Obtain the relation for the voltage applied to a series $LCR$ circuit.

(N/A) Consider a series $LCR$ circuit connected to an $AC$ source with electromotive force $\varepsilon = V_m \sin \omega t$.
Let $q$ be the charge on the capacitor and $I$ be the current in the circuit at any time $t$. According to Kirchhoff's voltage law (loop rule),the sum of potential drops across the inductor $(V_L)$,resistor $(V_R)$,and capacitor $(V_C)$ must equal the applied source voltage $V$.
$V = V_L + V_R + V_C$
Substituting the expressions for potential differences:
$V = L \frac{dI}{dt} + IR + \frac{q}{C}$
Where:
$V_L = L \frac{dI}{dt}$ is the potential difference across the inductor.
$V_R = IR$ is the potential difference across the resistor.
$V_C = \frac{q}{C}$ is the potential difference across the capacitor.