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An RC circuit puzzle

December 22nd, 2008 at 10:16 pm

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:

RC circuit

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 $V_{c}(t) = V_{R}(t)$ 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:

$I_{c}(t) = C\frac{dV_{c}(t)}{dt}=\dot{V}_{c}(t)$

Substituting it into the equation above and also recalling that $V_{R}(t)=RI_{R}(t)$ , we get:

$\frac{1}{C}I_{c}(t)=R\dot{I}_{R}(t)$

But the current through the capacitor and resistor is the same current, so this can be rewritten simply as:

$RC\dot{I}(t)-I(t)=0$

This is a simple first order differential equation, the solution of which is:

$I(t)=I_{0}e^{\frac{t}{RC}}$

For some initial current $I_{0}$ . 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.

This entry was posted on Monday, December 22nd, 2008 at 22:16 and is filed under EE / Embedded, Math. You can follow any responses to this entry through the RSS 2.0 feed. Both comments and pings are currently closed.

2 Responses to “An RC circuit puzzle”

Aaron Says:
December 23rd, 2008 at 02:33
Answer, or criticism:

From the exercise you say “But the current through the capacitor and resistor is the same current”

… (posting rest of response at following link as to not spoil the problem: http://codepad.org/LN4IvrHR)
bindoo ZIJA Says:
December 29th, 2008 at 21:17
Add me also to the list of people who lost some hair trying to solve your puzzle. Thank you for providing us the solution too, or you could have easily made my night miserable worrying about how can I possibly miss such a simple puzzle!
(Obviously I always consider myself as a giant in electronics, till some smart guys like you remind me)