Resistors and Ohm's law in AC circuits

Resistors and Ohm's law in AC circuits

The voltage v across a resistor is proportional to the current i travelling through it.
Further, this is true at all times: v = Ri. So, if the current in a resistor is

i = I_m . sin (ωt) ,           we write: v = R.i = R.I_m sin (ωt) v = V_m. sin (ωt)            where V_m = R.I_m

So for a resistor, the peak value of voltage is R times the peak value of current. Further, they are in phase: when the current is a maximum, the voltage is also a maximum. (Mathematically, φ = 0.) The first animation shows the voltage and current in a resistor as a function of time.

The rotating lines in the right hand part of the animation are a very simple case of a phasor diagram (named, I suppose, because it is a vector representation of phase). With respect to the x and y axes, radial vectors or phasors representing the current and the voltage across the resistance rotate with angular velocity ω. The lengths of these phasors represent the peak current I_m and voltage V_m. The y components are I_m sin (ωt) = i(t) and voltage V_m sin (ωt)= v(t). You can compare i(t) and v(t) in the animation with the vertical components of the phasors. The animation and phasor diagram here are simple, but they will become more useful when we consider components with different phases and with frequency dependent behaviour.