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Lab 3: Electric Fields II Introduction This week's lab has two parts. In the first part, you will use the EM Field computer program to continue Investing the properties of electric fields. In the second part, you will create, and measure the properties of a real two- dimensional electric field. Objectives • To explore the electric field produced by a line of charges. • To gain experience with the concept of electric flux. • To verify Gauss' Law. • To measure the properties of real two-dimensional electric field. Equipment • The EM Field computer program. • Two dimensional field apparatus. • DC Power Supply. • Digital Multimeter (DMM). Part I(a): Examining the field due to a line charge A. What Is a line charge? If you charge a conducting rod the charge will spread over the entire rod. If there are no other nearby charges, the charge will spread uniformly. We can simulate such a charge distribution in EM Field by placing point charges at equal distances along a line. Select File/Get charges or currents from file from the menu bar, then double-click on qline.emf. Note that there is one (+1) charge at each grid point. The line charge could be given as 1 charge per grid unit. In SI units this would be 1 C/m (one coulomb per meter). A line charge is specified in term of linear charge density, charge per unit length. The symbol to represent linear charge density is the Greek letter lambda (), where "l" suggests length. 1. How could you change the charge density? You increase each charge unit to 2 units, and thus have 2 C/m. To see a way of changing the density, open the charge file qline2.emf. What is the charge density for this configuration? ______________________. B. Electric field versus distance from a line charge You know that the electric field from a point charge is inversely proportional to the square of the distance from the charge. That is, E 1/r2. Use the field vectors to find out how the electric field depends on the distance from the line charge. Complete the table on the next page. 1 Electric Fields II qline.emf qline2.emf Distance from line Length of field Distance from line (in Length of field arrow (in grid units) arrow (in cm) grid units) (in cm) 1 2 4 1. Does E 1/r2? Use the program Graphical Analysis to determine the power law behavior by fitting a curve to E versus r using the functional form y = AxB. Report the power, B:__________. 2. How does E depend on ?______________________. EM field allows you to view a line charge from one end. Select Sources/2D charged rod from the menu bar. Place a rod near the bottom of the screen. Drop a few field vectors to find out how the electric field depends on the distance and direction from the rod. Are your results consistent with the qline results from above?_________________________________________________. Part I(b): Gauss’ Law Gauss' Law is one of the most powerful, yet mystifying, statements in physics. Its mathematical form is: E dA q / o . In words, Gauss' law states that if you enclose a charge in a closed mathematical surface, then the electric flux integrated over that surface is proportional to the charge enclosed by that surface. The electric flux is defined as the product of the component of the electric field perpendicular to the surface times the area of the surface. What is the meaning of flux and Gauss' law? EM Field will help you get a feeling for the meaning of both. For a brief explanation of how EM Field shows the flux through a surface, select Option/How we display Gauss' law from the menu bar. Read the text, clicking on the screen for each new graph. A. Flux from a single charged rod Place a 2D charge rod with charge density +3 in the middle of the screen. To draw a Gaussian surface around the rod, select Field and Potential/Flux and Gauss' law from the menu bar. Now use your mouse to draw a circle with a radius one or two grid units around the rod. When the cursor is near the starting point you can release the button and let the program complete the circle for you. Remember that this circle is actually the end view of a long cylinder enclosing the rod. Note that the electric field vector was always (approximately) perpendicular to the surface The gray bars outside the circle represent the flux though each strip of the cylinder. "Q=3" means that the value of the charge enclosed in the cylinder is three units. 1. Draw a second circle with a much larger radius around the rod. The program again says "Q=3," but the electric field was much smaller at the surface. How can the total flux be te 2 Electric Fields II same through these two surfaces? (Hint: Consider the definition of flux)_______________ ___________________________________________________________. 2. Draw a second circle with a much larger radius around the rod. The program again says "Q=3," Select Sources/Add more sources from the menu bar. Place a rod with charge density -3 in the center of the two circles and remove the +3 rod. What is different about the flux and how is it represented by the program?____________________________________________________________ ______________________________________________________________________. 3. Move the -3 rod to the edge of the screen and place a +3 rod in the center of the circles again. Select Display/Unconstrain so that you can place the rod anywhere on the screen. Now move it near the surface, but still inside the inner circle. The field, and flux, near the charge is much larger, but the total flux is still given by "Q=3." Explain how this can be: __________________ _________________________________________________________________________. 