AC voltage applied to a resistor

What happens when an ac voltage is applied to a resistor?

In plain words: a resistor does not care which way the current flows; it heats up either way. The current swings back and forth in step with the voltage, and the heating comes in pulses twice every cycle. The rms current is the steady current that would heat the resistor just as much.

Model used throughout: sinusoidal ac across an ideal, constant resistance, with negligible inductive and capacitive effects. AC need not always be sinusoidal. Bulb and iron examples use the resistance at their stated operating temperature.

Current in phase with the voltage

• A resistor is joined to an ac source, v = v_m sin ωt (NCERT Eq. (7.1), page 178, Figure 7.1).
• Kirchhoff’s loop rule gives v_m sin ωt = iR, so (NCERT Eqs. (7.2)–(7.3), page 178):

• This is Ohm’s law, which for resistors works equally well for ac and dc voltages (page 178).
• The voltage and current reach zero, their maxima and their minima at the same times: they are in phase (NCERT page 178, Figure 7.2).
• The current takes positive and negative values in each cycle, so its average over a cycle is zero (page 178). That does not mean no energy is used.

Average power over a cycle

• Joule heating depends on i², which is never negative. The instantaneous power is (NCERT Eq. (7.4), page 179):

• Using sin² ωt = ½(1 − cos 2ωt), and the fact that cos 2ωt averages to zero over a cycle, the average of sin² ωt is ½ (NCERT page 179). So the average power is (NCERT Eq. [7.5(c)], page 179):

RMS current and voltage

• To write ac power in the dc form P = I²R, NCERT defines the root mean square (rms) or effective current (NCERT Eq. (7.6), page 180):

• Then the average power is P = ½i_m²R = I²R (NCERT Eq. (7.7)). Likewise the rms voltage is V = v_m/√2 = 0.707 v_m (NCERT Eq. (7.8)), and V = IR (NCERT Eq. (7.9), page 180).
• So in rms values, power and Ohm’s law look just as they do for dc: P = I²R = V²/R = IV (page 180), the same forms as for electrical power in dc circuits.
• The rms current is the dc current that would produce the same average power loss as the alternating current (page 180).
• Ac quantities are customarily given as rms values: household mains of 220 V is an rms value, with a peak of √2 × 220 ≈ 311 V (page 180).
• The name means what it says: square the current, take the mean over a cycle, then the square root. The mean of i_m² sin² ωt is i_m²/2, whose square root is i_m/√2 (NCERT Eq. (7.6), page 180).
• NCERT slip: NCERT’s boxes on pages 178 and 179 spell the inventor’s name “Nicola Tesla”; it is Nikola Tesla (Nikola Tesla Museum, biography).

Solved examples

Common mistakes to avoid

Quick revision summary

• With v = v_m sin ωt across a resistor, i = i_m sin ωt with i_m = v_m/R: current and voltage are in phase.
• The current averages zero over a cycle, but the power does not: p = i_m²R sin² ωt, with average ½i_m²R.
• RMS values: I = i_m/√2 = 0.707 i_m and V = v_m/√2; then P = I²R = V²/R = IV and V = IR, as for dc.
• Ac ratings are rms values: 220 V mains has a peak of about 311 V.

Questions students ask

Why is the rms value of alternating current used instead of its average value?

The rms value of an alternating current is used because it measures what the current does in a resistor: a sinusoidal current averages zero over a full cycle yet heats the resistor, and the rms value is the steady current that heats it at the same average rate, so P = I^²R and V = IR keep their dc forms (NCERT pages 178–180). A half-cycle average suits effects of charge carried one way, such as the charge a full-wave rectified current gives a battery, and for a sine wave the rms value is 1.11 times it (beyond the NCERT text: charge = average current × time; US Department of Energy, DOE Fundamentals Handbook: Electrical Science, 1992, Module 7).

What is the rms value of an alternating current over half a cycle?

The rms value of a sinusoidal current over any half cycle, or over a quarter cycle from a zero to a peak, is the same as over a full cycle, i_m/√2 ≈ 0.707 i_m, because the mean of sin^² ωt over each of these intervals is ½ (reasoned from NCERT page 179). Over other intervals it differs: from t = 0 to T/8 the mean of sin^² ωt is ½ − 1/π ≈ 0.18, so the rms value over that eighth of a cycle is only about 0.43 i_m (beyond the NCERT text, reasoned from the definition of rms). The rms current that NCERT defines is the value over whole cycles; ordinary ac ratings use this convention, and a meter must be suitable for the waveform being measured (NCERT page 180).

What is the rms value of a square wave or a triangular wave?

A square wave that switches between +V₀ and −V₀ has an rms value equal to its peak, V₀, since its square is V₀² at every instant, while a triangular or sawtooth wave ramping uniformly between −V₀ and +V₀ (or between zero and V₀) has an rms value of V₀/√3 ≈ 0.577 V₀ (beyond the NCERT text, reasoned from the definition of rms, NCERT page 180). So NCERT’s 1/√2 cannot be assumed for other shapes: with the same 24 V peak, a square wave heats a resistor like 24 V dc, a sine wave like 17.0 V and a triangular wave like 13.9 V. A square wave at +V₀ for half of each cycle and zero for the rest gives V₀/√2 too.

What is the rms value of a current such as i = 3 + 4 sin(ωt + π/3) A?

A current made of a steady part I₀ and a sinusoidal part of peak i_m has the rms value √(I₀² + i_m²/2), because the mean of its square is I₀² + i_m²/2: the cross term 2I₀i_m sin(ωt + φ) averages to zero over a cycle (beyond the NCERT text, reasoned from NCERT pages 179–180). For i = 3 + 4 sin(ωt + π/3) A this gives √(9 + 8) = √17 ≈ 4.1 A, not 3 + 4 = 7 A or 3 + 4/√2 ≈ 5.8 A, and the phase π/3 makes no difference. Such a current heats a resistor as a steady 4.1 A would.

Are peak and rms values of alternating current in the JEE Main and NEET syllabus?

Yes, peak and rms values of alternating current are named in both syllabuses: Unit 14 of the JEE Main syllabus lists “Alternating currents, peak and RMS value of alternating current/voltage”, and Unit 14 of the NEET (UG) 2026 syllabus has the same words. NCERT covers them with the resistor in Section 7.2 of Class 12 Physics Part I (pages 178–180), where the average power ½i_m²R leads to the rms current I = i_m/√2. Neither syllabus names the average value over a half cycle, which NCERT does not define either: its only average current is the one over a full cycle, which is zero (NCERT page 178).

Practice questions

Try each question first, then open the answer to check your working.

1. Q1. What is the phase difference between the current and the voltage in a purely resistive ac circuit?

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   Zero: they are in phase (NCERT page 178).

2. Q2. A 110 V rms supply is connected to a 55 Ω resistor. Find the rms current and the peak current.

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   I = V/R = 110/55 = 2.0 A rms, and i_m = √2 × 2.0 ≈ 2.8 A (NCERT Eqs. (7.6) and (7.9), page 180).

3. Q3. Why is the rms value of an ac current called its effective value?

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   It is the dc current that would produce the same average power loss in a resistor as the alternating current (NCERT page 180).