Studio Session 3

Circuit elements

Our bodies separate and store charge as a power source to transmit signals along nerves.  An excess of positive ions on the outside of the cell membrane results in a potential difference across the membrane.  The inside of the cell is at a negative potential of ~100 mV with respect to the outside.  The membrane acts like a capacitor.

Electrical signals play a role in transmitting information through our bodies.  Sensory information is transmitted via nerves.  Each nerve consists of a bundle of nerve cells or neurons.  A neuron receives stimuli at the input end and produces a signal that is transmitted across the axon to the output end.  The axon membrane can be modeled as a charged capacitor.  When the neuron is stimulated, the voltage across the capacitor rapidly changes and the charge on the plates reverses, only to thereafter quickly return back to its original value.  For this to happen, a current must flow through some effective resistance.  The whole axon can be modeled as a chain of capacitors and resistors connected in series and parallel.  A voltage and current pulse propagates along this chain. 

The speed of propagation of the action potential depends on the electrical resistance R within the core of the axon and the capacitance C across the membrane.  A simple electrical circuit, consisting of a resistor in series with a capacitor, has a time constant τ = RC.   The time constant characterizes the time it takes for the capacitor to charge and discharge and therefore limits the maximum speed with which signals can travel through the circuit.

Nerves transmit electrical signals across the body with an RC time constants.  By the end of this lab, you will determine whether your own body's electrical resistance and a improvised capacitor could handle signal speeds fast enough to react to touching a hot stove.

Equipment needed:

• Digital multimeter
• Pasco RLC circuit board
• Pasco voltage sensor
• 3 patch cords  (1 black, 1 red, 1 yellow)

Open a Microsoft Word document to keep a live journal of your experimental procedures and your results.  Include all deliverables, (data, graphs, analysis, outcome).  Write a 'mini-reflection' immediately after finishing each investigation, experiment or activity, while the logic is fresh in your mind.

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The resistance of your body

Fatal electric shock occurs when a sufficiently large electric current flows through the body.  A fraction of such a current flows through the heart and may disrupt the cardiac cycle.  Typical effects are listed in the table below.

Shocking current:	Effect:
<1 mA	no observable effect
~1 mA - ~10 mA	tingling sensation
~10 mA - ~100 mA	muscular paralysis ("can't let go")
~100 mA	ventricular fibrillation
~1A - ~10 A	thermal damage to tissue

Paradoxically, brief currents of > 1 A may be less dangerous than lower currents.  Instead of putting the heart into ventricular fibrillation, these currents clamp the whole heart muscle at the same time.  When the current is turned off, a normal heart beat may resume on its own accord.  Indeed, currents of about 1 A are used clinically to defibrillate the heart.

Experiment 1

Before taking any measurements or looking up data, make some prediction.  Do you think your body's electrical resistance will be higher when your skin is completely dry, or when it is wet with tap water?  Write down your reasoning.

Do you expect your skin resistance to be closer to 100 Ω, 10 kΩ, or 1 MΩ?  Why?
Do you think your body's electrical resistance will be higher when your skin is completely dry, or when it is wet with tap water?  Why?

Use the digital multimeter to measure the resistance of your body.  Switch the meter on the 20 MΩ scale.  Make sure the leads are plugged into the Ω and COM connectors on the lower right of the meter. 

Note:  This multimeter is only for measuring components not connected to a power source.  Do not connect it to a circuit that has power.

Press the thumb of one of your hands against the black and the thumb of the other hand against the red lead.  How does contact area, pressure, or moisture affect skin resistance?  Record the values for each member of your group.

Based on your measured skin resistance, a 110 V wall socket might seem safe under dry conditions.  Why is it still potentially lethal if you are sweating or holding a metal tool?"
Based on the actual measured resistance, calculate what current would flow through you if you accidentally touched a 110 V wall outlet with a wet or dry hand while the other hand touches a grounded object.  Look at the Fatal Shock Table.  What physiological effect would this have on you?

