: Capacitors and Resistor-Capacitor Circuits Phy208 Spr 2008 Name Section Your TA will use this sheet to score your lab. It is to be turned in at the end of lab. You must use complete sentences and clearly explain your reasoning to receive full credit. What are we doing this time? You will complete three related investigations. PART A: Build capacitor circuits on circuit board, investigating current flow and voltages. PART B: Build resistor-capacitor circuits, and measure time-dependent phenomena. PART C: Use these ideas to measure and investigate a crude cell membrane electrical model, investigating propagation of an action potential down the cell membrane. Why are we doing this? Capacitors are almost as ubiquitous as dipoles, showing up almost everywhere there is an insulator. Actually, capacitors have some similarities to dipoles, with equal and opposite charges on the electrodes. And they almost always show up in combination with some non-insulator a resistor-capacitor circuit! What should I be thinking about before I start this lab? You should be thinking about the ideas of circuits you developed when looking at resistors last week, and the aspects of capacitors you looked at in lecture and discussion. In particular, how the voltage across the capacitor is related to the charge on it, and how the current in a circuit delivers charge to a capacitor.
For the first part of the lab, you use the same circuit board as you did last week. The board is shown below: Resistor or capacitor block goes here These 5 points connected together Holes connected by black lines are electrically connected by conducting wires, so all points connected by black lines are at the same electric potential. You build a circuit by plugging in resistors and capacitors across the gap between crosses. The resistors are built into plastic blocks with banana-plug connectors that exactly bridge the gaps. After you plug in a resistor, there will still be unused holes in each cross. You will use the remaining holes to connect the variable voltage source to supply your circuit with charge, and to connect the electrometer or Pasco interface to measure potential differences and currents at various points in the circuit. 2
A. Capacitor circuits. You use the electrometer to measure voltages. Connect the red probe to the electrometer input connection, and the black probe to the electrometer ground input with black coaxial cables (not the banana plug cables). You don t need any adaptors: the coaxial cable connects directly to the probe and to the electrometer. 1) Series capacitors: you will build the circuit below and measure it, but first predict how the 20V provided by the voltage supply will split among the two series capacitors. V across C1: V across C2 Build the circuit below (note that the power supply is not connected). DC voltage source 30V 1000V 0.47 µf C1 1.0 µf C2 a) First, use a bananna plug cable to temporarily short out each capacitor to make sure there is no charge separation between the plates. Make sure the cable is removed before proceeding. b) Set the voltage supply to 20V. Touch the black and red voltage supply leads across the series circuit, then disconnect both the black and red leads from the circuit. c) Use the electrometer to measure the voltage drops across each capacitor individually with the supply disconnected. Compare these to your prediction. 3
2) Parallel capacitors. Here you connect capacitors in parallel to see how they share charge. Start by discharging both capacitors. You first charge up C1 (the 0.5µF capacitor) to 20V, then disconnect the voltage supply. The electrometer will not measure correctly unless you disconnect the power supply. You then connect the 1 µf capacitor in parallel. a) Do a calculation here to predict the final voltage across the capacitors. Now do the experiment: DC voltage source 30V 1000V 0.47 µf Wire C1 Wire 1.0 µf C2 Plug 1 µf capacitor in after making sure it is discharged, and after disconnecting the power supply. b) Measure the voltage drop across each capacitor with the electrometer. Remember that the power supply is disconnected here. How does this compare with your predicted value? 4
B. Resistor-capacitor circuits. Now you will connect a resistor and capacitor to investigate how a capacitor charges and discharges. Turn the DC voltage source on and set it for zero volts output. Connect the DC voltage source and electrometer to the 10 MΩ resistor and 1 µf capacitor exactly as shown below. DC voltage source - + Electro meter 30V 1000V black red red black 10 MΩ 1 µf Put the electrometer on the 30V scale, and make sure the voltage is still at zero V. Quickly increase the voltage on the DC supply to 20 V and watch the electrometer. After the electrometer needle stops moving (wait at least a minute), quickly turn the voltage supply to zero volts, and watch the electrometer again. Note that the electrometer measures both the sign and the magnitude of the voltage across the resistor, proportional to the current through the resistor. Remember that the voltage supply keeps a constant voltage between its two output terminals. Zero volts means that there is no potential difference between the red and black terminals: it is as if a wire connects them. 1) What is the direction of the resistor current after increasing the voltage 0V to 20V. 2) What is the direction of the resistor current after decreasing the voltage 20V to 0V? 5
