Kirchhoff's Voltage Law does not hold for a free source RC circuit?

Question

Say I have a simple free source RC as depicted in the image below. The capacitor initially has a charge of Q on its positive plate and at time t=0 the switch is close. I use the mesh current as shown in the diagram and apply KVL to that loop.

But then the result I get does not have the minus sign infront of the t in the exponent indicating that the capicator is charging up not discharging. Is this because KVL does not apply here? I have seen some other unconvincing answers to this but none of them apply KVL and I ideally want to understand it in terms of KVL.
Any help on this would be most appreciated.

Yejus · Accepted Answer

K.V.L. does apply here. Your mistake lies in incorrectly substituting the current in the resistor for the time-rate-of-change of charge in the positive-plate of the capacitor. The $I$ in the expression is actually $I(t)$ and it relates to the charge on the positive plate of the capacitor. Notice that as the charge flows around the circuit in the form of current, the charge on the positive plate decreases, so $I(t) = -dQ/dt$. Replace that in your equation and get the right answer.

Correct answer by Yejus on June 5, 2021

Kirchhoff's Voltage Law does not hold for a free source RC circuit?

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