To solve this problem, we need to determine the displacement current $I_d$ between the plates of a capacitor connected to an AC source.$\\$Given:$\\$• Capacitance $C \\$• Voltage $V = V_0 \sin \omega t \\$The displacement current $I_d$ is related to the rate of change of electric flux, which is equivalent to the current $I$ through the capacitor.$\\$The current $I$ through a capacitor is given by:$\\ I = C \frac{dV}{dt} \\$Substitute the expression for $V: \\ I = C \frac{d}{dt}(V_0 \sin \omega t) \\$Differentiate $V_0 \sin \omega t$ with respect to $t: \\ \frac{d}{dt}(V_0 \sin \omega t) = V_0 \omega \cos \omega t \\$Substitute back into the expression for $I: \\ I = C (V_0 \omega \cos \omega t) \\$Simplify:$\\ I = V_0 \omega C \cos \omega t \\$Thus, the displacement current $I_d$ is given by:$\\ I_d = V_0 \omega C \cos \omega t \\$Therefore, the correct option is Option 1.