Electric Fields and Equipotentials

Transcription

1 OBJECTIVE Electric Fields and Equipotentials To study and describe the two-dimensional electric field. To map the location of the equipotential surfaces around charged electrodes. To study the relationship between the electric field and the equipotential surface. To interpret the relationship between conductor shape and distance from a charge to the electric field and equipotentials. INTRODUCTION Positive and negative charges exhibit a region of force about their surfaces. This region of force is designated by the electric field of that particular charge. By using a positive test charge, the region around each charge can be tested. The force on the test charge designates the electric force exerted by the charged particle. The region of space surrounding the charge and perpendicular to the electric field is denoted as an equipotential surface (surfaces of equal potential difference, emanating from the surface of the charge). In addition to being perpendicular to the electric field, the equipotentials are parallel to the surface of the conducting surface, thereby, taking the shape of the conductor. The electric field lines and equipotential surfaces will be studied with the aid of conducting electrodes, power supplies, and electrical test electrodes. APPARATUS Electric Field computer program Computer with printing capability THEORY As previously stated, a positive test charge placed in the electric field region around a charged particle will experience an electric force due to their mutual interactions. The relationship between the electric field E [N/C], electric force F [N] and the positive test charge q [C] is given by: E = F q Equation 1 Electric Fields and Equipotentials - Page 1

2 The magnitude of the electric force between the test charge and the charged particle is given by: F = F k k q q 1 r 2 q 1 2 E = = = 2 E = q Equation 2 Where, k [N m 2 /C 2 ] is Coulomb's constant, q1 [C] & q2 [C] are the two interacting charges, and r [m] is the distance between the two interacting charges. Combining Equations 1 & 2 gives the magnitude of the electric field at a distance r from a charged particle: r q q Equation 3 Note: The magnitude of the electric field is proportional to the magnitude of the charge present and inversely proportional to the square of the distance from the charge. Thus, the farther away from the charge you test, the weaker the electric field will become. Likewise, the density of these field lines will increase as the electric field strength increases. Experimentally, the magnitude of the average electric field between two points, separated by a displacement )x [m], can be calculated from the difference in the electric potential v [V] between the two points. This relation is given below. ( v2 v1 ) ( x x ) 2 1 Equation 4 2 k r q 2 v = x Where, v2 [V] is the potential difference (voltage) at point #2, v1 [V] is the potential difference (voltage) at point #1, x2 [m] is the position of potential difference #2, and x1 [m] is the position of potential difference #1. The negative sign in front of the potential difference quantity relates to the electric field. If v2 - v1 > 0, the electric field is in the opposite direction of the displacement, x2 - x1. If v2 - v1 < 0, the electric field is in the same direction as the displacement, x2 - x1. Electric Fields and Equipotentials - Page 2

3 Consider two conducting electrodes. One electrode is connected to the positive side of a DC power supply while the other electrode is connected to the negative side of the power supply. The current flow between the electrodes will be in the same direction as the electric field at any point. Additionally, a positive test charge released in the vicinity of the two electrodes will feel an attractive force toward the negative electrode and an equally repulsive force away from the positive electrode. Relating the two previous statements leads to the relation that the current flow is from the positive electrode to the negative electrode. Graphically, the electric field of the positive charge and the electric dipole (a positive and negative charge pair of equal magnitude) configuration is illustrated in Figure 1. Figure 1 As the test charge is caused to move, due to the presence of a force, we say that work is done to move the test charge. The work done W [J] to move any object a distance d [m] under the presence of a force F [N] is given by: W W = F d Equation 5 In general, the work done in moving a charged particle between two points (point #2 and point #1) in an electric field, is defined by the potential difference )v [V] between the two points, given by: = q ( v ) 2 v1 Equation 6 We generally assign the negative electrode to be the one at zero potential for the sake of measurements. Therefore, the largest potential difference that can be experimentally recorded is that measured directly between the two electrodes, as you would expect. However, other potentials (voltages) can be measured in the space between the "zero potential electrode" and the "high voltage electrode." There are an infinite number of measurable potential differences between zero and the maximum. Each of these potential locations is the same distance from the "zero potential electrode" in all directions, thus, defining a constant voltage surface that surrounds the electrode. We call these constant potential surfaces equipotential surfaces. They extend in a threedimensional pattern around the electrodes. However, experimentally we use a twodimensional mapping board and can measure only the lines of constant potential that exist between the two electrodes. Electric Fields and Equipotentials - Page 3

