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Alternating current: Class 12 Physics
Alternating voltage and current, rms and peak values, reactance of capacitors and inductors, series RLC circuits, impedance and phase, resonance, power factor, and electromagnetic waves with their uses.
Key points
Alternating current (AC) changes its direction and size regularly with time. A sinusoidal supply is written V = V₀ sin ωt and I = I₀ sin ωt, where V₀ and I₀ are the peak values and ω = 2πf. The frequency f is the number of cycles per second in hertz (Hz), and the time period is T = 1/f. In Pakistan the mains supply is 220 V at 50 Hz.
The root mean square (rms) value of AC is the value of the steady DC that would give the same heating in a resistor. For a sine wave Irms = I₀/√2 = 0.707 I₀ and Vrms = V₀/√2. The mean value over a full cycle is zero, so ammeters for AC are marked in rms. The mains 220 V is an rms value, so its peak is about 311 V.
In a purely resistive circuit the current and voltage are in phase. The resistor opposes both AC and DC in the same way, and the power is P = Vrms Irms.
A capacitor in an AC circuit passes current continuously as it charges and discharges. The current leads the voltage by 90°. Its opposition is the capacitive reactance XC = 1/(ωC) = 1/(2πfC), in ohm. It falls as frequency rises. For DC (f = 0) the reactance is infinite, so a capacitor blocks DC.
An inductor opposes the change in current because of the induced back EMF. The current lags behind the voltage by 90°. The inductive reactance is XL = ωL = 2πfL, in ohm. It rises with frequency. For DC it is zero, so an inductor passes DC easily but resists high frequency AC. This is why it is used as a choke.
Series RLC circuit:
- The same current flows through R, L and C. The voltage across R is in phase with the current, across L it leads by 90°, and across C it lags by 90°.
- The impedance is Z = √(R² + (XL − XC)²), in ohm. The current is I = V/Z.
- The phase angle φ between voltage and current is given by tanφ = (XL − XC)/R. If XL is greater than XC, the circuit is inductive and the current lags. If XC is greater, the circuit is capacitive and the current leads.
Resonance occurs in the series RLC circuit when XL = XC. Then Z = R is at its minimum and the current is at its maximum, in phase with the voltage. The resonant frequency is f₀ = 1/(2π√(LC)). A graph of current against frequency has a peak at f₀. The sharper the peak, the smaller the resistance, and the better the circuit selects one frequency. This is the tuning of a radio.
The average power in an AC circuit is P = Vrms Irms cosφ. The quantity cosφ = R/Z is the power factor. For a pure inductor or capacitor φ = 90°, the power factor is zero, and no power is used, even if current flows. This is called a wattless current. A low power factor wastes energy in supply lines, so industries try to bring it close to 1.
A rectifier changes AC to DC. A single diode gives half wave rectification and four diodes in a bridge give full wave rectification. A capacitor across the output smooths the ripple.
Electromagnetic waves are transverse waves of oscillating electric and magnetic fields, at right angles to each other and to the direction of travel. Maxwell predicted them and Hertz produced them. In vacuum they all travel at c = 3.0 × 10⁸ m/s. They need no medium and carry energy and momentum.
The electromagnetic spectrum, from long to short wavelength:
- Radio waves: broadcasting and communication.
- Microwaves: radar, mobile phones and cooking.
- Infrared: heating, remote controls and thermal imaging.
- Visible light: seeing, from red (longest) to violet (shortest).
- Ultraviolet: sterilising and detecting forged notes.
- X-rays: medical imaging and checking metal defects.
- Gamma rays: cancer treatment and sterilising equipment.
Study tip: Memorise the phase relationships and the three reactance formulas. In RLC numericals first find XL and XC, then Z, then current and phase angle, in that order. Draw the spectrum as an ordered list from radio waves to gamma rays, since questions often ask for the order.
Definitions and flashcards
Say the meaning first, then open the card to check.
Peak value
The maximum value of an alternating voltage or current.
RMS value
The steady DC value that produces the same heating as the AC.
Reactance
The opposition of a capacitor or inductor to AC, in ohm.
Impedance
The total opposition of a circuit to AC, combining R and reactance.
Phase angle
The angle by which the current lags or leads the voltage.
Resonance
The state in a series RLC circuit when XL = XC and the current is maximum.
Power factor
The ratio cosφ = R/Z, the fraction of apparent power that is used.
Choke
An inductor used to reduce AC without much loss of power.
Electromagnetic wave
A transverse wave of coupled electric and magnetic fields travelling at speed c in vacuum.
