 Phys 223 University Physics III Lab 4     Capacitor Networks

In this lab, we shall continue investigation of RESISTOR-CAPACITOR circuits, “RC” circuits, in order to study the behavior of networks of elementary capacitors.

In the course of performing this experiment, we shall reinforce

1. proper use of electrical meters,

2. notions of direct and indirect measurement,

3. proper employment of semi-log graph paper,

4. use of experiment to corroborate or falsify theory, and

5. error analysis.

GRAND IDEA:   By careful analysis, we will measure the charging and discharging time constant(s) [OR the DECAY RATE(S)] governing particular RC circuits. We will also infer the actual capacitance of the system, and the manners in which capacitances combine.

• CHARGING
The electric potential difference [VOLTAGE] across a charging capacitor is expected to have the following time-dependence: where is the asymptotic (long time) value of the voltage, and is the charging and discharging rate characteristic of the system.

• DIS-CHARGING
The predicted form of the electric potential across a discharging capacitor is: where Vo in this case is the starting (t = 0) value of the potential.
According to the theory describing the dynamical behavior of RC-circuits, where R, the resistance, is in Ohms [W], and C, the capacitance, is in Farads [ F ].

The effective capacitance of series and parallel combinations of elementary capacitors is determined by the theory of electrostatics. Our tools for this experiment include: [LARGE-valued] capacitors and resistors, a circuit breadboard, wires, batteries, meters, and a stopwatch.

 STEP ONE [Setting up]

1. Refamiliarize yourself with the circuit breadboard. Check the resistance by direct measurement. Measure the total voltage of the battery.
CAVEAT: Quoted values of capacitance are usually only about 25% precise.
2. Before doing the experiment, and between trials, one should completely discharge the capacitor. Here’s a strategy for doing just this:

1. With the capacitor and resistor connected, complete the circuit for discharge. Leave the system alone for a minute or two.
2. Take a wire and “short out” the leads on the capacitor. The capacitor should be fully discharged.

Don’t short out the capacitor, step 2, without allowing it to drain first, step 1. Should you short it out without first draining it, then you might trigger a spark which could harm you and/or the capacitor and/or the wire.

 STEP TWO [Timing is Everything]

Decide upon a strategy for measuring the electric potential across the capacitor as a function of time. Two possibilities present themselves:
– STROBE Record the voltage read off the meter at pre-determined intervals, (i.e., every three, four, five, or six seconds).

– EVENT-DRIVEN Record the times when the voltage reaches particular levels (i.e., every fifth of a volt).

It is probably best to refine the method that you employed last week.

 STEP THREE [CHARGE!]

Assemble and connect the circuit elements: batteries, switch, resistor, capacitor, and voltmeter, making sure that the switch is OFF.

1. At t = 0, throw the switch. Collect time and voltage readings, , until the system stabilizes.
[We expect that most of the fun will occur in the first 30 – 150 seconds, or so.]
2. Infer the asymptotic value that the potential approaches as t goes to infinity. .   Plot vs t on semi-log graph paper.
3. Determine the (MAXIMUM, BEST-FIT, MINIMUM) slopes of the lines which accommodate the data on the graph. Convert the base 10 decay-rate measured from the graph to its natural logarithm counterpart (×2.3026).
4. From the range of values for the decay-rate, infer the value of the net capacitance of the parallel combination of elementary capacitors.

 STEP FOUR [DIS-CHARGE!]

Allow the voltage to stabilize. This affords you time to finish recording the “charging” data, and prepare to collect the “discharging” data.

1. At t = 0, throw the switch to discharge the capacitor through the resistors. Collect time and voltage, V-(t), readings. Continue until the system stablizes.
[We again expect that the fun will occur in the first 30 – 150 seconds, or so.]
2. Plot V-(t) vs t on semi-log graph paper.
3. Determine the (MAXIMUM, BEST-FIT, MINIMUM) slopes of the lines which accommodate the data on the graph. Convert the base 10 decay-rate measured from the graph to its natural logarithm counterpart (×2.3026).
4. From the range of values for the decay-rate, infer the value of the net capacitance for the parallel combination of capacitors.
 STEP FIVE [Once More, With Feeling]
Repeat STEP THREE and STEP FOUR with the series combination of capacitors. Plot both the charging and discharging data on the same semi-log graph using different colors, and use the combined data to determine the (MAXIMUM, BEST-FIT, MINIMUM) slopes.
 STEP SIX [Reflect]