4. Finally, move the rod outside the inner circle. The program says that the enclosed charge, and thus the total flux, is now zero. Explain how that comes about:___________________________ _________________________________________________________________________. B. Flux from two charged rods Place both the +3 and -3 rods outside the smaller circle, but within the larger circle. Note that there is no charge inside the smaller circle, while there are charges, but no net charge within the larger circle. 1. Explain how the total flux through each of the two surfaces is zero, even though the charges Inside are different. You may find it helpful to print your screen (select File/Print screen from the menu bar) and point to the positive and negative regions of flux in order to make your point._________________________________________________________________________ ___________________________________________________________________________________. 2. Make notes on the printed page showing where the field direction is away from the positive charge and toward the negative charge, and the sign of the flux In each case. Detach pages 2 and 3 from your lab manual and attached them to the worksheet for Part II. 3 Electric Fields II Part II: Measuring a 2-dimensional electric field. Introduction You were first introduced to the field concept with the gravitational field. Gravity, you remember, is not a force that acts when two objects touch. It is a force that seems to act over a distance. Physicists have never been happy with the idea of “action-at-a-distance.” Rather, they prefer to think about a gravitational field that is created by one object, like the sun. The field exists at the location of the second object, like Earth, and acts on it there. Thus the force is a local, not a distance force. In the same way, physicists have found it useful to describe the electric interaction in terms of an electric field created by one charge acting on a second charge at the location of that charge. Defining the electric In principle, a field can be measured by measuring the force on an object located field in that field. The force is then divided by the object’s charge (E = F/q) , in the case of the electric field, or mass (g = F/m), in the case of the gravitational field. The units of the electric field are newtons/coulomb. Electric field lines The electric field can be visualized by means of field lines. An electric field line is a line which is at every point tangential to the electric field. That is, it always points in the direction of the field and it traces out the path that a charge would follow if it were allowed to move at constant speed in the direction of the electric force on it. Electric potential A direct measurement of a field is very difficult. We will use an indirect way of difference measuring the electric field that will involve the measurement of electric potential difference. The electric potential difference is defined as the work done on a unit positive charge in moving the charge from one point to another in an electric field. The units of potential are work per charge or joules/coulomb. One joule/coulomb is defined as a volt. As is shown in the textbook, the electric field, in one dimension, is the derivative of the potential. For a three-dimensional field, we must use the three dimensional rate of change of the potential. Thus, in component form: dV dV dV Ex = – dx , Ey = – dy , Ez = – dz (1) We can see that another unit for electric field is potential per unit distance or volts/meter. (You can check to see that V/m and N/C are equivalent units.) Because E is related to a spatial derivative of V, we can use a difference method of measuring E. For points a and b close together we approximate the derivative by a difference equation: 4 Electric Fields II dV V Vb – Va Ex = – dx = – =– x –x (2) x b a The potential difference, Vb – Va, can be measured with a potential-difference meter, commonly called a voltmeter. The quantity xb – xa can be measured with a ruler. From these two measurements we can calculate Ex. We can do similar measurements for the other components, Ey and Ez. The negative sign means that the electric field component along any axis points opposite to the direction in which the potential increases. Thus, the electric field vector at any point in space always points opposite to the direction of the greatest rate of change of potential and has magnitude equal to this rate of change. Equipotential lines It is also useful to define the concept of equipotential lines and surfaces. A point and surfaces charge can be moved without doing any work along an equipotential line (in 2 dimensions) or along an equipotential surface (in 3 dimensions). If no work is done then the potential must be the same everywhere. Clearly there must be zero electric force in the direction of motion along an equipotential. This is all you need to know about electric fields and potentials to begin the experiment. You will discover some of their properties from your observations. The Experiment In this experiment, we will set up a two dimensional electric field in a thin sheet of slightly conducting material (called teledeltos paper) by applying a potential difference between two electrodes. Everything we have said so far is the same for this configuration. The potential difference means there is an electric field, and the force due to the field causes a flow of charge (current) in the sheet. We can be sure that the direction of the electric field is parallel to the faces of the sheet, of, it were not, charge would flow to the faces and build up until the field became parallel. Potential differences are measured by a voltmeter. In this experiment, we can use a type of meter that does not require very much current to operate. By using this meter, we can be sure that the field patterns are not distorted by the presence of the meter. 