Lab 4 - Electric Potential - W24

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Apr 3, 2024

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Toronto Metropolitan University - PCS125 Distribution, sale or profit without the authorization of the owner of this material is expressly prohibited Electric Potential Physics Topics If necessary, review the following topics and relevant textbook sections from Serway / Jewett “Physics for Scientists and Engineers”, 10th Ed. • Electric Fields and Field Lines (Serway Sec. 22.4, 22.5) • Electric field from a plane of charge (Serway Example 23.8) • Electric Potential (Serway 24.1 - 24.4) Introduction Review of Fields Concept We have seen that for gravity and electromagnetism, it is helpful to understand the field as a way to quantify how masses or charges exert forces on each other without touching. Masses/charges feel a force due to the gravitational/electric field. The field is the force on a mass m (or a charge q ) divided by that mass (or charge), ⃗g = ⃗ F g /m (1) ⃗ E = ⃗ F E /q (2) Another way of saying this is that the gravitational field is the “gravitational force per mass”. The electric field is the“electric force per charge”. Introduction To Electric Potential There is another quantity called electric potential which is also defined on a “per charge” basis. Electric potential is defined as “potential energy per charge”. In physics, only changes in potential energy matter so we define the potential difference , or voltage difference between two points as the change in potential energy of a charge moved between those two points divided by the value of its charge ∆ V = ∆ U/q (3) Note, the units of electric potential are energy per charge or Joules/Coulomb. The name for this unit is the Volt . (1V = 1 J/C). Page 1 of 8

Toronto Metropolitan University - PCS125 Distribution, sale or profit without the authorization of the owner of this material is expressly prohibited A Gravitational Analogy The closest analogy to potential for gravity is “height”. Lifting up a 1kg mass by 1.0m will increase its potential energy by 9.8J. The change in potential energy per mass is 9.8J/1kg = 9.8J/kg. Lifting up a 2kg mass by 1.0m will increase its potential energy by 2 × 9 . 8J = 19.6J. The change in potential energy per kg is 19.6 J/(2 kg) = 9.8 J/kg. Note that even though the heavier mass has a higher potential energy than the lighter mass, because they are at the same height they have the same gravitational potential . We often represent heights on 2D maps with lines of constant elevation (a topographic map). Figure 1 - A Topographic Map. The top part of the image is the actual 3D landscape, the bottom topographic map is a projection of the image with lines representing surfaces of constant elevation. (Source: https://www.greenbelly.co/pages/contour-lines) We can do the same for electricity. Moving a charge +2C from point A (low potential) to point B (high potential) will require more energy than moving a charge +1C from point A to point B. Thus, the larger charge will have a bigger change in potential energy than the smaller charge, but they will both have the same change in electric potential since they start and end at the same points. When drawing a map for electricity, in this case, the lines are not constant height, but lines of constant voltage (electric potential). We call these lines “equipotential lines”. In this lab, you will use metal electrodes (which act like “charges” when they are hooked to a power supply) and conductive paper to try to map the electric potential between two “point charges” and between two metal bars which act like parallel plates. You will also investigate the potential difference as you move along the center line between the charges. Point charges In 3-dimensions, the electric potential V at a distance r from a point charge Q is given by V ( r ) = k e Q/r (4) If there are multiple charges nearby, the total potential a point P is found by summing the individual potentials V tot = k e Q 1 r 1 + k e Q 2 r 2 where r 1 is the distance between P and Q 1 and r 1 Page 2 of 8

Toronto Metropolitan University - PCS125 Distribution, sale or profit without the authorization of the owner of this material is expressly prohibited is the distance between P and r 2 . However, in this lab, we will be using conductive paper which is a 2-dimensional surface. On such a surface the electric potential at a point P away from a “point” charge is given by V ( r ) = 2 KQ ln( r 0 /r ) (5) where K is a constant, and r 0 is also a constant reference point (note that the potential at position r 0 = 0). To find the total potential, you can still add up the potential from two point charges V tot = V 1 + V 2 + ... . “Infinite” Plates In three dimensions, the field between two large plates of charge is approximately constant. When confined to a 2-dimensional plane, the field between two long lines of charge will also be approximiately constant. In such situations, the potential difference between the bar at lower potential and a point P a distance ∆ x away from it is ∆ V = | ⃗ E | ∆ x (6) where | E | is the magnitude of the electric field in between the bars/plates. Pre-Lab Questions Please complete the following questions prior to coming to lab. They will help you prepare for both the lab and the pre-lab quiz (Found on D2L). 1.) Suppose you are in the vicinity of two point charges in 3 dimensions + Q and − Q . The negative charge has ( x, y ) coordinates (0 , 0) while the + Q charge has coordinates (0 , d ). Using equation (4), write the total electric potential at a point in between the charges (along a line connecting them), a distance r away from the negative charge. 2.) Suppose you are in the vicinity of two point charges + Q and − Q in 2 dimensions . The negative charge has ( x, y ) coordinates (0 , 0) while the + Q charge has coordinates (0 , d ). Using equation (5), write the total electric potential at a point in between the charges (along a line connecting them), a distance r away from the negative charge. 3.) Simplify your result from the previous question using the rules for logs ln( AB ) = ln A + ln B and ln( A/B ) = ln( A ) − ln( B ). 4.) If you took data of V total and r , how could you plot your data so as to form a straight line? What would be the slope of this line? 5.) Suppose you moved a charge of +1C from point A (on a 10V equipotential line) to point B (also on a 10V equipotential line). (a) How much would its electric potential energy change? Explain. Page 3 of 8