4 Figure 2 shows the equipotential lines for a round positive charged electrode. Figure 2 (1) Lines of E are always perpendicular to the potential surfaces. (2) Lines of E do not intersect. (3) Density of lines is proportional to E strength. If the electric field is not perpendicular to the equipotential surfaces, then the component of the force along the surface would indicate that work was being done to move the charge along the equipotential surface. It can be shown that this is mathematically impossible based on Equation 6 and the assumption that point #2 = point #1: W = q ( v ) 2 v1 = q (0) = 0 Equation 7 Therefore, there is no work done along an equipotential surface. Experimentally, we will study the electric field and equipotential surfaces (lines) around three different charged electrode configurations. Electric Fields and Equipotentials - Page 4

5 THE SOFTWARE Electric Field is an interactive program that lets you select charges, arrange them how you want, and then view the electric fields & electric potentials around them. There is a link for the program on the desktop of your laboratory computer. Once the program is running you will see the following menu bar at the top of the screen: New Grid Field Save Grid Field Open Saved File Show Electric Field Lines Show Equipotential Surfaces Add a New Charge Move Existing Charge Erase Existing Charge Print Preview Print Electric Field & Electric Potential Meter Duplicate Existing Charge Tools Options Menu Draw Single Equipotential Surface Tool Clear All Electric Field Lines and Equipotential Surfaces a) To get started, select the "Tools Option Menu." Once the menu comes up, note there are five option tabs at the top, click on the "Grid" tab. In the "Charge Alignment" box, check the box entitled "Align charges To Grid"; click "Apply" at the bottom of the options box. Electric Fields and Equipotentials - Page 5

6 b) Next, click on the "Color" tab; you will see three slider switches. Slide the "Decreasing visibility of field lines" and the "Decreasing visibility of equipotential lines" completely to the "Slow" setting (as far left as they will slide). Slide the "Grid color" most of the way to the "Light" setting...but NOT all the way; click "Apply" at the bottom of the options box. c) Finally, click on the "Charge" tab. In the "Charge Range" box, click the button entitled "MilliCoulomb"; click "Apply" at the bottom of the options box and then click "OK" at the bottom of the options box. d) You are now ready to start using the program! Electric Fields and Equipotentials - Page 6

7 EXPERIMENTAL PROCEDURE Getting Used to the Software a) Click on the "New Grid Field" menu button and create a new 700x600 grid field. Be sure to expand the new window to full screen or your illustrations will not be centered correctly! b) Click on the "Add a New Charge" button. Move the cursor down to the middle of the screen and click the mouse. A box will appear asking you enter the type (positive...shows up in RED or negative...shows up in BLUE) and the magnitude of the charge you are interested in creating. Go ahead and create any charge you'd like to experiment with for the time being. While we earlier changed the unit default to Coulombs, you will not see the actual units on your charge! Things to Notice The 700x600 grid is divided into "Grid Blocks." Each Grid Block is 10x10 units. Thus the 700x600 grid is made up of 70x60 actual Grid Blocks. The exact 700x600 location is always indicated by number pair in the lower left of the screen; based on the position of the cursor's tip. These are measured starting from the upper leftmost corner of the grid. For the image shown, note that the cursor's tip is located at 207 units from the left and 27 units from the top. c) Click the "Electric Field & Electric Potential Meter" button; notice that the cursor now changes to a plus-sign and a black & yellow vector arrow (the center of the + sign is the intersection of the black & yellow vector arrow). Now move the cursor somewhere around the charge; notice that the vector arrow changes to show the magnitude (length) and direction of a electric field vector for your charge. ALL measurements made with this meter tool are referenced from the center of the charge to the "+"; NOT the arrowhead!! Electric Fields and Equipotentials - Page 7

8 d) If you click the mouse at any location you will get an "On This Point" box to pop up that indicates the electric field strength and electric potential value for the location. Drag your vector arrow around a few more places around the charge (near, far) and investigate the relative magnitude of the electric field and electric potential at those locations. What should the magnitude of the field be at twice any original distance away from the charge? Why? Try it; were you right? ** From here, play around with as many other features of the software as you wish until you feel comfortable enough to move to the next part ** Electric Field Lines & Equipotential Surfaces of a Point Charge a) Click on the "New Grid Field" menu button and create a new 700x600 grid field. b) Click on the "Add a New Charge" button and create a positive charge (any size) near the center of the grid. c) According to Coulomb s Law, the magnitude of the electric field of a point charge Q is given by E = k Q/r 2. The software assumes that the distance r is measured in units of grid boxes (again, starting at the center of the charge to the "+" sign), and that the charge Q does not have any units (it is just a number). Now find at the magnitude of the electric field "strength" at some location near your charge (I'd suggest horizontally or vertically); recall that you can use the "Electric Field & Electric Potential Meter" to find this value. Record the values of Q and r for your point charge, and the electric field strength, in the provided data table in order to determine the value of the constant k used by the software. **DO NOT expect to find your answer to be 8.99 x 10 9 Nm 2 /C 2. You are calculating the internal value of "k" that the software uses; not the actual Coulomb's Constant** Repeat this calculation for at least two different additional distances around the SAME charge. d) Using the "Show Electric Field Lines" button, fill in the electric field lines for this charge. e) Using the "Show Equipotential Surfaces" button, fill in the equipotential surfaces for this charge. ** In the "File" menu option, select "Export to image...". Name the file, give it the extension ".bmp" (actually type this extension at the end of the filename), and save it to the desktop on the computer. Find and double-click on this saved file to print this final illustration for your laboratory report ** Electric Fields and Equipotentials - Page 8

