If you're interested in electronics, you'll find the following simple "paradox" amusing. It's the usual case of "proving that 2+2=5". The fun is finding where the mistake in the reasoning is.
Consider the following circuit:
![](https://eli.thegreenplace.net/images/2008/12/cap_resistor.png)
Assume that the capacitor is charged to some initial voltage before the switch is closed. At time 0, the switch is closed. What is the current in the circuit as a function of time ?
Let's solve it using the familiar RC circuit methods. We know that
![](https://eli.thegreenplace.net/images/math/5e23343bb687c00a0eb8ce9ef60e95b356568127.gif)
because of Kirchoff's voltage law. We'll differentiate both sides by time:
$\dot{V}_{c}(t) = \dot{V}_{R}(t)$
We know that for a capacitor, the relation between current and voltage is:
![](https://eli.thegreenplace.net/images/math/928acbef4f8f2eeb39d3c51ca68ab3e08279393f.gif)
Substituting it into the equation above and also recalling that
![](https://eli.thegreenplace.net/images/math/6809dfc8324feb51c746bc469c8bd7dbbe3ea32e.gif)
, we get:
![](https://eli.thegreenplace.net/images/math/73fa9296a1e93050c9ba41b7bd8d5ddeaa1d84a6.gif)
But the current through the capacitor and resistor is the same current, so this can be rewritten simply as:
![](https://eli.thegreenplace.net/images/math/eba020de23be652ba084b91a27aa173aade8a360.gif)
This is a simple first order differential equation, the solution of which is:
![](https://eli.thegreenplace.net/images/math/46c7da1f88d1806982454f784d37742fbfa0c332.gif)
For some initial current
![](https://eli.thegreenplace.net/images/math/7dd1d81670e79a2861ab8214c079d2f03ee310a0.gif)
. But wait a second, how can the exponent be positive, won't it grow to infinity with time ? There's obviously a mistake here, somewhere. Can you find it ?
This problem gave me some headache last night, and today I've successfully stumped a few co-workers with it. I'll post a solution in a couple of days.