An ideal resistor obeys Ohm's law at every instant:

$$v(t)=Ri(t).$$

If the voltage is sinusoidal, the current is sinusoidal at the same frequency.

Voltage and current are in phase

Suppose

$$v(t)=V_m\cos(\omega t+\phi).$$

Then

$$i(t)=\frac{V_m}{R}\cos(\omega t+\phi).$$

Voltage and current therefore reach maxima, minima and zero crossings together: their phase difference is zero.

The peak amplitudes obey

$$\boxed{V_m=RI_m}.$$

RMS relation

The same resistance relates RMS values:

$$\boxed{V_{\mathrm{rms}}=RI_{\mathrm{rms}}}.$$

Example

A resistor of

$$R=100,\Omega$$

is connected to a sinusoidal source with

$$V_{\mathrm{rms}}=12.0,\mathrm V.$$

Then

$$I_{\mathrm{rms}}=\frac{12.0}{100}=0.120,\mathrm A.$$

The current waveform is in phase with the voltage waveform.

Power

Instantaneous resistor power is

$$p(t)=v(t)i(t)=Ri^2(t),$$

so it is never negative under the passive sign convention. The resistor continuously converts electrical energy into thermal energy.

For sinusoidal steady state,

$$\boxed{P_{\mathrm{avg}}=V_{\mathrm{rms}}I_{\mathrm{rms}} =I_{\mathrm{rms}}^2R}.$$

Unlike ideal capacitors and inductors, an ideal resistor does not store energy and return it later in the cycle. Its AC behavior is therefore the simplest case: voltage and current remain proportional and in phase at every instant.