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I am trying to derive the equation for the charge in an RC circuit with respect to time.

I have drawn the circuit as follows:

schematic

simulate this circuit – Schematic created using CircuitLab

Using a KVL, I determine the following:

\$iR - \frac{q}{C} = 0\$

\$\frac{dq}{dt} * R = \frac{q}{C}\$

Solve for q and t:

\$q = q(0)\exp(\frac{t}{RC})\$

This indicates exponential growth and not decay, which doesn't make sense. However, it is evident to me that the first line must potentially be this instead:

\$iR + \frac{q}{C} = 0\$

...which will solve my problems. But I do not understand why. Could someone help explain to me please?

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I think I may have answered it myself.

The issue relates to the fact that dq / dt refers to the current through the resistor (a positive quantity) and therefore dq refers to the differential charge related to the resistor, whereas I am "mixing" my charges when also referring to charge on the capacitor in the same equation (q and dq to not refer to the same charges in the same system).

If I want the quantity dq / dt to refer to the current through the capacitor, which is a negative quantity when it is discharging due to the passive sign convention, (so that I do not "mix" my q's) then I need to apply a negative sign in front of it, since dq / dt for the capacitor itself is also negative and the negative's "cancel" to yield the correct KVL equation

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  • \$\begingroup\$ Suppose a counter-clockwise current (one opposite anticipated): KVL would suggest \$0\:\text{V} - I\,R -\frac1{C}\int I\,\text{d}t=0\:\text{V}\$. Taking the derivative, \$-R\frac{\text{d}\,I}{\text{d}t}-\frac{I}{C}=0\$, re-arrangement yields \$-\frac{\text{d}\,I}{I}=\frac{\text{d}\,t}{R\,C}\$. Or, \$I=I_0\,e^{^\left[\frac{-t}{R\,C}\right]}\$, which is a declining current (negative, since \$I_0\$ will be negative to show it moves in the direction opposite the assumption above.) To arrive at a charge equation, solve \$q=\int \text{d}\,q=I_0\,\int e^{^\left[\frac{-t}{R\,C}\right]}\:\text{d}t\$. \$\endgroup\$ – jonk Oct 12 '20 at 15:37

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