University of Puget Sound Introductory Physics Laboratory

7.Time-dependent circuits

Name:______Date:______

Objectives

To investigate the time-dependent behavior of circuits involving capacitors and resistors.

Equipment

Capacitors, resistance boxes, voltmeter, light bulbs, and semi-log graph paper.

Helpful Reading

Review the introduction to potential difference, current, and resistance in either Hecht Chapter 17 an 18 or Rex/Jackson Chapter 21.

Background

In the last lab session you analyzed circuits involving resistors using the node and loop equations. For circuits with other kinds of elements (capacitors, inductors, transistors, etc.) we can apply the same analysis provided we know how to describe the voltage across the various circuit elements. For resistors, the potential difference across the resistor is given by Ohm’s Law V=IR, where R is the resistance and I is the current through the resistor. Today we experiment with capacitors, which are simply parallel metal plates separated from each other by some non-conducting material. The circuit symbol for a capacitor is two parallel lines with a small separation between them (this is to remind you of parallel plates).

When current flows into a capacitor, it can't jump to the other plate so charge piles up on the plate. This charge creates an electric field between the plates, and hence a voltage difference across the capacitor. The potential difference across a capacitor is proportional to the charge on the plates

V = Q/C

where Q is the charge on the capacitor (+Q on one plate, -Q on the other plate) and C is the capacitance. Capacitance is measured in Farads, which is another name for a Coulomb per volt.

It is important to note that there are two kinds of capacitors, polar and non-polar. The non-polar capacitors, shown above on the left, can be used in any orientation. Their terminals are equivalent. Polar capacitors, on the other hand, have a polarity (+, -) associated with the terminals and must be hooked up correctly in order to work (and not explode). We will be using polar capacitors. Make sure that the negative (low potential) terminal of your battery or power supply will be connected to the negative terminal of the capacitor (an arrow with a negative sign indicates the capacitor’s low potential terminal).

Charging and discharging experiments with light bulbs

Assemble the charging circuit shown below with two batteries (but don't connect the batteries yet), a light bulb, and a 4700 F polar capacitor. Now connect the batteries and watch the bulb. What happens?

Description/discussion of charging behavior

The capacitor should now be charged to 3 V - right? Disconnect the batteries and use the charged capacitor to make a discharging circuit with the light bulb. What do you observe? Does this differ from what you observed while charging? Repeat the charging/discharging process a few times; see if you can fine-tune your observations.

Description/discussion of discharging behavior

Think about what you have observed in terms of the movement of charge in the circuit. Assume that you have a capacitor C initially charged to a potential difference V0. Now connect it across a resistor R. Why does the current flow? Why does the current stop?

Capacitors store energy. Starting with a charged capacitor, identify the forms that the energy takes from fully charged state to the fully discharged state.

Discussion of charge movement and forms of energy

Derivation of the potential difference across a discharging capacitor

We can formalize your observations mathematically to model the discharging circuit behavior. The rate at which charge is removed from the capacitor must equal the current that flows through the resistor:

I = -dQ/dt

From the loop equation we know that the voltage difference across the capacitor and the resistor must be equal. Using V=IR for the resistor and V=Q/C for the capacitor, the above equation becomes:

This is a differential equation for the potential difference across the capacitor as function of time. We can solve this by rewriting the equation as

,

which can be integrated from 0 (initial time) to t (later time) on the right side and from Vo (initial potential difference) to V (potential difference at the later time) on the left to yield

V(t) = V0 e-t/RC

The voltage (and hence the charge) on the capacitor decreases exponentially in time, with a characteristic time  = RC. This is amazing: R is in Ohms and C is in Farads. In the space below, prove that an Ohm-Farad is indeed equal to a second. By what factor is the voltage reduced after the first  seconds? After the second  seconds?

dimensional analysis

Discharging experiment

By charging a capacitor and measuring the potential across the capacitor while it discharges, we can quantify the discharging behavior and compare it to the exponential decay model. Construct the following circuit, taking special care about the polarity of the capacitor. Exploding a 4700 F capacitor would be about as memorable as the earthquake. Be sure the negative terminal of the capacitor is connected to the negative terminal of the power supply.

The resistor R and capacitor C comprising our RC circuit are on the right. The knife switch ABD is used to charge and then discharge the capacitor. The power supply is used to charge up the capacitor initially by putting the switch in the “AB” position. (The resistor, r =10 , is there to prevent the capacitor from being ruined by a large current from the supply.) After the capacitor is fully charged, the switch is moved to the “AD” position and the voltmeter is used to measure the potential difference across the capacitor as a function of time.

  • Use a variable resistance box set at 4000  for R and one of the 4700 F capacitors for C. What decay time would you predict?
  • Put the switch to “AB” and adjust the power supply to charge the capacitor up to 5 V.
  • Move the switch to “AD” and measure the voltage across the capacitor as a function of time.

Do a few practice runs to get a sense of how long a run you will need and how often you will be able to take data. Work out a plan with your group members as to who will do what and when. Blank data tables are provided on a following page. You may need to do this more than once to get a good data set. Make sure to take data over several 'e-foldings', that is, over a time longer than several RC times.

Plot your data on semi-log paper, not on the computer, with voltage on the logarithmic axis and time on the linear axis. Why are we plotting with log-linear axes? Take the natural logarithm of the V(t) expression given above. If you were to plot ln(V) vs. time, what would it look like? Does your data look like this?

  • Obtain voltage vs. time data setsfor the following resistor /capacitor combinations:

4000  / 4700 F (just completed)

2000  / 4700 F

4000  / 9400 F

You do not have a 9400 F capacitor. How can you make one with two that are 4700 F? Plot these results on the same semi-log paper that contains your other graph.

What are the physical significance of the slope and intercept? Do the measured values of slope and intercept agree with your experiment? (You will need to be careful when obtaining slope and intercept information from your plot.)

Check-in with your instructor as you leave:

  1. Voltage/time data.
  2. Completed voltage/time graphs.
  3. Explain the concept of a decay time, and how you read one from a semi-log graph.

Time (sec) / Voltage (volts)
/ Time (sec) / Voltage (volts)

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