9 DO NOT erase the current illustration in the program as you're going to use it to collect more data shortly! Answer the following two questions on the back of this printout: Is the configuration of the electric field lines what you expected them to look like? Explain. Is the configuration of the equipotential surfaces what you expected them to look like? Explain. f) Using the current illustration for your positive charge, collect a series of at least five potential AND electric field values at different known distances from your charge and create a graph using Logger Pro of the electric potential and the electric field (each on the y-axis) vs. distance from the charge (the x-axis). You re going to have a SINGLE graph showing plots of V vs. r and E vs. r. o Once in Logger Pro, go to the data menu, select new manual column, and add a second y-column. At this point, three data columns will appear on the left. Clicking on the vertical column labels will allow you to change the axis name for the data column as well as on the graph. Additionally, click on the Y vertical label ON THE GRAPH and select all of the above. This will place both y-columns on the graph together. ** Print a copy of this graph for your laboratory report ** g) On the printout of the graph, connect the respective dots (they may or may not be linear) and on the back of the printout, explain what the significance of the shape of each of the graph lines is; i.e. what do they represent? Electric Fields and Equipotentials - Page 9

10 Electric Field Lines & Equipotential Surfaces of a Dipole Two Identical Positive Point Charges a) Click on the "New Grid Field" menu button and create a new 700x600 grid field. b) Near the center of the grid, place two equal positive point charges a distance of approximately 30 grid blocks apart. c) Using the "Show Electric Field Lines" button, fill in the electric field lines for this configuration. d) Using the "Show Equipotential Surfaces" button, fill in the equipotential surfaces for this charge. You will also want to use the "Single Equipotential Surface" tool to add more surfaces near AND around the charges to enhance the detail there. ** In the "File" menu option, select "Export to image...". Name the file, give it the extension ".bmp" (again, actually type this extension at the end of the filename), and save it to the desktop on the computer. Find and double-click on this saved file to print this final illustration for your laboratory report ** Considering the region between and around the charges shown, answer the following questions on the back of this printout: Is the configuration of the electric field lines what you expected them to look like? Explain. Based on the illustration and using your "Electric Field & Electric Potential Meter," where, aside from on top of one of the charges, is the field strongest? How can you tell? Is this expected? Explain. Where, aside from on top of one of the charges, is the field weakest? How can you tell? Is this expected? Explain. No numbers are needed here, only a visual representation of the electric field strength vector! Label these points on your printout! Is the configuration of the equipotential surfaces what you expected them to look like? Explain. Electric Fields and Equipotentials - Page 10

11 Two Identical Opposite Point Charges a) Click on the "New Grid Field" menu button and create a new 700x600 grid field. b) Near the center of the grid, place two equal but opposite point charges a distance of EXACTLY 40 grid blocks apart. L = 20 grid blocks DO NOT add the electric field vectors or potential lines to this right now leave it blank!! ** In the "File" menu option, select "Export to image...". Name the file, give it the extension ".bmp" (again, actually type this extension at the end of the filename), and save it to the desktop on the computer. Find and double-click on this saved file to print this final illustration for your laboratory report and label the exact position of point P on this printout. ** c) Using this charge printout in which you labeled point P, and the value of "k" you determined earlier, calculate the magnitude of the electric field due to the positive point charge acting on the location P. Remember that the units for the magnitude of the charge are MilliCoulombs. You will need to use the grid blocks for your distance and some trigonometry to calculate the distance to point P from the charge. Show your measurements and the electric field vectors at point P on the printout of the illustration along with your calculations on the back of the same printout. d) Using the same methodology, calculate the magnitude of the electric field due to the negative point charge acting on the location P. e) Draw the magnitude and direction of the two electric field vectors at P on the printout. On the back of the printout, calculate the magnitude and direction of net electric field vector at point P. If you illustrated the diagram correctly, you will note that the two electric field vectors should wind up at 90 o to each other. As such, vector components are not necessary...you can just apply Pythagorean Theorem directly! Indicate this magnitude and direction on the illustration printout. f) Now, referring back to the on-screen configuration, use the "Electric Field & Electric Potential Meter" button, verify the numerical values you calculated for the magnitude (quantitatively) and direction (qualitatively) of the electric field at P. Electric Fields and Equipotentials - Page 11

