In this lab, we shall investigate RESISTOR-CAPACITOR circuits, or as they are commonly known, “RC” circuits. Unlike typical situations involving batteries and networks of resistors, in which steady-state behavior is quickly realized, RC circuits evince more readily observable exponentially-limited growth, or decay, of salient properties.
In the course of performing this experiment, we shall reinforce
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.
CHARGINGAccording to the theory describing the dynamical behavior of RC-circuits,
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 V_{o} in this case is the starting (t = 0) value of the potential.
where R, the resistance, is in Ohms [W], and C, the capacitance, is in Farads [ F ].
Our tools for this experiment include: [LARGE-valued] capacitors and resistors, a circuit breadboard, wires, batteries, meters, and a stopwatch.
WARNING: the capacitors that we use are oriented, that is, they are engineered, so that positive charge accrues to one plate only. The orientation is indicated by a dark arrow residing in a light-grey band on the barrel of the capacitor. The positive terminal of the battery is connected on the POSITIVE side, the base of the arrow, while the negative terminal is connected to the NEGATIVE capacitor lead, at the arrowhead.
ONE | [Setting up] |
- Refamiliarize yourself with the circuit breadboard.
- Identify one large resistor (i.e., 1.0×10^{5} Ohms or 2.2×10^{5} Ohms), and one large capacitor (i.e., 100 µF or 330 µF) by their markings and labels.
- Check the resistance by direct measurement.
- Measure the voltage provided by the battery.
- Before doing the experiment, and between trials, one should completely discharge the capacitor. Here’s a strategy for doing just this:
- With the capacitor and resistor connected, complete the circuit for discharge. Leave the system alone for a minute or two.
- Take a wire and “short out” the leads on the capacitor. The capacitor should be fully discharged. Don’t short out the capacitor without first allowing it to drain. Otherwise, you might trigger a spark which could harm you and/or the capacitor and/or the wire.
TWO | [Timing is Everything] |
Come up with 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).
THREE | [CHARGE!] |
Assemble and connect the circuit elements: batteries, switch, resistor, capacitor, and voltmeter, making sure that the switch is OFF.
- 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.]
- Infer _{}. Plot _{} vs t on semi-log graph paper.
- Determine the (MAXIMUM, BEST-FIT, MINIMUM) slopes of the lines which accommodate the data on the graph.
base-e Employ the natural logarithm,
base-10 Should one prefer to think base-10, the decay rate calculation takes place in two steps:to obtain the decay-rate directly.
and _{}.
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.
- At t = 0, throw the switch to discharge the capacitor through the resistors. Collect time and voltage, V-(t), readings, until the voltage approaches zero. [We again expect that the fun will occur in the first 30 – 150 seconds, or so.]
- Plot V-(t) vs t on semi-log graph paper.
- Determine the (MAXIMUM, BEST-FIT, MINIMUM) slopes of the lines which accommodate the data on the graph. [Do this directly with the natural log, or via [_{ }.]
FIVE | [Reflect] |
Reflect upon your observations.Q: Does the voltage across the capacitor experience exponentially-limited growth on charging and exponential decay on discharging, as predicted?
Q: Are the charging and discharging rate constants equal?
Combine the two decay rates to obtain a single range for _{}, and from this and the measured value of the resistance, infer the capacitance.CAVEAT: Stated values of capacitance are usually about 25% precise.