Electric Fields and Potential

Transcription

1 General Physics Lab 2 Siena College Object Electric Fields and Potential This experiment further explores the electrostatic interaction between charged objects. The concepts of electric field and potential are illustrated primarily through exercises with the EM Field visualization program. Various stations provide additional hands-on demonstrations of electric charge and force. Equipment This manual EMField 6.4 Program Theory Fundamental forces interact via fields. In the case of electrostatics, fields arise from the potential for a source charge to exert a force on a test charge, by virtue of the quantity of source charge and the distance from it. We can re-write Coulomb s Law for point charges in terms of the force F c between a test charge q 0 and an electric field E; FC q 0 E (Eq. 1) where both F and E are vectors. From Coulomb s Law and Eq. 1, the expression for the electric field of a point charge is given by E k q 2 r. (Eq. 2) r As this is a vector quantity, the field associated with a distribution of charges is determined from the trigonometric vector sum of the fields associated with each source charge. The force and field equations are analogous to those describing gravitational interactions. Continuing the analogy, the work W associated with moving a charge q 0 along an electric field line is the scalar product of the Coulomb force with the distance: W Fd q 0 Ed. (Eq. 3) Recall that work is also defined as the change in energy of the test particle. In the case of electrostatics, the particle is not moving and thus the energy is only potential energy, similar to changing the gravitational potential energy by raising or lowering an object. The change in potential energy is thus: PE W q 0 Ed. (Eq.4)

2 We define a scalar quantity to describe the potential for interaction associated only with the source charges, independent of the properties of any test charge. The difference in electric potential, ΔV, is thus defined by dividing the difference in potential energy by the test particle charge, V PE q 0. (Eq. 5) This difference is more commonly known as the voltage between two points in an electric field. The potential and potential energy are closely related but easily confused. The major difference is that potential is associated with just the source independent of the test charge, whereas potential energy involves both source and test charges. NOTE: Eq. 3 and the right-hand equality in Eq. 4 are applicable only when E is constant or the distance d is very small. Eq. 5 and the left-hand equality in Eq. 4 are correct in all cases. At any distance r from a point charge, the potential is given by V k q r (Eq. 6) assuming that V approaches zero as r approaches infinity. The potential energy between two charges is thus PE q 2 V 1 k q 1q 2 r (Eq. 7) Finally, points surrounding a charge at the same potential constitute an equipotential surface. In the case of a point charge, the equipotential surfaces would simply be spheres enclosing the charge at various distances. Such surfaces can be drawn like a topographical map to indicate the strength of the electric field surrounding the source charge. Part I: EM Field Program In this exercise, you will explore the fields and potentials associated with various source charges. Keep in mind that the field lines represent the force that a positive test charge experiences. Electric field - definition in terms of force on a test charge The electric field due to one or more charges is defined in terms of the force produced on a positive separate charge. In the diagram below, a point charge of + 4 units is placed at a grid point as shown. The "X" indicates the location of a positive test charge with charge = +1 unit. 2

3 o o o o o X o +4 o o o o o 1. On the diagram, draw a vector to indicate the force on the test charge when located at point x. Explain briefly how you arrived at the answer. 2. How would your answer change if the test charge was +2 units (instead of 1 unit)? Explain. (Hint: consider Coulomb's law). 3. How would your answer change if the test charge was +3 units (instead of 1 unit)? Explain. 4. For each case considered, determine the ratio of the magnitude of the force felt by the test charge to the size of the test charge. This ratio is (by definition) the magnitude of the electric field due to the point charge. 5. Does the magnitude of the electric field depend on the size of the test charge used to measure it? NOTE: The direction of the electric field is (by definition) the same as the direction of the force on a positive test charge. 3

4 Electric field due to a single point charge 1. A point charge of +4 units is placed at a grid point in the diagram below. In the diagram, draw a vector at each point marked "X" to indicate the electric field E at that point due to the point charge. NOTE: Your vectors must properly display both the relative magnitudes and directions of the electric field. o o o o o o o o o o PREDICTION o o o o o o o X o o o o o o o o o o o X +4 o o X o o 2. How would your E vectors change if the point charge had charge of -4 units (rather than +4 units)? Explain briefly. 3. How would your E vectors change if the original charge (of +4 units) was moved one grid point to the right? 4