12 Please indicate this actual value on the back of the printout along with your calculated value. Make a statement regarding the actual qualitative direction of the electric field vector based on its predicted direction. g) Using the "Show Electric Field Lines" button, fill in the electric field lines for this configuration. h) Using the "Show Equipotential Surfaces" button, fill in the equipotential surfaces for this charge. ** In the "File" menu option, select "Export to image...". Name the file, give it the extension ".bmp" (again, actually type this extension at the end of the filename), and save it to the desktop on the computer. Find and double-click on this saved file to print this final illustration for your laboratory report ** Answer the following questions on the back of this printout: Is the configuration of the electric field lines what you expected them to look like? Explain. Based on the illustration, where is the field strongest? How can you tell? Where is the field weakest? Is the configuration of the equipotential surfaces what you expected them to look like? Explain. Two Like Point Charges of Different Magnitude a) Click on the "New Grid Field" menu button and create a new 700x600 grid field. b) Near the center of the grid, place two like-signed point charges, one with four times the magnitude of the other, a distance apart of approximately 40 grid blocks apart. c) Using the "Show Electric Field Lines" button, fill in the electric field lines for this configuration. If your charges are too close together or too far apart for you to clearly see the electric field line pattern, clear the screen and move the charges to new locations. d) Using the "Show Equipotential Surfaces" button, fill in the equipotential surfaces for this charge. You will also want to use the "Single Equipotential Surface" tool to add more surfaces near AND around the charges to enhance the detail there. Electric Fields and Equipotentials - Page 12

13 ** In the "File" menu option, select "Export to image...". Name the file, give it the extension ".bmp" (again, actually type this extension at the end of the filename), and save it to the desktop on the computer. Find and double-click on this saved file to print this final illustration for your laboratory report ** Answer the following questions on the back of this printout: Is the configuration of the electric field lines what you expected them to look like? Explain. Based on the illustration, where is the field strongest? How can you tell? Where is the field weakest? Is the configuration of the equipotential surfaces what you expected them to look like? Explain. Two Unlike Point Charges of Different Magnitude a) Click on the "New Grid Field" menu button and create a new 700x600 grid field. b) Near the center of the grid, place two unlike-signed point charges, one with four times the magnitude of the other, a distance apart of approximately 40 grid blocks apart. c) Using the "Show Electric Field Lines" button, fill in the electric field lines for this configuration. If your charges are too close together or too far apart for you to clearly see the electric field line pattern, clear the screen and move the charges to new locations. d) Using the "Show Equipotential Surfaces" button, fill in the equipotential surfaces for this charge. ** In the "File" menu option, select "Export to image...". Name the file, give it the extension ".bmp" (again, actually type this extension at the end of the filename), and save it to the desktop on the computer. Find and double-click on this saved file to print this final illustration for your laboratory report ** Answer the following questions on the back of this printout: Is the configuration of the electric field lines what you expected them to look like? Explain. Based on the illustration, where is the field strongest? How can you tell? Where is the field weakest? Is the configuration of the equipotential surfaces what you expected them to look like? Explain. Electric Fields and Equipotentials - Page 13

14 Electric Field Lines & Equipotential Surfaces of Multiple Charge Configurations 8-Charge Square a) Click on the "New Grid Field" menu button and create a new 700x600 grid field. b) Using the "Add a New Charge" button, and as accurately as possible, create the configuration shown below near the center of the screen: L = 20 grid blocks c) Using the "Show Electric Field Lines" button, fill in the electric field lines for this configuration. If your charges are too close together or too far apart for you to clearly see the electric field line pattern, clear the screen and move the charges to new locations. d) Using the "Show Equipotential Surfaces" button, fill in the equipotential surfaces for this charge. For any underrepresented regions, you will also want to use the "Single Equipotential Surface" tool to add more surfaces near AND around the charges to enhance the detail there. ** In the "File" menu option, select "Export to image...". Name the file, give it the extension ".bmp" (again, actually type this extension at the end of the filename), and save it to the desktop on the computer. Find and double-click on this saved file to print this final illustration for your laboratory report ** Electric Fields and Equipotentials - Page 14

