Explain $AC$ voltage applied to a resistor and explain it with a necessary graph.

(N/A) Consider a resistor of resistance $R$ connected to an $AC$ voltage source. The source produces a sinusoidally varying potential difference across its terminals,given by:
$V = V_{m} \sin \omega t$ .....$(1)$
where $V_{m}$ is the amplitude of the oscillating potential difference (maximum voltage) and $\omega$ is its angular frequency.
Applying Kirchhoff's loop rule to the circuit:
$V - IR = 0$
$\therefore IR = V_{m} \sin \omega t$
$\therefore I = \frac{V_{m}}{R} \sin \omega t$
Since the current amplitude $I_{m} = \frac{V_{m}}{R}$ (Ohm's law),we can write:
$I = I_{m} \sin \omega t$ .....$(2)$
From equations $(1)$ and $(2)$,we see that the voltage and current are in phase,meaning they reach their maximum and minimum values at the same time. The graph below shows the variation of voltage $v$ and current $i$ as a function of $\omega t$.