Lab 5: EC-3, Capacitors and RC-Decay Lab Worksheet Name_____________________________ 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 four 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! The 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.

Lab 5: EC-3

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For the first part of the lab, you use the same circuit board as you did last week. The board is shown below:

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 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 computer to measure potential differences and currents at various points in the circuit.

These 5 points connected together

Resistor or capacitor block goes here

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A. Capacitor circuits. You use the electrometer to measure voltages, not the Keithley DMM. You will need to connect the red and black needle probes to the electrometer with coaxial (BNC) cables. Important: the 10MΩ input impedance of the DMM is not enough here: the capacitors would discharge through that resistance while they are being measured.

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 Explain: Lead students into figuring out that both capacitors must have the same charge. i.e. “look at the connection between them. Whatever charge going onto the right plate of C1 had to come from the left plate of C2”. Then the potentials must add up to 20V, and the potential across C1 is twice that across C2 since the capacitance is half.

!

2V2

+V2

= 20V "V2

= 6.67V , V1

=13.33V Build the circuit below (note that the power supply is not connected).

Measure the voltage drops across the capacitors. 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, and briefly touch the black and red leads across the series circuit, then disconnect both the black and red leads from the circuit.

c) Use the electrometer (not the Keithley DMM) to measure the voltage drops across each capacitor individually. Compare these to your prediction.

30V

DC voltage source

1000V

1.0 µF 0.47 µF

C1 C2

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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. You then connect the 1 µF capacitor in parallel. a) Do a calculation here to predict the final voltage across the capacitors. The capacitors must come to the same potential difference because their terminals are connected together. Also, the charge is conserved, it just gets redistributed across the capacitors. So

!

Vfinal =Q1/C

1=Q

2/C

2, and

!

Q1

+Q2

=Qorig = C1Vorig

!

C1Vorig = C

1Vfinal + C

2Vfinal "Vfinal =

C1

C1+ C

2

Vorig #1

3Vorig

Now do the experiment:

b) Measure the voltage drop across the parallel combination with the electrometer (not the DMM). Remember that the power supply is disconnected here. How does this compare with your predicted value?

This part will not work unless the power supply is disconnected when measuring with the electrometer. This is because, when plugged into the wall, the electrometer ground and the power supply ground are connected. For instance, touching the black electrometer lead to the red power supply lead shorts out the power supply.

30V

DC voltage source

1000V

1.0 µF

0.47 µF

C1

C2

Wire

Wire

Plug 1 µF capacitor in after making sure it is discharged, and after disconnecting the power supply.

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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 as shown below. For the electrometer, use a BNC to banana plug adapter connected to the ‘input’:

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 they are connected by a wire. 1) What is the direction of the resistor current after increasing the volts 0 to 20V. In steady state, there will be no current flow through the capacitor. This means no current flow through the resistor. There is no voltage drop across the resistor, so it must all be across the capacitor, which is now charged. This charge came through the resistor from the voltage supply. When the voltage is turned back to zero, the charge goes back into the supply 2) What is the direction of the resistor current after decreasing the volts 20V to 0V?

Electrometer

+ -

10 MΩ 1 µF

30V

DC voltage source

1000V

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3) What happens to this charge after it goes through the resistor? It builds up on one plate of the capacitor. Charge is pulled from the other plate, giving the impression that current has flowed through the capacitor. 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). Capacitor voltage increases as charge builds up on it. 5) How does the capacitor voltage affect the current through the resistor? Voltage drop across the resistor must then decrease, which means that current through it will go down. 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 (don’t take any measurements) to approximate the behavior of the circuit as you increase the voltage 0V to 20V. Time ΔQR=IRprev

!

"t QC VC VR IR 0 0 0V 20V 2 µA 2 sec 4 µC 4µC 4V 16V 1.6µA 4 sec 3.2 µC 7.2µC 7.2V 12.8V 1.28µA 6 sec 2.56µC 9.76µC 9.76V 10.24V 1.02µA 8 sec 2.04µC 11.8µC 11.8V 8.2V 0.82µA 10 sec 1.64µC 13.44µC 13.44V 6.56V 0.66µA 12 sec 1.32µC 14.76µC 14.76V 5.24V 0.52µA 14 sec 1.04µC 15.8µC 15.8V 4.2V 0.42µA 7) Now replace the 10MΩ resistor with a 100KΩ resistor. Measure the circuit again with the electrometer. How has the behavior changed? Capacitor charges up much more quickly, because current is larger.

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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 Pasco interface measures the voltage drop across the resistor, hence the current through the circuit.