15 Answer the following questions on the back of this printout: Is the configuration of the electric field lines what you expected them to look like? Explain. Based on the illustration, where is the field strongest? How can you tell? Where is the field weakest? Is the configuration of the equipotential surfaces what you expected them to look like? Explain. Without resorting to the software program discuss how you would answer the following (think about any symmetry you see in the illustration): o What would be the net direction of the instantaneous force experienced by an electron if it were released from rest from positions A, B and C (respectively)? Point Charge & A Like Line Charge a) Click on the "New Grid Field" menu button and create a new 700x600 grid field. b) Using the "Add a New Charge" button, place a point charge to the right of the center of the screen. Then, create a VERTICAL line of at least 10 charges approximately 40 grid blocks to the left of that charge that all have the same sign and magnitude as your point charge. The point charge should line up with the center of the line Be sure to place the charges close enough together to simulate a line; they should be touching but not overlapping. c) Using the "Show Electric Field Lines" button, fill in the electric field lines for this configuration. If your charges are too close together or too far apart for you to clearly see the electric field line pattern, clear the screen and move the charges to new locations. Electric Fields and Equipotentials - Page 15

16 d) Using the "Show Equipotential Surfaces" button, fill in the equipotential surfaces for this charge. You will also want to use the "Single Equipotential Surface" tool to add more surfaces near AND around the charges to enhance the detail there. ** In the "File" menu option, select "Export to image...". Name the file, give it the extension ".bmp" (again, actually type this extension at the end of the filename), and save it to the desktop on the computer. Find and double-click on this saved file to print this final illustration for your laboratory report ** Answer the following questions on the back of this printout: Is the configuration of the electric field lines what you expected them to look like? Explain. Based on the illustration, where is the field strongest? How can you tell? Where is the field weakest? Is the configuration of the equipotential surfaces what you expected them to look like? Explain. Point Charge & An Unlike Line Charge a) Click on the "New Grid Field" menu button and create a new 700x600 grid field. b) Using the "Add a New Charge" button, place a point charge to the right of the center of the screen. Then, create a VERTICAL line of at least 10 charges approximately 40 grid blocks to the left of that charge that all have the opposite sign but the same magnitude as your point charge. Again, the point charge should line up with the center of the line and be placed close enough together to simulate a line. Electric Fields and Equipotentials - Page 16

17 c) Using the "Show Electric Field Lines" button, fill in the electric field lines for this configuration. If your charges are too close together or too far apart for you to clearly see the electric field line pattern, clear the screen and move the charges to new locations. d) Using the "Show Equipotential Surfaces" button, fill in the equipotential surfaces for this charge. ** In the "File" menu option, select "Export to image...". Name the file, give it the extension ".bmp" (again, actually type this extension at the end of the filename), and save it to the desktop on the computer. Find and double-click on this saved file to print this final illustration for your laboratory report ** Answer the following questions on the back of this printout: Is the configuration of the electric field lines what you expected them to look like? Explain. Based on the illustration, where is the field strongest? How can you tell? Where is the field weakest? Is the configuration of the equipotential surfaces what you expected them to look like? Explain. COVER PAGE REPORT ITEMS (To be submitted and stapled in the order indicated below) (-5 points if this is not done properly) Completed Laboratory Responsibility and Cover Sheet DATA (worth up to 50 points) Printouts of the EIGHT required charge illustrations Coulomb's constant calculation data (including the Excel spreadsheet) Dipole Electric Field P (Illustration & Calculations) DATA ANALYSIS (worth up to 20 points) Coulomb's Constant Calculation Dipole Electric Field Calculation for P Electric Fields and Equipotentials - Page 17

18 GRAPHS (worth up to 5 points) Electric Potential & Electric Field vs. Distance from Charge (with data table included) GRAPH ANALYSIS (worth up to 5 points) For your Electric Potential & Electric Field vs. Distance graph, describe the nature of the relationship between electric potential and distance from the charge. CONCLUSION (worth up to 0 points) NOT required for this laboratory QUESTIONS (worth up to 10 points) DO NOT forget to include the answer to the ONE question that was asked within the experimental procedure. 1) In your illustrations, the directions of the electric field lines are indicated, but not the direction of the corresponding equipotential surface lines. Was this an oversight on the part of the software? 2) What can you say, in general, about the orientation of the electric field and that of its equipotential? Electric Fields and Equipotentials - Page 18