Unformatted text preview: Lab 2: Mapping of Electric Fields and Equipotentials Introduction The purpose of this experiment is to examine the nature of electric fields by mapping the equipotential lines, electric field lines and calculating the electric field. Background, Theory and Application An electric field is a region in which forces of electric origin are exerted on any electric charges that may be present. If a force, F , acts on a charge, q, at some particular point in the field, the electric field strength, E , at that point is defined as the force per unit charge, and the magnitude is given by (1) Remember for a point charge, Coulomb’s law tells us that F qQ 4 0 r 2 (2) This means that at a distance r from an isolated charge, Q, we would expect an electric field of E Q
4 0 r 2 (3) ,where 0 is the permittivity of free space, 8.85x10‐12 C2/Nm2. Since E is a vector quantity, it also has direction, and we arbitrarily define the direction of an electric field as the direction of the force on a positive test charge placed at the point in the field. When there is more than one electric charge present, the total electric force and electric field are found by adding the forces/fields due to each charge as vectors. The English scientist Michael Faraday introduced the concept of lines of force as an aid in visualizing the magnitude and direction of an electric field. A line of force is defined as the path traversed by a free test charge as it moves from one point to another in the field. Figure 1 shows several possible paths that a test charge might take in going from the positively charged body to the negatively charged body. The relative magnitude of the field intensity is indicated by the spacing of the lines of force and the arrows indicate the direction. Figure 1 Figure 2 Electric Potential Since a free test charge would move in an electric field under the action of the forces present, work is done by the field in moving charges from one point to another. If external forces act to move a charge against the electric field, then work is done on the charge by the external force. If the charge, q0 (Figure 2) is placed at a point very far from the charge, q, where the repulsion is essentially zero, the work per unit charge to move it from this point to the point P is called the absolute potential at P. The ratio of the work, W, to the charge, q0 is called the potential difference between the two end points of the path traversed. Hence we may write V W q0 (4) ,where V is the difference in potential. For an isolated pointed charge, the electric potential varies inversely with distance from the charge. V q
4 0 r (5) Note that the electric potential is a scalar, and thus has NO direction. Its sign indicates the sign of the charge, q. Electric potentials due to multiple charges can be added directly. However, if the charge is moved along a path at all times perpendicular to the lines of force, there is no force component along the path, and hence, there is no work done. Then the points are said to be at the same potential, and the path traveled is called an equipotential line. One can readily see that many equipotential lines, or surfaces, are possible in an electric field. Experimentally, it is much easier to trace the path of equipotential lines than to trace the line of force. When a network of equipotential lines have been mapped out, the lines of force, being everywhere normal to the equipotential lines, can then be plotted. The magnitude of the electric field between two equipotential lines can be estimated using the relationship E V s (6) , where V is the difference between the potentials of the two lines, and s is the distance between them. The negative sign is an indication of direction of the electric field vector. Procedure Go to ‐and‐fields and click the play button. You should see a screen as below. Tick “grid” and “show numbers” in the green box. Part 1: Fields and Potentials due to Single Point Charges 1. Drag a single positive (red) 1nC charge into the center of the grid, making sure to drop it right on an intersection of two dark yellow grid lines. 2. Move the equipotential box (gray with blue rim) so that the cross hairs are exactly 0.5m to the right of the 1nC point (one set of dark yellow grid lines). What voltage do you get? 3. Now move the box 0.5m above the 1nC point: what is the voltage? 4. Click the light green “plot” button in the equipotential box. This will draw a line that passes through all points with the same voltage as in your box. What is the shape of this line? 5. Move the cross‐hairs to a distance of 1.0m (two dark yellow grid lines) to the right of the 1nC charge What is the new potential? Again plot the equipotential line. Does it have the same shape as in step 4? 6. Compare the potentials in step 2 and step 5. The distance from the point charge has doubled. What do you expect to have happened to the electric potential according to equation 5—does your prediction agree with the experiment? Use equation 5 and the size of the charge (1 nC) to calculate what the potential should be 2.0 m away from the charge (four large grid squares?) Check if your prediction is correct. 7. Calculate the electric field between the 0.5m equipotential and the 1.0 m equipotential using equation 6. Which direction should it point in? Now grab an “e‐field sensor” (orange dot) from the box and drop it halfway between the 0.5m and 1.0m equipotentials. Were your predictions correct? 8. Click “Clear all” and drag a single blue ‐1nC charge into the center of the screen. Repeat steps 1 through 7 for this configuration. Part Two: Fields and Potentials from Electric Dipole Configurations 9. Click “Clear all” and drag in a one red and one blue charge. Arrange them exactly as shown below. (8 large grid squares apart and along the same line. Be as careful as you can to get the charges exactly in the corners). 10. Use the equipotential box to record the voltage / electric potential every 0.5m (one large grid spacing) between the two, filling in Trial 1 in Table 1 below. At each location, click plot. Describe the shape of the final curves. Click “show E‐field”. The red arrows that come up point along the lines of charge, and their intensity indicates how large the field is. What do you notice about the relative orientations of the red arrows to the equipotential lines when they cross? Is this what you expect? Table 1: Voltages for Dipole Configurations Trial [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] 1 2 3 4 11. Repeat steps 9 and 10 but with two POSITIVE charges at 8 large grid squares apart. (Trial 2) 12. Repeat steps 9 and 10 but with two NEGATIVE charges at 8 large grid squares apart. (Trial 3) What do you notice about the voltage values for trial 3 compared to trial 2? Does this make sense? 13. Try pulling another negative charge on top of each of the negative charges in step 12. You now have ‐2nC at each location. This is trial 4. What happens to the voltage values? What happens to the shape of the electric potential lines? Part 3: Fields and Potentials from Multiple Point Charges 14. Make a square 4 grid lines by 4 grid lines. At two of the corners put positive charges, and at two of the corners put negative charges. Put the cross‐hairs of the equipotential box exactly in the center of the box of charges, and draw an orange E‐field sensor into the center as well. Make each of the six possible configurations shown below, one at a time. What configurations of charges have a zero total potential at the center? What configurations have a zero electric field at the center? Explain. (You may need to move the equipotential box to allow you to read the E‐field sensor properly.) A B C D E F 15. Hit “Clear all” and make the same shape, but with 4 positive charges. What are V and E in the center? 16. Repeat step 15 with 4 negative charges. What do you notice about V and E compared to step 15? ...
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