The salty fluids within the human body are electrical conductors.  Salt water conducts electricity because it has mobile electrons and ionic states via the salt atoms.  Salt water provides a large surface area of contact for the conductive element and it connects with the sweat glands so electricity can flow past the skin and into your body, which has low electrical resistance.  The internal resistance of an arm (from hand to shoulder) is less than 100 Ω.  If there is a voltage across this internal resistance, a current will flow and heat will be generated.  If the current is large or the connection time is long enough, this heat will cause burns and destroy tissue.  Fortunately the resistance of dry skin is high.  The dry protein of your skin is an insulator.  Using a typical contact area, the skin acts like an approximately (10 - 100) kΩ resistor in series with the internal resistance of the body.  At voltages below about 50 V the dry skin provides safe current limiting protection.

Be extremely careful not to have electrical contact with a voltage source if you have wet or sweaty skin.

Experiment 1 Deliverables: (to be included in the your journal)

• Analysis:  Your prediction with your reasoning before the measurement.  Your reasoning about the safety of touching a 110 V wall socket, based on your measurements.
• Outcome:  Your calculation of the current that would flow through you if you accidentally touched a 110 V wall outlet with a wet or dry hand.

Measuring capacitance

Assume you connect two identical metal plates of area A, separated by a non-conducting material which has a thickness d to a battery and a switch, as shown.  When the switch is open, there is no excess charge on either plate.

Discuss what happens when the switch is then closed. 

Experiment 2

The goal of this experiment is to determine the mathematical relationship between plate separation d, overlapping area A, and capacitance C.
Before measuring, sketch qualitative graphs predicting how C changes with
(a) increasing separation d and
(b) increasing area A.

Construct a parallel plate capacitor out of two rectangular pieces of metal foil.  The sheets have small "handles".  Attach the leads of the multimeter to those handles and make sure the leads are plugged into the + and - connectors on the lower left of the meter.  Switch the multimeter to the 20 nF scale.  Slip the two foil sheets between the pages of a heavy textbook and separate them by pages of the book.  Make sure the foil sheets do not touch each other and "short out".  The areas of the sheets should overlap and the "handles" should stick out on opposite sides.  Weight the book down with another heavy textbook.

Note:  The metal foil has sharp edges.  Be careful not to cut yourself.

Choose 5 different page counts and 3 different overlap areas.  Just move the foils, do NOT fold them.  Record your data in a table in your log.  Measure the capacitance of your "parallel plate capacitor" using your multimeter.  Record your measurements in a clean table.
Plot C vs. d, C vs. 1/d, and C vs. A.  Which plot yields a straight line?  Derive an empirical equation for C(A, d) from your slopes.

Experiment 2 Deliverables: (to be included in the your journal)

• Visuals:  Plot C vs. d, C vs. 1/d, and C vs. A.
• Analysis:  How do your slopes support your empirical equation for C(A, d)?

Ohm's law

Ohm discovered that when the voltage (potential difference) across a conductor changes, the current flowing through the conductor changes.  He expressed this as I = V/R or V = IR.  As the voltage increases, so does the current.  For many conductors R is approximately constant.  These materials are called ohmic.  If the voltage across an ohmic resistor is increased, a plot of voltage versus current shows a straight line.  The slope of the line is equal to R.  For non-ohmic materials, R is not constant and a plot of voltage versus current will not show a straight line.

Experiment 3

Use the Pasco RLC board to investigate the relationship between current and voltage in ohmic and non-ohmic materials.

• Make sure the Pasco 850 interface is turned on.  Open the Capstone program.  The icon for this program is on the desktop.
• In the Hardware Setup window click on the yellow circle over the two output jacks on the image of the PASCO 850 interface. 
  
  Select the Output Voltage - Current Sensor.
• Plug a black patch cord into the left output and a red patch cord into the right output.
• Click Hardware Setup again to close this window.  In the Control Panel at the bottom set the sample rate to 1000 Hz = 1 kHz.
• Click on the Signal Generator icon in the tool panel, (4 icons below Hardware Setup), and choose the 850 Output 1.
• Choose the Triangle Wave, set the amplitude to 3 V, the frequency to 1 Hz.  Click AUTO to turn it on automatically.
• Close the signal generator window and click on Classic templates, two small and one large display, graphs.
• For the vertical axis of one small graph choose Output Voltage (V).  For the vertical axis of the other small graph choose Output Current (A).  For both small graphs choose time for the horizontal axis.
• For the large graph choose Output Voltage (V) for the vertical and Output Current (A) for the horizontal axis.
• Click the Recording Conditions icon in the control panel.
  Choose Start Condition, Measurement Based, Output Voltage (V) is above 0.2 V.  Choose Stop Condition, Time Based, 3 s.
• Connect the 10 Ω resistor to the output of the interface as shown below.
  