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Toronto Metropolitan University - PCS125 Distribution, sale or profit without the authorization of the owner of this material is expressly prohibited (b) How much would its electric potential change? Explain. (c) How (if at all) would the answers to the previous two questions cange if the charge had magnitude 2C? Explain 6.) Suppose you moved a charge of +3C from point A on a 10V equipotential line to point B on a 20V equipotential line. (a) How much would its electric potential change? Give a value and explain whether it increased or decreased. (b) How much would its electric potential energy change? Give a value and explain whether it increased or decreased. (c) How (if at all) would the answers to the previous two questions change if the charge had magnitude 2C? Explain. . Apparatus • PASCO Conductive paper • Power Supply • Cylindrical magnet posts (to be used as point charges) • Metal bars (with embedded magnets to ensure good contact) • Banana to banana cables • Magnetic board (on which the conduc- tive paper will be fixed) • Digital Multimeter (DMM) Procedure Preliminary Setup and Exploration 1.) Connect the banana-banana cables to your digital multimeter. Turn the meter on and set it to measure DC voltage (20V). Connect the ground wire (COM) from your DMM to the ground plug on the power supply. 2.) Connect two banana cables to the power supply, red to the positive terminal and black to the negative terminal. You are effectively stacking the black cable on top of the negative banana cable from the DMM. Part I - “Point-like” electrodes Plotting equipotential Lines 1.) Place two of the cylindrical magnet posts on the conductive paper and secure in place they will act as your electrodes. Your electrodes should be at least 20 cm apart, lying on the same horizontal line. Record the ( x, y ) coordinates of each magnet. Page 4 of 8

Toronto Metropolitan University - PCS125 Distribution, sale or profit without the authorization of the owner of this material is expressly prohibited 2.) Using the banana cables you connected to the power supply, connect the (+) side of the power supply to the right magnet, and the ( − ) side of the power supply to the left magnet. The magnets are now acting like point charges. Important: we are using magnets as a simple way to attach a conductive piece of metal to the conductive paper, but this lab does not have anything to do with magnetic fields. We are measuring electric potential. The lab would still work with non-magnetic metals attached to the paper in a different way . 3.) Set the voltage on your power supply to a value in the range of 4-5 V. 4.) Touch the red lead from the DMM to the positive (+) metal electrode. The voltage reading should match your power supply setting. If this is not the case, make sure all contacts are secure and try again. If your readings are significantly different than this, ask your lab TA for help. 5.) Touch the red lead from the DMM to the ( − ) metal electrode. Record your observation. 6.) Download and open the Excel Spreadsheet file Electric Potential Worksheet.xlsx . The file is available on D2L. 7.) Immediately save the file with a new file name which includes your name or group number, for example: PCS125LabPotentialSmith Group12.xlsx 8.) Touch the red lead from the DMM to a point directly below the ( − ) electrode. Record the voltage at this point and enter it into the orange box on the spreadsheet. Enter the (x,y) coordinates of the point in green area below. 9.) By moving to the right and upward, find another point which has the same electric potential as you found in the previous step. 10.) Do your best to estimate the coordinates of the point you found if it does not exactly lie on a “dot” on the conductive paper. Enter the coordinates into the the same column as previously 11.) By systematically moving around the electrode in a counterclockwise fashion, repeat steps (9) - (10), recording the coordinates of at least 5 points which have the same potential as the original point in the Excel spreadsheet. As you enter each of the coordinates into the spreadsheet you should see a line being plotted on a graph. Try to get a good range of points so as to make clear what the shape of the equipotential line is. 12.) Go back to a point below the ( − ) electrode and find a different starting point which differs from your previous voltage by at least 0.5V. You can find this new point by moving right on the sheet (towards the positive electrode). Repeat steps (8) - (10) for this next equipotential line. Page 5 of 8

