Lab 3: Electric Field Mapping Lab Last updated 9/14/06 Lab Type: Cookbook/Quantitative Concepts Electrostatic Fields Equi-potentials Objectives Our goal in this exercise is to map the electrostatic equi-potential surfaces of the following configurations: 1. Parallel Plate Capacitor 2. Cylindrical Capacitor 3. Electric Dipole and deduce the corresponding electrostatic field lines. In addition, we will see quantitatively how the electric field varies inside and outside of a parallel plate capacitor. Introduction You should know from lecture that electrostatic equi-potential surfaces are perpendicular to electrostatic field lines. You should also know that you can gauge the strength of the field by the distance between two equipotential surfaces ("r) and the voltage difference ("V ), the average electrostatic field strength in that region is E r " #V r and the direction of the E points from high potential to #r low potential. We will use these relationships between equi-potential surfaces and electrostatic field lines to learn about the electrostatic field for three common potential configurations: parallel-plate capacitor, cylindrical capacitor, and electric dipole. Although a charge produces electrostatic potential everywhere in space, it is not possible to measure such electrostatic potential using a common voltmeter (which is essentially an ampere meter with a large series resistor). If we did use such voltmeter to measure the electrostatic potential difference between two points in space, what happens is that initially the potential difference across the two leads will cause an electrical current to go through the voltmeter for a very brief moment of time (perhaps mirco-seconds depending on the resistance and capacitance of the voltmeter). This electrical current allows the charges in the voltmeter to rearrange themselves to establish electrostatic equilibrium, now the 1
two leads of the voltmeter are at the same electrostatic potential, hence the voltmeter will always read zero voltage difference at equilibrium. The bottom line is that a voltmeter is a conductor (not a very conductor but nonetheless conducts); at electrostatic equilibrium, the charges on the conductor arrange themselves such that the conductor is an equi-potential surface. In this lab, we have to cheat a little. In order to sustain the voltage difference at two points, we need to establish a steady current situation rather than electrostatic equilibrium situation. We will fill the space between the charge objects with a resistive material which allows a current to flow from one charge object to the other and we connect a voltage source (an electromotive source (EMF) such as a battery or power supply) to the charge objects to maintain the charge on the objects (if we don t, the current will deplete all the charges from these objects). Instead of dealing with 3-dimensional objects, we create 2-D analogs of these charge configuration by using silver paint (conductor) and a special paper called the Teledeltos paper (a resistive medium). Referring to Fig. 2, the white circle and the center dot are made of silver paint and they are connected to a voltage source. The black background is the Teledeltos paper. If you use a voltmeter and touch two points on the paper, you will get reading of voltage difference between those two points. There is one drawback with this direct method. The voltmeter will draw a small amount of current (the voltmeter has a high resistance but it is not infinite), and thereby distorts what we try to measure. We have a better method, read on. (Note: Since we are dealing with 2-D, two lines represents the cross-section of an infinite parallel plate capacitor; a circular dot is the cross-section of an infinitely long cylinder. Do you see why I put quotation marks around electric dipole?) We construct another circuit (see Fig. 1) with a voltage source and a resistor with an adjustable contact called the potentiometer (the arrow in the figure represents that it is adjustable). The potentiometer is a voltage splitter (or called voltage divider); the potential at the center tap of the potentiometer depending on the position of the contact which is control by a knob. The dial on the knob is calibrated to register the potential of the center tap. The center tap of the potentiometer is connected to an ammeter (a galvanometer for reading current). If you place your probe on a point on the paper that s at a different potential then the potential of the center tap, there will be a flow of current and the ammeter will deflect. If you move the probe to a location of the paper where the potential is the same as the center tap, the ammeter will not deflect. To find a particular potential, all we need to do is set it on the potentiometer and then find a point on the paper where the ammeter needle doesn t deflect! Since no current is drawn, this measurement does not distort what we want to measure. This method essentially compares an unknown potential with a known potential. 2
Figure 1: Abstract Apparatus Procedure Obtain: 1 sheet of carbon paper 1 of each of the Teledeltos papers 1 sheet of white copy paper Black or red wires with the fork on the end (in the cardboard box) Banana connectors 3
Part I: Cylindrical Capacitor Figure 2: Lab setup 1. Construct the circuit as instructed by the TA. 2. Place the white copy paper on your board. On top of that, place the carbon copy paper (black) face down, and place one of the Teledeltos papers face up (with the silver lines on the top). 3. Place the 0 V lead on one of the silver lines. Set the 10 V lead on the other silver line. 4. Turn on the power supply and set it to 10 V. 5. Turn the potentiometer (helipot) to 2.0 V on the dial. BE SURE IT IS SET TO COARSE, C. Place the probe on the paper at any point (other than the silver lines). 6. Determine where the current is approximately zero. 7. Fine tune the location by switching over to the fine setting, F. 8. Mark the location by tapping lightly on the probe with a pen. 9. Repeat steps 8-11 for the 4V, and 6V settings on the potentiometer. Part II: Electric Dipole Repeat for the electric dipole, but only find the 2, 4, 6, and 8 V lines. Part III: Quantitative Repeat for the parallel plate capacitor, but find the 2, 4, 5, 6, and 8 V lines. Also be sure to take the 4 and 6 V lines very carefully (about 1 cm between measurements) from one end of the paper to the other. Move the paper holders if you need to. Measure the length (l) of the parallel capacitor plate and the distance (t) between them. 4
Finally, we ll graph the x dependence of E y. We know that: E = "!V r Equation 6 Ey =! dv / dy Equation 1 We also know dv = 2 V (the change in voltage) and dy = the distance between the 4 V and 6 V line. The distance should be the distance traced out by the electric field line, which is a curve line, especially when you are near the edge of the capacitor plates. For this lab, you may approximate the distance by the vertical distance. Calculate Ey for each vertical line (1cm spaced) on your paper. Graph Ey as a function of distance from the center of the parallel plate. You need only do this for one half of the parallel plate (e.g. x > 0 ). Note we are ignoring E x in this case since it is much smaller than E y. Questions 1. At what distance from the edge of the plate does the Ey fall off to ½ and 1/10 its value at the center? 2. If an electron were to travel through the parallel plates, would the field outside the plates appreciably deflect it? In other words, is it justifiable to ignore the fields outside the plates? 5