# ELECTRICAL FIELDS AND POTENTIAL

SUMMARY You will measure voltages in the space between two electrical terminals and use your

measurements to determine the lines of equal potential. Based on these voltages, you will then calculate the strength and direction of the electric field between the terminals.

THEORY Electric charges accelerate when placed into electric fields. The kinetic energy of the charge

increases as it accelerates, and therefore (due to conservation of energy) the electric potential energy of the charge must simultaneously decrease. Positive charges accelerate in the direction of the electric field, and therefore the field points from areas of high potential energy to areas of low potential energy.

The electrical potential V – often called voltage – is defined as V = EPE/q, where EPE is electric

potential energy. V is measured in units of J/C, also called volts. Only a difference in voltage, V, can be measured and in many cases (including this lab) we usually measure voltage with respect to an arbitrarily set ‘zero point’ (ie. a place where we assume V = 0).

The force on a charge q from an electric field E is F qE

If the electric field is constant, so is the force and therefore

EPE = qV = qEx

which can be rewritten in terms of V as

V = E x Since like charges repel, the field lines of a positive charge point

straight out from the charge, as shown in the diagram at right. At equal distances from the charge the voltage (potential) is the same, so the surface of any sphere centered on the charge is all at the same potential. These are called equipotential surfaces and in general will differ in shape depending on the geometry of the charge distribution (they are the dotted lines in the diagram). Because electric field lines are always perpendicular to equipotential surfaces, and voltages are easier to measure than electric fields, determining the shape of the equipotential surfaces is a method of calculating the electric field.

REFERENCE: Cutnell and Johnson chapters 18 and 19, specifically sections 18.6, 18.7 and 19.4.

Electric field and equipotential lines for a positive point charge.

Photograph of the lab setup. The fixed probe (the one at the left) touches the negative (black) terminal of the power supply and should never be moved. You will use the other probe to take voltage measurements at every point between the two parallel lines (simulated plates of a capacitor). The fixed probe corresponds to the V = 0 point, so the hand-held probe is measuring the difference in voltage between the point you are touching and the negative terminal.

PHY108 Lab 2 Electrical Fields and Potential

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EQUIPMENT AND PROCEDURE

1. SETUP THE MEASUREMENT SYSTEM Purpose: to see how the voltmeter measures voltage differences, and see how the voltage varies within the parallel plate geometry

1.1 POWER SUPPLY AND VOLTMETER

Check that the multimeter is set to measure voltage on the 20 Volt DC scale. Connect one cable to the ‘V’ input of the multimeter and another cable to the ‘Com’ input. (We have several models of multimeter that all function identically but may have slightly different labeling. Check with your TA if you are unsure exactly how to check the setup of the multimeter at your setup.) Turn the power supply on, and set the voltage to about 5 Volts. Touch the two cables from the multimeter to the two terminals of the power supply. It should read about 5 volts. Switch the two cables at the power supply and check to see the voltage reverses on the multimeter. Touch both cables to the positive (red) terminal of the power supply. You should read a voltage of 0 V.

This is because the multimeter measures V (not V), and both leads are connected to the same voltage.

1.2 EXPERIMENTAL APPARATUS SETUP

1.2.1 Set up The electric field lines will be found between a charge distribution outlines by silver conductive paint on black resistive paper. The charges move easily through the silver paint, but have difficulty moving along the black paper. When the silver circuit is connected the power supply, charge will build up on the two parallel lines separated by the dots. Always turn the power supply off when making connections. Check that your lab is set up as follows:

1) The positive terminal of the power supply is connected to one side of the paper using a red cable.

2) The negative side of the power supply is connected to the other side of the paper using a black cable.

3) The stationary probe (the one with a little stand to hold the metal probe) is connected to the ‘com’ side of the multimeter, and is placed so it is touching one of the parallel sides of the pattern.

4) The hand held probe is connected to the ‘V’ side of the multimeter.

If the above is set up correctly, turn on the power supply to about 5 V.