Toronto Metropolitan University - PCS125 Distribution, sale or profit without the authorization of the owner of this material is expressly prohibited 13.) Continue this process until you have plotted at least 3 equipotential lines. Try to keep the interval between your equipotential lines constant (for example, every 1V). 14.) Save your excel file after you have finished taking data and upload it to the Lab 4 folder on D2L to ensure that data does not get lost. V ( x ) along center line for point like electrodes 1.) Continue to the next page in your excel file “Point Charges - Center Line”. 2.) Keeping the black lead of the DMM attached to the ground terminal of the power supply, place the red lead of the DMM along the center line between the two electrodes and 1.0 cm to the right of the (-) electrode. 3.) Record the voltage reading, and the distance r from the ( − ) electrode. 4.) Now increase the distance between the read lead and the ( − ) electrode by 1-2 cm. Record the corresponding voltage. Continue taking data until you have a set of data r vs. Voltage. Stop when you are 1cm away from the (+) electrode. 5.) Save your file and re-upload it to D2L. Part II - “Bar-like” electrodes Plotting equipotential lines 1.) Remove the magnet posts from the conductive paper and attach them at the top of the metal sheet (in the area indicated by electrical tape). 2.) Place the two metal bars (oriented vertically) on the conductive paper and firmly secure in place. Your electrodes should be at least 20 cm apart, with their centers on the same horizontal line. Keep the bars the same distance apart as you had your “point charges”. Record the ( x, y ) coordinates of centers of the bars in your notebook or excel file. 3.) Using the same banana cables as Part I, connect the (+) side of the power supply to the right electrode, and the ( − ) side of the power supply to the left electrode. 4.) Touch the red DMM lead to the (+) metal bar/electrode. The voltage reading should match the reading on your power supply. If this is not the case, make sure all contacts are secure and try again. If your readings are still significantly outside of this range, ask your lab TA for help. 5.) Touch the red DMM lead to the ( − ) metal bar/electrode. Record your observation. 6.) Proceed to the third page of the excel file “Bar Charges - Equipotential” Page 6 of 8

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Toronto Metropolitan University - PCS125 Distribution, sale or profit without the authorization of the owner of this material is expressly prohibited 7.) Repeat all steps as you did previously for the “point-like” electrodes for these “bar” electrodes. Plot at least 3 equipotential lines. Save your file after you have finished taking data and make sure to upload it to the lab folder on D2L. V ( x ) along center line for bar electrodes 1.) Proceed to the final page of the excel file “Bar Charges - Center Line”. 2.) Keeping the black lead of the DMM attached to the ground terminal of the power supply, place the red lead of the DMM along the center line between the two bar electrodes and 1.0 cm to the right of the (-) bar electrode. 3.) Record the voltage reading, and the distance r from the ( − ) electrode. 4.) Now increase the distance between the read lead and the ( − ) electrode by 1-2 cm. Record the corresponding voltage. Continue taking data until you have a set of data r vs. Voltage. Stop when you are 1cm away from the (+) electrode. 5.) Save your file after you have finished taking data and make sure to re-upload it to the lab folder on D2L. Make sure that your excel file has been successfully uploaded to D2L before proceeding to Analysis Analysis 1.) On the same graph, plot the V vs. r (distance away from the negative electrode) data for the point-like electrodes and the bar electrodes. 2.) Using your answers to the pre-lab questions, on a different graph , plot your “point like” electrode voltage vs. distance graph in such a way as to form a linear graph. 3.) Fit the graph you made in the previous part with a line. Assuming K has the same nu- merical value as Coulomb’s constant in 3D (but different units) ( K = 9 × 10 9 N · m / C 2 ), how much positive charge Q is on the (+) electrode? 4.) Fit the graph of the “bar” electrodes voltage vs. distance with a line. Determine the approximate value of the electric field between the bars. Wrap Up The following questions are designed to make sure that you understand the physics implica- tions of the experiment and also to extend your knowledge of the physical concepts covered. Your report should answer these questions in the noted section in a seamless manner. 1.) [Theory] Equipotential lines cannot cross. Explain why. Page 7 of 8

Toronto Metropolitan University - PCS125 Distribution, sale or profit without the authorization of the owner of this material is expressly prohibited 2.) [Discussion] Compare your voltage vs. distance graphs for your point electrodes and your bar electrodes. Which is more linear? Does this make sense according to equations (5) and (6)? 3.) [Discussion] Looking at your equipotential lines for the bar electrodes, what would the lines look like if you used bars which were twice as long? What would the lines look like if the bars were extremely (infinitely) long? Explain. 4.) [Discussion] Set up your bar electrodes again and hook them to the power supply as before. Now, using your circular conductor, create a closed “loop” in the middle of the conductive paper. With the black lead of the probe on the ( − ) electrode measure the electric potential at various points inside the ring. Record your observations. Do you have any explanation for what you see? Report Labs will be completed in groups, you will enroll in a group with your lab partner at the beginning of each lab session. Each group will submit a single report through the assignment section on D2L. • Introduction – What is the experiment’s objective? • Theory – You may be able to show a derivation of the physics you’re investigating, or you may want to reference a source that provides a description/equation representing the physics you’re investigating. – You may want to provide graphs that illustrate or predict how you expect the system under study to behave. • Procedure – Explain the systematic steps required to take any measurements. • Results and Calculations – Tabulate your measurements in an organized manner. – Based on your procedure, you should know what your tables – Provide examples of any calculations. • Discussion and Conclusions – Discuss the main observations and outcomes of your experiment. – Summarize any significant conclusions. • References • (Appendices) Page 8 of 8