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## Homework Statement

You have a capacitor, of capacitance C farads, with charge Q coulombs. It is connected in series with a resistor of resistance R ohms. Derive an expression for the potential difference over the capacitor at any time t.

**2. Homework Equations and theorems**

[tex]I_{c} = C\frac{dV_{c}}{dt}[/tex]

[tex]V = IR[/tex]

*Kirchhoff's voltage law

## The Attempt at a Solution

Using KVL:

[tex]V_{c} - I_{c} R = 0[/tex]

[tex]V_{c} = RC\frac{dV_{c}}{dt}[/tex]

[tex]\frac{1}{RC} dt = \frac{1}{V_{c}} dV_{c}[/tex]

then:

[tex] V_{c} (t) = V_{initial} e^{\frac{t}{RC}}[/tex]

where

[tex]V_{initial} = \frac{Q}{C}[/tex]

**3. My concern**

as t approaches infinity the potential difference over the capacitor also approaches infinity. This is definitely not right - the capacitor is discharging. Every textbook/website I look at comes up with the equation:

[tex] V_{c} (t) = V_{initial} e^{\frac{-t}{RC}}[/tex]

For the life of me I cannot figure out what I'm doing wrong. I know what's supposedly wrong with my solution, but I cannot see the mathematical proof of the minus sign on the power of e.

I'd greatly appreciate any help or insight anyone could give.