1.2.2 Measure the Voltage between the bars Touch the hand held probe to the positive bar (this is the part of the pattern connected to the red terminal of the power supply). You should get a fairly steady measurement of about 5 V everywhere along the bar (the slight difference between your measured V and the voltage of the power supply is due to resistance loss in the wires). Record the voltage in your spreadsheet.

1.2.3 Measure the distance between the bars Measure x, the distance between the bars, with a ruler and record it in your excel spreadsheet.

1.2.4 Compute the field If the electric field is constant between the bars, then the electric field would be V/x (this is explained in the theory section of this lab). Compute the electric field for the values of V and x you just measured and record this result. In your report you will be comparing this estimate to the value you measure.

It doesn’t matter to the circuit what

color the wires are, but using red for

positive and black for negative is a

useful convention that greatly simplifies

troubleshooting when things go wrong.

For the more complex circuits later this

semester this will be invaluable.

PHY108 Lab 2 Electrical Fields and Potential

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2. CALCULATING THE ELECTRIC FIELD BETWEEN THE TWO BARS

2.1 MEASURING VOLTAGES

An array of 13 by 7 dots (actually ‘+’ signs, but we’ll call them dots) has been printed on the paper in the area between the bars, as shown in the photo. You will measure voltages by touching the handheld probe to dots between the two bars, while keeping the fixed probe on the negative bar. Before you start, hold the handheld probe on any dot closest to the positive bar and check to see if the voltage displayed on the voltmeter is good (steady and non-zero). If it fluctuates or is zero, adjust the fixed probe to get a good voltage display.

2.2 MEASUREMENT

Move the handheld probe to leftmost upper dot (near the positive bar). The voltage at that point will be displayed on the multimeter, and should be greater than 0 but less than 5 V.

2.3 SET UP EXCEL FILE

Since you have done much of the following in the prelab, you can just open your prelab Excel file and use it as the basis for your lab data. You will be replacing the data values used in your prelab with the values you measure now.

2.3.1 The first voltage value You are going to enter voltages corresponding to the 13×7 array of silver dots into a corresponding pattern of cells in Excel similar to the example shown in the prelab. The top left voltage (which is the one you are reading right now) will go into the top left cell in your data, and the bottom left voltage will go into the bottom left cell.

2.3.2 The first row Move the handheld probe one dot to the right and enter the value into the next cell to the right. Repeat eleven more times to finish the row.

2.3.3 The average and standard deviation

To the right of your voltage values, calculate the average and standard deviation for the first row. (These statistics give a profile of the consistency of your equipotential lines.)

This is the positive (red) bar, which is at about 5 V.

This is the negative (black) bar, which is at 0 V. Your fixed probe will always be touching this terminal.

You’ll measure voltages by touching the handheld probe to the 91 dots between the two bars.

To compute the average of a set of data, use the

“average” function in Excel

To compute the standard deviation of a row of

data, use the “stdevp” function in Excel

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2.3.4 Voltages for the remaining rows Repeat 2.3.2 and 2.3.3 for the remaining six rows, so you have measured and recorded a total of 91 different voltage values. To save time you can just copy and paste the average and standard deviation formulas down alongside each row. When complete, your data should resemble the table in the prelab (but obviously your exact values will differ).

2.3.5 The E-Field between the first two rows The magnitude of the electric field is

V

E x

where x is the distance between the rows, and V is the voltage change between the rows. Using the

ruler, measure x and (in your spreadsheet) calculate the electric field between the first and second rows (putting your result to the right of the average and standard deviation, stating from the second row).

2.3.6 The E-field for the rest of the pairs of rows. Copy the formula down to find the electric field between each of the rows.

2.3.7 Average field and uncertainty Use Excel to calculate the average and standard deviation of the electric field. Compute the percentage uncertainty (standard deviation/average x 100%) of your electric field value. In your report, you will be discussing your electric field results and comparing them to the estimate you obtained earlier.

3. MAKE A SURFACE PLOT A surface plot shows the areas of constant potential. Assuming you have done the lab correctly the equipotential lines should be very clear on your plot.