5 Electric field due to a single point charge using EM Field Procedure: Start the program EM FIELD 6.9. When the information screen appears, click anywhere to run the program. To control this program, you will use the mouse to choose items from the menu bar at the top of the white screen. In the Sources menu, choose 3D point charges. Choose Show Grid from the Display menu, and then Constrain to Grid from that same menu. Drag a single point charge of +4 units (the solid circles) to the center of the grid. 1. While holding the mouse button down and not releasing it, slide the mouse across the screen and around the point charge. Describe what you see on the screen. (NOTE: Do not click with the mouse at this point. When you are finished, drag downward with the mouse so that the cursor (actually an arrow) goes off the screen at the bottom.) 2. What does the vector indicate? Explain both magnitude and direction. 3. Refer to the previous grid diagram where you predicted the electric field at various points (Step 1 of the previous section). At each location indicated by X, click with the mouse button. Describe what you see on the screen. Is it what you predicted? If there are differences between what you predicted and what you see on the screen, explain and resolve those differences. 4. Replace the +4 point charge with a -4 point charge at the same location. Refer to the previous grid diagram where you predicted the electric field at various points (Step 2 of the previous section). At each location indicated by X, click with the mouse button. Describe what you see on the screen. Is it what you predicted? If there are differences between what you predicted and what you see on the screen, explain and resolve those differences. 5. Replace the -4 point charge with a +4 point charge at the same location. Refer to the previous grid diagram. At each location indicated by X, click with the mouse button. This should duplicate the electric field vectors observed in step 2 above. Now move the +4 point charge one grid point to the right by clicking, dragging, and releasing it at the desired spot. Compare what you see on the screen with your prediction in question 3 of the previous section. If there are differences between what you predicted and what you see on the screen, explain and resolve those differences. 5

6 6. Choose Clean up screen from the Display menu in EM Field. This will remove electric field vectors previously displayed. Identify three points that are 1, 2 and 3 grid units from the +4 point charge. Evaluate the expression for the magnitude E of the electric field of a point charge: E = kq/r 2 to predict the magnitudes of E at the three points selected. Assume that k = 1 and that r is expressed in units equal to the spacing between adjacent grid points. 7. Now click at the three points on the screen where you just predicted the magnitude of E. For each point, measure the length of E in cm. How does this compare to the value of E you predicted in the previous step? 8. Choose Field lines from the Field and potential menu. Click on the three points where the electric field is displayed. EM Field will draw the electric field line through each of those points. What is the relation between these electric field lines and the electric field vectors displayed in the previous step? Click on a few more points surrounding the point charge, so that you have a symmetrical pattern of electric field lines. Print the figure you have created on the screen by choosing "Print screen" from the File menu. 6

7 Electric Potential: Work and Potential Difference o o o o o o o o o o X D o o o X A o o o o o X B o o +9 o o o o o o o o o o o o X C o o A. Suppose a charge q 0 is located at position x A. The charge is able to move along any path to positions x B, x c, and x D. Make a prediction about the work done on the charge q 0 to move it from rest at x A to rest at x B, x c, or x D. Explain how you arrived at your answer. If there is not enough information for you to answer the question, explain what information you would need. B. Suppose the charge q 0 is moved slowly by an external agent (e.g. a hand) along a straight path from rest at x A to rest at x B. 1. Consider the force exerted by the external agent to move the charge. Is the work done by this force positive, negative, or zero? Explain. Compare the sign and magnitude of the work done by the electric force to the sign and magnitude of the work done by the external force. Explain. 2. Write an equation that describes the work done by the electric field of the point charge on the charge q 0 as it moves from rest at x A to rest at x B. Explain. 7

8 3. Write an equation that describes the work done by the external agent on the charge q 0 as it moves from rest at x A to rest at x B. Explain how you arrived at your answer. 4. How, if at all, would your answer to question 3 change if the charge had a magnitude of 2q 0? How, if at all, would your answer to question 3 change if the charge had a magnitude of 3q 0? Compare the ratios of the work done by the external force on the charge to the magnitude of the charge. The ratio you have found is the electric potential difference, ΔV. The electric potential difference between two points describes the ratio of the work done on a test charge to move it from rest at one point to rest at the second point to the magnitude of the test charge. 5. Does the potential difference depend on the magnitude of the test charge? Explain. Consider that the charge is moved from rest at x A to rest at x B along the shortest possible path. 6. Compare the sign and magnitude of the potential difference for this path to the sign and magnitude of the potential difference when the charge is moved from x B to x A. Explain. 7. Find a value (in terms of k) for ΔV between locations x A (9 grid points from the q = +9 charge) and x B (3 grid points from the q = +9 charge). Assume the units of charge and distance are Coulomb and meter, respectively. Show all work. 8

