AC Voltage 🔌 Applied to a Resistor

Connect a resistor R to a sinusoidal (AC) supply. The voltage produced by the source is

$$v = v_m \sin \omega t$$

where vm is the peak (maximum) voltage and \( \omega \) is the angular frequency. :contentReference[oaicite:0]{index=0}

Current through the Resistor

Because the resistor obeys Ohm’s law at every instant, the current is

$$i = i_m \sin \omega t$$

with the peak current given by

$$i_m = \dfrac{v_m}{R}$$

Notice that voltage and current reach their maxima, minima, and zeros together—so they are in phase. ⚡ This means the waveforms overlap perfectly in time. :contentReference[oaicite:1]{index=1}

Power and Heating 😊

  • Instantaneous power converted to heat is $$p = i^2 R = i_m^2 R \sin^2 \omega t$$
  • Even though the average current over one cycle is zero, the resistor still gets warm because \(i^2\) is always positive.
  • Average (mean) power over a cycle becomes $$\overline{p} = \frac{1}{2}\, i_m^2 R$$

Half the peak-power value is steadily dissipated as heat. :contentReference[oaicite:2]{index=2}

Effective (rms) Values 🌟

To treat AC like steady DC, we define root-mean-square (rms) or effective quantities:

  • Current: $$I = I_{\text{rms}} = \frac{i_m}{\sqrt2} = 0.707\, i_m$$
  • Voltage: $$V = V_{\text{rms}} = \frac{v_m}{\sqrt2} = 0.707\, v_m$$
  • Average power in a resistor: $$P = I^2 R = I V = \frac{V^2}{R}$$
  • Relation between rms values: $$V = I R$$  — identical in form to DC Ohm’s law.

Everyday mains rated “220 V” is actually an rms value; the peak swings up to \(v_m \approx 311\text{ V}\). ⚡ :contentReference[oaicite:3]{index=3}

Worked Example 📝

Data: A light bulb is marked “100 W, 220 V”.

  1. Resistance $$R = \frac{V^2}{P} = \frac{(220\text{ V})^2}{100\text{ W}} \approx 484\,\Omega$$
  2. Peak supply voltage $$v_m = \sqrt2\, V \approx 311\text{ V}$$
  3. rms current $$I = \frac{P}{V} = \frac{100\text{ W}}{220\text{ V}} \approx 0.454\text{ A}$$

This matches the idea that rms current delivers the same heating power as a DC current of the same value. :contentReference[oaicite:4]{index=4}

High-yield NEET Ideas 🚀

  1. For a pure resistor, voltage and current are exactly in phase.
  2. rms values: \(I = i_m/\sqrt2\) and \(V = v_m/\sqrt2\); memorize the 0.707 factor.
  3. Average power in AC: \(P = I^2 R = V I\) (same form as DC).
  4. Peak–rms link: household 220 V rms corresponds to ≈ 311 V peak.
  5. Ohm’s law applies instantaneously in AC circuits: \(i_m = v_m/R\).