3) The current represents a charge flow. What happens to this charge after it goes through the resistor? 4) What do you think is happening to the voltage across the capacitor as a function of time? (you can t measure this with the electrometer because of the way it is grounded). 5) How does the capacitor voltage affect the current through the resistor? 6) You saw that the current through the resistor changes smoothly with time. But it can be easier to think about this in short time steps. Fill out the following table to approximate the current as a function of time as you increase the voltage 0V to 20V. ΔQ R is the charge that flowed through the resistor during the previous 2 sec. Q C is the charge on the capacitor V C is the voltage drop across the capacitor; V R is the voltage drop across the resistor I R is the current through the resistor. I R prev is the current during the previous 2 sec. Time ΔQ R =I prev R "t Q C V C V R I R 0 2 sec 4 sec 6 sec 8 sec 10 sec 12 sec 14 sec Sketch a plot of the voltage across the capacitor as a function of time. V C TIME 5) Now replace the 10MΩ resistor with a 100KΩ resistor and repeat the experiment of increasing and decreasing the supply voltage. How does the behavior change? 6
Now you do use the computer to measure the time-dependent current through the resistor. Use the 1µF capacitor, and the 100 kω resistor. To make an accurate measurement, the voltage needs to be switched very quickly from 0V to 10V, more quickly than you can do it by turning the knob. Set up the circuit below in order to quickly switch the voltage. The switch is in the parts tray. Flipping the switch changes the applied potential from 20V to 0V and vice versa. DC voltage source Pasco interface A 30V 1000V 100KΩ Switch 1.0 µf Wire The Pasco interface measures the voltage drop across the resistor, and hence the current through the circuit. Start DataStudio by clicking on the Lab6Settings1 file on the Laboratory page of the course web site. Use DataStudio to measure the time-dependence of the current through the circuit when you close the switch and then open it, repeating several times. 6) Describe the results here. Explain what is happening. 7
In this part you determine how much total charge has flowed through the resistor. You have measured voltage vs time, but want current vs time. The conversion factor is the resistance. You can directly scale the plot in DataStudio, or just account for this factor in your area calculation. 7) Use the mouse to select the decay from maximum to zero current on your Current vs Time (sec) graph, and find the area under the curve by selecting area from the statistics (capital sigma) pull-down menu at the top of the graph. Area under curve Value Units 8) In the space below, calculate the expected value of this area from basic principles of the capacitor (you don t have to do any integration). How does your value compare to the measured one? 9) The Pasco interface is acting like a voltmeter, but not a perfect one. Its electronic circuitry is equivalent to a 1MΩ resistor between its terminals. This means that although you plugged in a 100KΩ resistor to the circuit above, it is really 1MΩ in parallel with the 100KΩ when the Pasco is connected. Use this value and recalculate the area above. Does this improve the agreement? 8
Now think about this circuit, but don t build it or measure it. Why would you ever care about this circuit? Be patient in the next section you use this as a basis for a model of a nerve signal propagating down a cell membrane. R 1 =100 KΩ R 2 =100 KΩ C 1 =1 µf C 2 =1 µf 10) The capacitors start out discharged, and you apply 10V across the circuit. At that instant, what are the currents through R 1 and R 2? 11) Where does the charge flowing at this instant end up? 12) Think about later time intervals. Why does C 1 charge up sooner than C 2? If the voltage across each capacitor triggered something to happen, then that event would occur sooner at C 1 and some time later at C 2. In this way you can think about this as a signal propagating down the circuit. In the next section you watch a voltage pulse propagate down a similar circuit. 9