Start DataStudio by clicking on the Lab5Settings1 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 again. 6) Describe the results here. Note: the Pasco interface has an input impedance of only 1 MΩ. This means that connecting the Pasco interface puts a 1 MΩ resistance in parallel with the 100kΩ resistor, making the effective value 10% less. This is discussed in the next question.

100KΩ

1.0 µF

Wire

Pasco interface A

30V

DC voltage source

1000V

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You have measured voltage vs time, but want current vs time. The conversion factor is the resistance. You can directly scale the plot, or just account for this factor in your area calculation.

7) Use the mouse to select the decay from maximum negative 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?

The total charge delivered must be that which charges the capacitor to the supply voltage. This is

!

Q = CV = 10"6F( ) 10V( ) =10"5C .

The value is a little off because of the Pasco input impedance. 9) The Pasco interface is acting like a voltmeter, but not one with infinite resistance

between its terminals. It’s 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?

Agreement is then good.

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C. Electrical model of a cell membrane F Cell membrane equivalent circuit

As discussed in class, a cell membrane has a potential difference between the interior and exterior of the cell. The low-resistance cell exterior is modeled by a low-resistance wire through which charges move. The medium resistance cell interior is modeled by 100 KΩ resistors. The capacitors model the capacitor formed by the interior and exterior conducting fluids that sandwich 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 can move around in the fluids in response to electric fields, with the resistance to their motion given by the resistors/wires above. The result is that 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 voltage-triggered ion channels distributed throughout. They

open and close in response to varying potential differences across the membrane. That is, their resistance to charge motion varies with voltage – they don’t obey Ohm’s law. We don’t model them here because we don’t have suitable nonlinear resistors.

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 artificially introduce the pulse. 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.

Pulse generator

Interior of cell

Exterior of cell

Lipid Bilayer

You don’t need to use the differential amplifiers here, just plug right into the Pasco interface.

30V

DC voltage source

1000V

100 KΩ 100 KΩ 100 KΩ 100 KΩ 100 KΩ

WIRE

1 µ

F

1 µ

F

1 µ

F

1 µ

F

1 µ

F

WIRE WIRE WIRE WIRE

A1

B1

A2

B2

A3

B3

A4

B4

A5

B5

A6

B6

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Making the measurement to determine signal propagation

To start the action potential, you want to introduce a voltage pulse at A1,B1, and watch it propagate down the cell membrane, similar to an action potential.

This pulse then propagates down the cell membrane. If you could measure the potential differences Ai,Bi at a particular instant in time, you might see something like the following.

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 a continuous version of the model. The triangles correspond to the voltage differences at slightly later time ( t2 ) than the squares ( t1 ). Data such as this would indicate that the voltage pulse is traveling to the right at some speed.

In DataStudio we don’t measure all the voltages at a particular instant in time, but measure

each voltage as a function of time. For instance, at position 2, the voltage is large at time t1 and then smaller at time t2. The voltage at position 3 is small at time t1, and has gotten larger at time t2. Your job here is to reconstruct a graph like the figure above from the voltage vs time data acquired with DataStudio.

POSITION

PO

TEN

TIA

L D

IFF.

( V

)

2 3 4 5 6

Time t1

Time t2

Lab 5: EC-3

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Directions for taking data. You investigate the propagation by measuring the potential differences between the pairs

Ai,Bi with DataStudio. Start DataStudio by clicking on the Lab5Settings3 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 (A1,B1) 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. First, connect (A2,B2) and (A3,B3) to the B and C inputs. You don’t need to use the differential amplifier here, because we don’t need a quantitative answer like we did in the previous section.

You produce the pulse with a timing circuit that produces a single 22 milli-second pulse whenever you push the button. We only have one of these right now, so ask your TA after you have built the circuit, connected the voltage probes, and have DataStudio started with the Lab5Settings3 file from the course web site.

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. (A1,B1) will show you the input pulse, (A2,B2) is the first capacitor charging/discharging, and (A3,B3) is the second capacitor charging/discharging. Now connect the B and C inputs to (A4,B4), (A5,B5) and repeat the measurement. Finally, connect B and C to (A5,B5) and (A6,B6) 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.

How fast does the pulse propagate?

POSITION 1 2 3 4 5 6

PO

TEN

TIA

L D

IFFE

RE

NC

E

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Here is the raw data I obtained, and a plot of pulse shape vs time.

0

0.5

1

1.5

0 0.5 1 1.5 2

RC Cell Membrane

VO

LT

AG

E

( V

)

TIME ( S )

0

0.4

0.8

0 1 2 3 4 5

PULSE SHAPE AT DIFFERENT TIMES t

t=0.05t=0.1t=0.15t=0.20t=0.25t=0.30t=0.35t=0.40t=0.45t=0.50

VO

LT

AG

E

( V

)

POSITION