Response of Resistor to a Sinusoidal Voltage

Now that we are familiar with the characteristics of the derivative of a sinusoidal function, we can investigate the response of the basic elements R, L, and C to a sinusoidal voltage or current. For power-line frequencies and frequencies up to a few hundred kilohertz, resistance is, for all practical purposes, unaffected by the frequency of the applied sinusoidal voltage or current. For this frequency region, the resistor R of Fig. 1 can be treated as a constant, and Ohm's law can be applied as follows. For $v = V_m \sin wt$, $$i = {V \over R}={V_m \sin wt \over R} = I_m \sin wt$$ where $$\bbox[10px,border:1px solid grey]{I_m = {V_m \over R}} \tag{1}$$ In addition, for a given i, $$v=iR = (I_m \sin wt)R = V_m \sin wt$$ where $$ \bbox[10px,border:1px solid grey]{V_m = ImR} \tag{2}$$ A plot of v and i in Fig. 2 reveals that