- #1
Ahmedbasil
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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.