Circuits: Of Special Interest to Physics C Students

Practice problems for these concepts can be found at:

Circuits Practice Problems for AP Physics B & C

RC Circuits: Transitional Behavior

Okay, the obvious question here is, "what happens during the in-between times, while the capacitor is charging?" That's a more complicated question, one that is approached in Physics C. It's easiest if we start with a discussion of a capacitor discharging (see Figure 21.12).

Consider a circuit with just a resistor R and a capacitor C. (That's what we mean by an RC circuit.) The capacitor is initially charged with charge Q₀. Apply Kirchoff 's voltage rule:

–IR + V_c = 0,

where V_c is the voltage across the capacitor, equal to Q/C by the equation for capacitors. By definition, current is the time derivative of charge,

So substituting this value for I into the Kirchoff equation we wrote above, and rearranging a bit, we get

This is a differential equation. On the AP exam you will only rarely have to carry out the algorithmic solution to such an equation; however, you must recognize that the solution will have an exponential term, and you should be able to use limiting case reasoning to guess at the precise form of the solution.

Here, the charge on the capacitor as a function of time is Q = Q_0^e–t/RC. What does this mean?

Well, look at the limiting cases. At the beginning of the discharge, when t = 0, the exponential term becomes e⁰ = 1; so Q = Q₀, as expected. After a long time, the exponential term becomes very small (e gets raised to a large negative power), and the charge goes to zero on the capacitor. Of course—that's what is meant by discharging.

And in between times? Try graphing this on your calculator. You get a function that looks like exponential decay:

What's cool here is that the product RC has special meaning. The units of RC are seconds: this is a time. RC is called the time constant of the RC circuit. The time constant gives us an idea of how long it will take to charge or discharge a capacitor. (Specifically, after one time constant the capacitor will have 1/e = 37% of its original charge remaining; if the capacitor is charging rather than discharging, it will have charged to 63% of its full capacity.)

So there's no need to memorize the numerous complicated exponential expressions for charge, voltage, and current in an RC circuit. Just remember that all these quantities vary exponentially, and approach the "after a long time" values asymptotically.

What does the graph of charge vs. time for a charging capacitor look like? (see Figure 21.13). Think about it a moment. At t = 0, there won't be any charge on the capacitor, because you haven't started charging it yet. And after a long time, the capacitor will be fully charged, and you won't be able to get more charge onto it. So the graph must start at zero and increase asymptotically.

The charge asymptotically approaches the maximum value, which is equal to CV (V is the final voltage across the capacitor). After one time constant, the charge is 1/e = 37% away from its maximum value.

Practice problems for these concepts can be found at:

Circuits Practice Problems for AP Physics B & C