AC through a resistor

Alternating current (ac) keeps reversing direction. The mains voltage follows a sine wave: v = v_m sin ωt, where v_m is the peak value and ω = 2πν. In India the frequency is ν = 50 Hz: the current changes direction 100 times a second.

We use ac for mains supply because its voltage is easy to change with transformers, which makes long-distance transmission cheap.

In a resistor, the current rises and falls exactly with the voltage: i = (v_m/R) sin ωt. They are in phase: both peak together and both pass zero together.

Over a full cycle, the average current is zero (as much one way as the other). But the heating is not zero, because i²R is always positive. So we use the rms (root mean square) value:

I = i_m / √2 ≈ 0.707 i_m and V = v_m / √2

The rms value is the steady (dc) value that would give the same heating in the same resistor. That is why it is also called the effective value.

Example: a supply with peak 100 V has rms value 100/√2 ≈ 70.7 V.

With rms values, the dc formulas work again: P = I²R = VI = V²/R gives the average power.

Every ac rating is rms. '220 V mains' means V_rms = 220 V; its peak is about 311 V. A 60 W, 220 V bulb draws I = 60/220 ≈ 0.27 A rms. Ammeters and voltmeters for ac read rms values.