9 Potential Difference of a Point Charge A. Choose Clean up screen from the Display menu in EM Field. Select Sources/3DPointCharges from the program s menu. Place a single positive charge of magnitude +9 at a grid point near the right side of the screen (about 2/3 of the way over) and about half way down. Select Potential Difference from the Field and Potential menu. 1. WHILE HOLDING THE MOUSE BUTTON AND NOT RELEASING IT, slide the mouse from the left edge of the screen toward the charge of +9 units. RELEASE the mouse at a point close to but not on top of the charge. Describe what you see. Choose Display/CleanUpScreen. Click and hold the mouse button and from x A to x B along a straight path, releasing the mouse at x B. o o o o o o o o o o X D o o o X A o o o o o X B o o +9 o o o o o o o o o o o o X C o o 2. Compare the value for the potential difference displayed on the computer with the value you computed in question 7 on the previous page. Resolve any discrepancies. 3. Clear the screen again. Now click and hold and drag the mouse from x B to x A, releasing the mouse button at x A. How does the value for ΔV between x B and x A compare to the value between x A and x B? Explain. How is the sign of ΔV reflected in the color of the line drawn on the screen? 4. Does the value of the potential difference between x B and x A depend on the path along which you move? Explain how you arrived at your answer. 9

10 B. Let s now consider positions x C and x D as well. 1. Is the potential difference between positions x A and x C greater than, less than, or equal to the potential difference between positions x A and x B? Explain. Choose Display/CleanUpScreen. Use the mouse to find the potential difference between positions x A and x C. Compare the value of ΔV between x A and x C to the value of ΔV between x A and x B. Resolve any discrepancies with your answer above. Does the value of ΔV between x A and x C depend on the path you take to move from x A and x C? 2. How, if at all, could you move a particle with charge q 0 so that the work done on the particle is always zero? Explain. How, if at all, could you draw a path between points x B and x C along which the potential difference is always zero? Explain. Choose Display/CleanUpScreen. Use the mouse to find a path along with ΔV is always zero. Compare this to your prediction. Resolve any discrepancies. 3. Predict the potential difference between points x A and x D (in terms of k). 4. Use the program to check your prediction. Resolve any discrepancies. 10

11 Field and Potential of a Dipole A. Place a +9 and -9 charge two grid units apart near the center of the screen. 1. Draw the two charges and roughly sketch a circle around them. Predict the direction of the dipole field at the top, bottom and sides of the circle. Also predict the field between the charges. 2. Use the program to check your answer. Choose FieldAndPotential/FieldLines. Click to draw field lines. Compare these field lines to your prediction. Resolve any discrepancies. 3. Now exam the electric potential around the dipole. First clean up field lines by selecting Display/CleanUpScreen. Choose FieldAndPotential/Equipotentials With Number. Click to draw equipotential lines between and around the charges. Explain qualitatively the shape and relative strength of equipotential lines. How are the equipotential lines oriented relative to the field lines? Field and Potential of a Charged Plane A. To create a charged plane, choose Sources/2DChargedRods. Create a horizontal line of charges across the middle of the screen by dragging charged rods of charge equal 1. Adjacent rods should be touching. The length of the rods is into the screen (in the z direction if x and y are the plane of the screen), so you now have a plane of charge. 1. Choose FieldAndPotential/FieldLines. Click to draw field lines above and below the charged plane. Print a picture and explain the pattern of the field lines. 2. Now examine the electric potential around the charged plane. Choose FieldAndPotential/Equipotentials With Number. Click to draw equipotential lines above and below the charged plane. Explain qualitatively the shape and relative strength of equipotential lines. How are the equipotential lines oriented relative to the field lines? 11

12 A capacitor consists of two oppositely charged plates. To examine the field and potential inside a capacitor, create another horizontal line of charge 3 units above your first line. Again, choose Sources/2DChargedRods. Use -1 charge to create this second line. 3. Now examine the electric field between these two charged planes. Choose FieldAndPotential/FieldLines. Click to draw field lines above, below, and between the charged planes. Print a picture and explain the pattern of the field lines. 4. Now examine the electric potential between the charged planes. Choose FieldAndPotential/Equipotentials With Number. Click to draw equipotential lines above, below, and between the charged planes. Provide a qualitative explanation for the shape and relative strength of the equipotential lines. How are the equipotential lines oriented relative to the field lines? 12