170 V, V_rms ≈ 120 V. This is a quick numerical question.
⚠️ The Poynting vector S = (1/μ₀)(E x B) gives the direction of energy propagation. Do not confuse it with E or B alone.
True or False: At resonance in a series RLC circuit, the impedance is at its maximum. (False. Impedance is at its minimum, equal to R.)
Fill in the blank: According to Lenz's Law, the induced current produces a magnetic field that __________ the change in flux. (opposes)
True or False: A step-up transformer increases both voltage and current. (False. It increases voltage but decreases current.)
Fill in the blank: V_rms = V_max / __________. (√2)
True or False: In a capacitor, the voltage leads the current by 90°. (False. In a capacitor, the current leads the voltage by 90°.)
Q: According to Lenz's Law, the direction of an induced current is such that it does what?
A: It produces a magnetic field that opposes the change in magnetic flux through the circuit. If flux is increasing, the induced field opposes the increase; if flux is decreasing, the induced field opposes the decrease.
Q: In an AC RLC series circuit, at what condition does resonance occur?
A: When the inductive reactance X_L equals the capacitive reactance X_C. At this point, Z = R, the current is maximised, and the voltage and current are in phase.
Q: Which component in an AC circuit causes the current to lead the voltage by 90°?
A: A capacitor. (Remember: ICE, current I leads voltage E in a capacitor C.)
Q: What is the RMS voltage of an AC source with peak voltage 170 V?
A: V_rms = 170 / √2 = 170 / 1.414 ≈ 120 V.
Q: A step-up transformer is designed to do what?
A: Increase the voltage and decrease the current. The turns ratio N_s/N_p > 1.
Q: Which vector represents the direction of energy propagation in an electromagnetic wave?
A: The Poynting vector, S = (1/μ₀)(E x B). It points in the direction the wave carries energy.
Q: In the thermodynamics lab, if a 5.0 kg block of water absorbed 1960 J of heat and its temperature increased by 0.094°C, what is the experimental specific heat?
A: c = Q/(mΔT) = 1960 / (5.0 × 0.094) = 1960 / 0.47 ≈ 4170 J/(kg·°C).
Q: For an RLC circuit with R = 50 Ω, L = 0.2 H, C = 10 μF, V_rms = 120 V, f = 60 Hz, calculate the average power dissipated.
A: X_L = 2π(60)(0.2) = 75.4 Ω. X_C = 1/[2π(60)(10⁻⁵)] = 265.3 Ω. Z = √[50² + (75.4 – 265.3)²] = √[2500 + 36062] ≈ 196.4 Ω. I_rms = 120/196.4 ≈ 0.611 A. P_avg = (0.611)² × 50 ≈ 18.7 W.
AC circuit analysis is where the DC concepts (Ohm's law, Kirchhoff's rules) meet the time-varying world of electromagnetic induction. The impedance framework extends naturally to filters and signal processing in electronics courses. Lenz's Law is the physical principle behind eddy current braking in trains and the operation of induction cooktops. The Poynting vector connects to radiation pressure and antenna theory, both of which appear in more advanced E&M courses.
AC circuit, RLC circuit, series RLC, impedance, inductive reactance, capacitive reactance, resonance, resonant frequency, RMS voltage, RMS current, root mean square, phase angle, power factor, average power, Lenz's Law, Faraday's Law, electromagnetic induction, Poynting vector, electromagnetic wave, EM wave, energy propagation, transformer, step-up transformer, step-down transformer, turns ratio, specific heat, calorimetry, thermodynamics lab, ELI the ICE man, PHYS 212, University Physics E&M