• Click the Record button.
  • You will record data for 3 seconds.
  • Click the "autoscale" button in the graph displays to rescale the graphs.
  • In V versus A graph choose a linear fit and click on Fit.  The slope of the large V versus A graph will be displayed.
• Give your large graph of voltage versus current a title and copy it into your journal.
• Repeat the measurement with the 100 Ω resistor connected to the output of the interface.
  
• Now connect the light bulb on the RLC board to the output of the interface and take data for 3 s. 
  
  Repeat the measurement a few times, and alternately watch the bulb and the trace on your large graph. 
• Give your large graph of voltage versus current a title and copy it into your journal.
• Now erase all your data and connect the LED on the RLC board to the output of the interface and take data for 3 s.
  
  • Repeat the measurement a few times, and alternately watch the LED and the trace on the graph 3.
• Change the signal generator amplitude to 5 V and repeat.

Experiment 3 Deliverables: (to be included in the your journal)

• Visuals:  Your voltage versus current graphs (with titles)
• Analysis:
  Are the 10 Ω and 100 Ω resistor ohmic resistors?   Why or why not?  Base your conclusions on your measurements.
  At low currents, what is the dynamic resistance ΔV/ΔI of the light bulb?   How does it change at high currents?
  Why does the bulb's resistance increase as it glows brighter?  Think about atomic lattice collisions and temperature.
  What feature of the LED's V vs. I curve explains why LEDs require current-limiting resistors in practical circuits?

RC circuits

Find out how the voltage across a capacitor varies as it charges and discharges.

Experiment 4

• Before making a measurement, sketch the expected voltage across the capacitor and the current through the circuit vs. time during charging and discharging on the same time axis.
• Connect the 330 μF capacitor and the 100 Ω resistor in series to the output of the interface as shown below.
  Note:  You have to short out the coil.
  
• In the Signal Generator window choose the square wave.
  • Choose an amplitude of 2 V and a frequency of 2 Hz.
    Under Offsets and Limits adjust the voltage offset to be 2 volts.
• In the Capstone program, click the Hardware Setup button on the left, double click Analog Channel A and choose to add the Voltage Sensor.
• Plug the Voltage Sensor into Analog Channel A.
• Connect the voltage sensor across the capacitor as shown below.
  
• Change the axes of your large graph.  Choose Voltage (Ch A) for the vertical axis and time for the horizontal axis.
• Click the Recording Conditions icon and choose Stop Condition, Time Based, 1 s.
• Click the Record button. You will record data for 1 s.  The power supply will turn on for 1/4 s, then turn off for 1/4 s then turn on again for 1/4 s and so on.  During the time the power supply is off the RC circuit is shorted out.  The capacitor will charge when the power supply is on and discharge when it is off.
• Click the "autoscale" button in your graphs to rescale the display.  Magnify a region of the plot of voltage versus time that shows the voltage rising from zero volts to the maximum voltage.
  • Use the smart cursor to find the time t₁ when the voltage begins to rise and the time t₂ when the voltage has reached ½ of its maximum value.
    Find t_½ = t₂ - t₁.
  • Use t_½ = τ*ln2 = 0.693*τ to find the time constant τ.

Experiment 4 Deliverables: (to be included in the your journal)

• Visuals:  Your voltage versus time graph (with title)
• Analysis:  
  Suppose you double the resistance R in your circuit.  Predict what will happen to
  (a) the time required to charge the capacitor to 50% and
  (b) the peak charging current.
  An axon membrane has an equivalent capacitance per unit length C and internal fluid resistance R.  If an axon's radius increases (reducing internal resistance R), what happens to the propagation delay τ?  How does this explain why large nerves conduct signals faster?
• Outcome: 
  What value did you find for τ?   Does this value agree with the nominal value τ = RC?

Convert your journal into a lab report.

Name:
E-mail address:

Laboratory 3 Report

• Make sure you completed the entire lab and answered all parts.  Make sure you show your work and inserted and properly labeled relevant tables and plots in your journal.
• Add a summary reflection at the end of your report in a short essay format.

Save your Word document (your name_lab3.docx), go to Canvas, Assignments, Lab 3, and submit your document.