C. Electrical model of a cell membrane DC voltage source 30V 1000V Interior of cell A 1 A 2 A 3 A 4 A 5 A 6 F 100 KΩ 100 KΩ 100 KΩ 100 KΩ 100 KΩ Pulse generator 1 µf 1 µf 1 µf 1 µf 1 µf Lipid Bilayer WIRE WIRE WIRE WIRE WIRE B 1 B 2 B 3 Exterior of cell Cell membrane equivalent circuit As discussed in class, a cell membrane has a potential difference between its interior and exterior. The low-resistance cell exterior is modeled by a low-resistance wire. The medium resistance cell interior is modeled by 100 KΩ resistors. The capacitors model the insulating lipid bilayer. The potential difference arises from charges in the conducting fluids that form the electrodes of the cell-membrane capacitor. These charges move around in the fluids, making the potential difference vary with position. This makes an action potential (generated at the left end here) can propagate down the cell membrane. The switch models an ion channel that is triggered by some external stimulus, mechanical, chemical, or electrical. It could be a channel opening in response to a pinprick in your finger. This causes a change in potential difference at that location. No other ion channels are modeled here. In a real cell membrane, there are voltagetriggered ion channels distributed throughout. They open and close in response to varying potential differences across the membrane. We don t include them here because these are not simple resistors. but one of their effects is to bring the potential difference across the cell membrane back to its resting state after reaching a threshold value. This contributes to making the action potential a short pulse. Here we use a pulse generator to both initiate the pulse, and also to bring the potential back to the resting value. The voltage-sensitive ion channels also serve to propagate the pulse down the cell membrane, keeping a sharp shape via their non- Ohmic behavior, and contributing to determining the propagation speed. Here only the contribution from the RC network is modeled, so the pulse broadens and decays. Hodgkin and Huxley shared the 1963 Nobel prize in physiology and medicine partly in recognition of their quantitative analysis of the full nonlinear electrical circuit model. B 4 B 5 B 6 10
To start the action potential, you will (on the next page) introduce a voltage pulse at A 1,B 1, and watch it propagate down the cell membrane, similar to an action potential. If you could measure all the potential differences A i,b i at two different instants in time (t 1 [squares] and t 2 [triangles]), you might see something like the following. POTENTIAL DIFF. ( V ) Time t 1 Time t 2 2 3 4 5 6 POSITION You can only measure potential differences across the capacitors the squares and triangles represent potential differences measured at these locations. The dashed line is what the voltage pulse might look like in an actual, continuous, cell membrane. The triangles correspond to the voltage differences at slightly later time ( t 2 ) than the squares ( t 1 ). This data indicates that the voltage pulse is traveling to the right at some speed, since the peak voltage position has moved. You won t be measuring the voltages across all the capacitors at a particular instant in time, but measure the voltage across each capacitor as a function of time. For instance, at position 2, the voltage is large at time t 1 and then smaller at time t 2. The voltage at position 3 is small at time t 1, and has gotten larger at time t 2. Your job here is to reconstruct a graph like the figure above from the voltage vs time data acquired with DataStudio. Directions for taking data. You investigate the propagation by measuring the potential differences between the pairs A i,b i with DataStudio. Start DataStudio by clicking on the Lab6Settings3 file on the Laboratory page of the course web site (We are not using LabSettings2 right now). DataStudio has three analog inputs A, B, and C. The potential difference at (A 1,B 1 ) measures the pulse before it propagates down the cell membrane. This must always be connected to input A because DataStudio watches this input to determine when to start recording data. You measure the other potentials with inputs B and C. To start, connect (A 1,B 1 ) to the A input, and (A 2,B 2 ) and (A 3,B 3 ) to the B and C inputs. 11
You start a pulse moving down the membrane by generating a single 22 milli-second pulse at one end. After you have the circuit, connected the voltage probes, and have DataStudio started with the Lab6Settings3, go get a pulse generator and plug it into your board. Set your voltage supply to 12V, and connect it to the red and black terminals of the pulse generator (if you don t get the polarity right you ll burn out the chip but its easy to replace.) If you re not sure you can set it up correctly, ask your TA. Take a measurement by clicking Start on DataStudio, then push the button on the pulse-generator board. DataStudio stops acquiring data automatically after 2 seconds you don t have to click stop. (A 1,B 1 ) will show you the input pulse, (A 2,B 2 ) is the first capacitor charging/discharging, and (A 3,B 3 ) is the second capacitor charging/discharging. Now connect the B and C inputs to (A 4,B 4 ), (A 5,B 5 ) and repeat the measurement. Finally, connect B and C to (A 5,B 5 ) and (A 6,B 6 ) and take the last measurement. You will now have in DataStudio voltage vs time for all of these positions, 1-6. Now you should pick particular times (use the cross-hair tool), and plot voltage vs position for these times on the plot below. You can do this in Excel if you like. POTENTIAL DIFFERENCE 1 2 3 4 5 6 POSITION How fast does the pulse propagate? 12