3.1 ADD A SURFACE PLOT Using your voltage data, create a surface plot with each value of the voltage range displayed in a different color. Label the chart. If you have time (and you should) you can individually adjust the colors of the plot to give it a smooth gradient.

3.2 FIX THE DISPLAY

Print your data and your surface plot. On top of the plot, hand draw arrows showing the direction of the electric field. (If you are unsure how to determine field direction, check the theory section of this lab.)

To make a Surface plot: Highlight the data. From the ‘Insert chart’ option in Excel choose a ‘contour’ plot. This may be in the ‘surface charts’ category depending on how Excel is configured.

Example surface plot. The black arrows show the electric field direction and are perpendicular to the equipotential lines. Make sure your (hand drawn) arrows are pointing in the correct direction, which may not be the same as this example depending on how Excel plots your data.

Series1

Series2

Series3

Series4

Series5

Series6

Series7

1 2 3 4 5 6 7 8 9 10 11 12 13

4.5-5.0

4.0-4.5

3.5-4.0

3.0-3.5

2.5-3.0

2.0-2.5

1.5-2.0

1.0-1.5

0.5-1.0

0.0-0.5

To copy values, click on the cell you wish to copy,

type ctrl-c, highlight the cells you wish to copy

into, then type ctrl-v.

PHY108 Lab 2 Electrical Fields and Potential

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LAB REPORT QUESTIONS

Each question needs to be answered in a separate paragraph in the results section of your report. If you haven’t yet, you should read the lab report guide on Blackboard before starting your report.

1. What was the potential difference between your positive and negative bars? What was the distance between the two bars? Based on these values, what was your estimate of the electric field between the two bars?

2. Based on the analysis of your 91 voltage values, what was your average electric field value, and the percent uncertainty of your field? How does your result compare to the electric field value you estimated based just on the positive and negative bar voltages? Which of the two would you say is more accurate and why?

3. How consistent were the voltages of your equipotential lines? Would you say the electric field between the positive and negative bars is constant? If so, what is the magnitude and direction of the field? If not, why not?

PRE-LAB QUESTIONS Before starting this prelab you should read the lab carefully. You are essentially doing the entire lab here using simulated voltage values, and if you understand this prelab and do it correctly the actual lab will run very smoothly when you do it.

Use this example voltage data to complete this prelab:

Follow these steps exactly to complete this prelab: a) Enter the data into an Excel spreadsheet arranged exactly as it is above. b) Calculate the average and standard deviation of each row. c) Calculate the electric fields between each row, as well as the average value and uncertainty of the

field, as described in sections 2.3.6 and 2.3.7. Assume the spacing between rows is 1 cm. d) Use Excel to create a surface plot, as described in part 3 of the lab. e) On one page, print your entire spreadsheet – including the surface plot – and hand-draw arrows

showing the electric field direction over the surface plot. You will be handing in this printout as your prelab. (You are advised to take the time to make sure your spreadsheet is well formatted and correctly labeled before printing it out.)

Your prelab must be printed and handed in, but you should keep a copy of the Excel files so you can use them again during the lab to speed up sections 2 and 3.

4.47 4.55 4.55 4.37 4.15 4.34 4.30 4.44 4.68 4.81 4.72 4.51 4.44

3.81 3.84 3.85 3.77 3.80 3.83 3.77 3.82 3.77 3.85 3.82 3.83 3.78

3.16 3.14 3.10 3.17 3.14 3.14 3.14 3.16 3.19 3.11 3.19 3.15 3.18

2.41 2.40 2.46 2.47 2.44 2.46 2.41 2.42 2.47 2.46 2.46 2.48 2.42

1.73 1.71 1.75 1.74 1.73 1.72 1.78 1.73 1.76 1.73 1.75 1.75 1.79

1.06 1.10 1.13 1.04 1.14 1.05 1.06 1.11 1.13 1.13 1.06 1.08 1.07

0.49 0.49 0.47 0.49 0.49 0.49 0.42 0.44 0.50 0.50 0.45 0.50 0.50