Formula sheet
| Name | Formula | What it means |
|---|---|---|
| RMS values | Irms = I₀ / √2, Vrms = V₀ / √2 | For sinusoidal AC. |
| Capacitive reactance | XC = 1 / (2πfC) | Falls as f rises. |
| Inductive reactance | XL = 2πfL | Rises as f rises. |
| Impedance | Z = √(R² + (XL − XC)²) | Series RLC circuit. |
| Phase angle | tanφ = (XL − XC) / R | Gives lead or lag. |
| Resonant frequency | f₀ = 1 / (2π√(LC)) | At this frequency XL = XC. |
| Average power | P = Vrms Irms cosφ | cosφ is the power factor. |
| Wave speed | c = fλ | For electromagnetic waves, c = 3.0 × 10⁸ m/s. |
Short questions with model answers
These are the kind of questions asked for 2, 3 and 5 marks. Try each one before opening the answer.
What is the rms value of an alternating current? (2 marks)
It is the value of the steady direct current that produces heat at the same rate in a given resistor as the alternating current does. For a sine wave it equals the peak value divided by √2.
Why does a capacitor block DC but pass AC? (2 marks)
A capacitor has an insulating gap, so a steady DC cannot flow after it is charged. Its reactance 1/(2πfC) is infinite when f is zero. With AC the capacitor charges and discharges repeatedly, so the current flows in the circuit.
Define impedance and write its formula for a series RLC circuit. (2 marks)
Impedance is the total opposition offered to AC by a circuit, the ratio of rms voltage to rms current. Z = √(R² + (XL − XC)²).
What is resonance in a series RLC circuit? Give the resonant frequency. (3 marks)
When the supply frequency makes XL equal to XC, the two voltages cancel, the impedance falls to its minimum value R and the current becomes maximum and in phase with the voltage. The resonant frequency is f₀ = 1/(2π√(LC)).
Why is the power consumed by a pure inductor zero? (3 marks)
In a pure inductor the current lags the voltage by 90°, so cosφ = 0. The average power P = VI cosφ is zero. The inductor takes energy from the supply for a quarter cycle and returns it in the next quarter, so there is no net loss. The current is called wattless.
State four properties of electromagnetic waves and give one use of microwaves and one of X-rays. (5 marks)
Electromagnetic waves are transverse. They consist of electric and magnetic fields at right angles to each other and to the direction of travel. They need no material medium. They all travel at 3.0 × 10⁸ m/s in vacuum. They carry energy and are not deflected by electric or magnetic fields. Microwaves are used in radar and mobile communication. X-rays are used to take images of bones and to find faults in metal.
Numericals with worked steps
The peak voltage of a supply is 311 V. Find the rms voltage.
- Vrms = V₀/√2.
- √2 = 1.414.
- 311 / 1.414 = 219.9.
Answer: Vrms ≈ 220 V
A 2.0 H inductor is connected to a 50 Hz supply. Find its reactance (use π = 3.14).
- XL = 2πfL.
- 2 × 3.14 × 50 = 314.
- 314 × 2.0 = 628.
Answer: XL ≈ 628 Ω
A series circuit has R = 30 Ω, XL = 80 Ω and XC = 40 Ω, with a 100 V rms supply. Find Z, the current and the power factor.
- XL − XC = 80 − 40 = 40 Ω.
- Z = √(30² + 40²) = √(900 + 1600) = √2500 = 50 Ω.
- I = V/Z = 100 / 50 = 2 A.
- cosφ = R/Z = 30 / 50 = 0.6.
Answer: Z = 50 Ω, I = 2 A, power factor = 0.6
Find the resonant frequency of a circuit with L = 0.1 H and C = 10 μF.
- C = 10 × 10⁻⁶ F = 1.0 × 10⁻⁵ F.
- LC = 0.1 × 1.0 × 10⁻⁵ = 1.0 × 10⁻⁶.
- √(LC) = 1.0 × 10⁻³.
- f₀ = 1 / (2π × 10⁻³) = 1 / (6.283 × 10⁻³) = 159.2 Hz.
Answer: f₀ ≈ 159 Hz
Common mistakes
- Using peak values in power calculations. Use rms values unless the question says otherwise.
- Confusing which one leads: in a capacitor the current leads the voltage, and in an inductor it lags.
- Adding R, XL and XC directly. They add as vectors through Z = √(R² + (XL − XC)²).
- Forgetting that reactance depends on frequency, while resistance does not.
- Writing that power is used in a pure inductor or capacitor. Only the resistance uses power.