QuickGuide to Capacitance

CAPACITANCE

Objectives

  • To become familiar with RC circuits;
  • To practice constructing an electronic circuit and making voltage measurements;
  • To interpret the voltage vs. time graph of a charging and discharging capacitor;
  • To observe how capacitors add in series and parallel;
  • To predict the behavior of an RC circuit.

Introduction
Capacitance (C) is measured in Farads (F). The Farad his is the ratio of the electric charge (Q) on each plate to the potential difference (V) between them. For example, a one-farad capacitor would store one coulomb of charge when one volt is applied across the plates. However, this charge cannot be put on the plates instantly. It takes a certain amount of time for the plates to charge. Examine the circuit below:

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When you first throw the switch (S) which connects the battery to the capacitor, the voltage across the capacitor is not immediately equal to as the battery voltage. The charge moves onto the plates quickly at first, and the capacitor voltage increases quickly. However, as the capacitor becomes charged, the charging rate begins to taper off. Eventually, the capacitor voltage is the same as the battery voltage, and no more charge can be forced onto the plates. At this point, the capacitor is no longer charging.

If you analyze voltage of a charging of the charging capacitor over time, you might notice that it models an inverse exponential curve. Incidentally, the following formula describes the charging of a capacitor over time:

VC = VB (1-e-t/RC)

VC is the voltage across the charging capacitor at any time (t). VB is the battery voltage, R is the resistance of the circuit, and C is the capacitance.

After a capacitor is charged, it will hold its charge even when the power supply is removed. The capacitor can now be used as a voltage source that stores charge until you need it to use it. When a capacitor is used, it will discharge. Its rate of discharge will be great at first. However, that rate tapers off exponentially with time. Eventually, the voltage across the capacitor will be near zero, at which time the capacitor is no longer discharging. Below is an example of a circuit that will discharge a capacitor:

discharging capacitor

If you analyze the voltage of a discharging capacitor over time,you might notice that it models a natural exponential curve. Incidentally, the following formula describes the discharging of a capacitor over time:

VC = VB (e-t/RC)

PRELAB QUESTIONS (to be checked off at the beginning of lab)

  1. According to the above inverse exponential (charging) equation, what does VC equal when t = 0? What about when t approaches infinity? Explain.
  2. According to the above natural exponential (discharging) equation, what does VC equal when t = 0? What about when t approaches infinity? Explain.
  3. According to the charging and discharging equations, what would be the effect of changing R or C? For example, if you doubled R or C, how would that change the graph?

MATERIALS

  • Battery
  • Resistors and Capacitors
  • Wires and connectors
  • Switch
  • Vernier Differential Voltage Probe
  • Multimeter
  • Vernier Lab Pro or GO! Connector

ACTIVITIES

Activity 1: Modeling charge and discharge of a capacitor
You will use the Vernier voltage probe to measure the charge and discharge of the capacitor. Connect the probe to the computer and open Logger Pro. To "zero" the probe, connect the ends of the probe together and press "ctrl-0". Logger Pro should display a readout of very close to zero.

Begin by discharging the capacitor (see below). The discharging circuit for the capacitor is very simple. A capacitor is discharged simply by connecting the ends together with a conductor. Leave it connected for about 10 seconds to make sure it is completely discharged.

short a capacitor

In the above discharge, resistance will be effectively zero, and the time constant will be very short. Placing a resistor is in the loop will slow the current, increasing the amount of time it takes to charge and discharge. This is important because you are going to analyze these charge and discharge curves.

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Now construct the RC circuit above using the resistor and one of the capacitors on your table, but do not connect the battery yet! Do not twist or bend the wires of the resistor or capacitor; use the alligator cables to make the connections. Make sure that the negative end of the capacitor is going to connect to the negative end of the battery! If you don't, the capacitor might overheat and explode! Do not close the switch yet. Attach the voltage probe so that it will measure the voltage across the capacitor. Ask your instructor to check your circuit.

When you have received permission from your instructor and the circuit is complete, click "collect." You may now close the switch. You should see the charging curve of the capacitor voltage.

After the capacitor is fully charged, you may disconnect the battery and discharge the capacitor using the circuit below.

discharging capacitor

When the circuit is constructed, press the switch and discharge the capacitor while Logger Pro is collecting.

With a little practice and some adjustment of the time and voltage scales, you should be able to charge and discharge the capacitor during one collection cycle, so that both curves are displayed, one after another, on the same graph.

Examine the equation that models the voltage of a charging capacitor, then use the curve fitting function of Logger Pro to apply that equation to the charging portion of the graph. Next examine the equation that models the voltage of a discharging capacitor, and use the curve fitting function of Logger Pro to apply that equation to the discharging portion of the graph. Print this graph.

Repeat for the second capacitor. Be sure to keep track of which capacitor was used for each graph! You will need this information for Activity 2.

Activity 2: Accurately determining capacitance
Although multimeters read resistance very accurately, most do not have the capability to read capacitance. Compounded with the fact that capacitors are very loosely labeled (easily varying by 25% in many cases), it is very hard to know to any good precision what the actual value of your capacitor is. In this activity you will examine charge and discharge graphs in order to accurately calculate the capacitance.

resistance

First, use the multimeter on the "Ohms" setting to measure the value of the resistance (see above). Make sure the battery is not connected!

For this activity, only the discharging curve of each capacitor will be analyzed. This is because the charging circuit contains not only the resistance of the resistor which you measured, but also the internal resistance of the battery, which you did not measure.

Examine the discharge equation used in activity 1, and compare it to your curve fit parameters to determine the value of the two capacitors. Using percent difference, compare these to the value that is actually printed on the capacitors.

Activity 3: Capacitors in series and parallel
This activity examines the way in which capacitors combine in series and parallel. Before beginning this activity, rank the following three situations, in order of least to greatest capacitance. (Assume that the capacitors have the values found in Activity 2.) Your instructor must check off this prediction before you continue.


A)  Alone

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B)  In series

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C)  In parallel

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After you have made your predictions, repeat the Activity 2 experiment to determine the total capacitance of the two capacitors placed in series. Repeat for the parallel configuration. Were your predictions correct? Explain.

POSTLAB QUESTIONS

  1. Imagine you have an RC circuit in which the capacitor is fully charged to 12 V. The capacitance is 10.0 μF and the resistance is 3.0 MΩ. Using the discharge equation, create a theoretical voltage vs. time graph which plots VC of a discharging capacitor. Starting at t = 0, sample VC once a second for 10 seconds. Plot in Logger Pro and print.
  2. When adding capacitors in series, 1/Ctotal = 1/Ca + 1/Cb. Using the values of the capacitors found in Activity 2, calculate their total capacitance when placed in series. Compare this to the series capacitance found in Activity 3 using percent difference.
  3. When adding capacitors in parallel, Ctotal = Ca + Cb. Using the values of the capacitors found in Activity 2, calculate their total capacitance when placed in parallel. Compare this to the parallel capacitance found in Activity 3 using percent difference.
  4. Using the value capacitance you determined for the first capacitor in Activity 2, use the charging equation to determine the resistance of the charging circuit. How does this resistance compare to the value of the resistor you measured with the multimeter? How can you use this information to make an inference about the internal resistance of the battery pack?
  5. Do you find it strange that the units of the time constant is Ω·F? Shouldn't a time constant have units of seconds? Use the following equations to show that an Ohm multiplied by a Farad is equal to a second!

Voltage = Resistance · Current
Charge = Capacitance · Voltage
Current = Charge / time


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