RC circuit (DC: charge/discharge)

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The red dots represent the electrons.
The color of the capacitor is yellow (red+green) to represent neutral charge.
When electron enter one side of the capacitor, color for part of the region turn into red.
Another side of the capacitor lack of electron so it color turn into green.
You can charge the capacitor or discharge it.

For the charging cycle: math_failure (math_image_error): V_s=V_R+V_C=I R + \int \frac{I dt}{C} where V_s is the voltage from the power supply.
$0=R \frac{dI}{dt}+ \frac{I}{C}$ , $\frac{dI}{dt}=-\frac{I}{RC}$ , so the solution is $I(t)=I_0 e^{-t/(RC)}$
at $t=0, V_c=0$ so $I(t=0)=I_0=V_s/R$
The result is $V_R(t)=I(t) R =V_s e^{-t/(RC)}$ , V_c(t)=V_s-V_R(t)= V_s (1- e^{-t/(RC)})

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