5 Electric Fields II 1. Equipotential Lines Procedure • Place a sheet of the conducting teledeltos paper on the mapping board that has two bolts through it. These two bolts provide two point-like electrodes. • Connect the power supply, multimeter and potential probe as shown in Figure 1. (The potential probe is the one with only one point.). + – Power Supply Probe Vol ts V COM Figure 1. Using the potential probe • Now touch the probe gently to the left-hand electrode. Adjust the power supply to read 12V on the multimeter. • You are now ready to look for equipotential lines. Gently probe for a point where the multimeter reads 10V. Mark this point with a pencil. Now find enough other points at 10V to be able to draw a smooth equipotential line connecting the points. You can probably make a closed curve. • Go back and do the job again, but this time measuring the equipotential line at the 8V potential. Make sure you measure enough points so that you can draw a smooth curve through the points. Can you make a closed curve this time? • Repeat this procedure for equipotentials at 6, 4, and 2 volts. Each time, make sure you determine enough points so that you have an accurate indication of shape and spacing of the equipotential lines. • Label the equipotential lines with their potentials. An equipotential line was defined as the line along which a charge can be moved without doing work. Charges can move freely along a conductor (much more freely than on the teledeltos paper). If you were to place a good conductor along such a line, then, even though charges would be free to move, there would be no force available to move them. Therefore, the potentials at other locations on the sheet would not be changed. Let’s check this. 6 Electric Fields II Procedure • Place the potential probe on the 8-volt line. Press the edge of a short piece of a ruler covered with aluminum foil along the 6-volt line. Hold it down with your fingers. Does the voltmeter show any change? • Rotate the ruler until it is at right angles to the 6-volt line and press it down. Does the voltmeter show any change now? If it does, find one or two points along the new 8-volt and 4-volt lines. 2. Field Lines Method • The field probe consists of two points spaced about 1 cm apart. The two wires of the electric field probe will be connected to the DMM (one wire to the black (com) terminal, one wire to the red (“V”) terminal). Connected in this manner, the field probe measures the potential difference between the two points. • You can find the direction of the field by first touching the black probe to the paper as before. Next, without removing this probe, choose a location for the second, gently touch the point to the paper and read the potential difference. Rotate the probe, trying different directions until you get the maximum potential difference. • The DMM reads the potential difference between two points one cm apart. To find the electric field at the midpoint between the two points, divide the voltage by 0.01 m. Procedure • Connect the field probe to the meter as discussed above and illustrated below. + – Power Supply Probe Vol ts V COM • Starting with one probe a few millimeters from the left-hand electrode, find the direction of the maximum potential difference and make pencil marks at both probe points. Record this maximum potential difference V. • Move the probe so that the first point is where the second was, and repeat this procedure. Repeat this all the way out to the second electrode. • By connecting the dots, you can draw a field line. Place arrows on the field line in the direction of the field. • Follow one other field lines, starting from the left-hand electrode at a different direction, as suggested below. One field line should be very 7 Electric Fields II nearly a straight line from one electrode to the other. 8 Electric Fields II Names Section Electric Fields II Data Analysis • Choose the line that is closest to a straight line from one electrode to the other. For that line you have a series of measurements of V and x , where x is the distance from the left-hand electrode to the midpoint of the two points marks of the electric field probe. Fill in the table with the values of V, and x . • Calculate E for each point. Because x = 1 cm = 0.01 m, E is just V/0.01 m, (see Equation 2). • Plot a graph of E versus x . (Review your graph plotting technique from the Physics 150 Laboratory Manual if necessary.) 1 2 3 4 5 6 7 8 9 10 11 12 13 V (volts) E (volts/m) x (m) Hand in your teledeltos paper and the electric field graph with these worksheets. Questions 1. Based on your graph, is the electric field constant between the two electrodes? If not, is there a region in which it is very nearly constant? Describe the region. 9 Electric Fields II 2. Where is the component of the electric field along your line strongest? Is this component of the field zero anywhere? If so, where? 3. Draw the field and potential lines between the point and straight-line electrode shown below. Hint: you need only consider the data you have taken and the symmetry of the electrodes. 6V 0V 4. Explain, in your own words, why the field probe method gives you the magnitude and direction of the field. 5. On the basis of your observations, what angle is made between field lines and equipotential lines? 6. Define, in your own words, an equipotential line. 7. Do equipotential lines ever cross each other? Explain. Hint: What would it mean? Think of being a charged particle on an equipotential when you came to a crossing. 8. Can field lines ever cross each other? Why? 9. Why don't charges move along an equipotential line? Hint: What causes a force on a charge? 10 Electric Fields II 10. Suppose a thin straight wire were placed between the two electrodes as shown. Draw the resulting field and equipotential lines. Wire 11. Suppose a thin straight wire were placed as shown below. Draw the field and potential lines that would result. Remember, the wire is an equipotential line. Wire 11 Electric Fields II This page intentionally left blank 12

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