Elizabethtown College

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1 Elizabethtown College Department of Physics and Engineering PHY 104 Laboratory Lab # 6 Electric Flux Lines and Equipotential Surfaces 1. Introduction In this experiment you will study and actually trace electric flux lines and equipotential surfaces/lines due to an electric field and/or a charge distribution. You will be able to directly test the properties of such lines and surfaces, as studied in class and in the text. 2. Background and Theory (a) Electric Fields An electric field E exists in the space surrounding any set of stationary charges. The field is defined at any point (x,y,z) as the quantification of external electric forces which would act on a unit of positive charge located at point (x,y,z.) (b) Electric Potential The electrostatic potential difference ( V) between two points located in an electric field region is defined as the work required to move a unit of positive charge from one point to the other. The electrostatic field being conservative, the work done on moving any charge does not depend upon the path taken. Furthermore, the work done by the mover of the charge is equal in magnitude and opposite in sign the work done on that charge by the electric field. The electric field (a vector field) is the gradient of the electric potential (a scalar field.) Thus, E V V V iˆ x y ˆ V j z kˆ (1)

2 Surfaces which have a constant electric potential are referred to as equipotential surfaces. The fact that the electric field is the gradient of the electrostatic potential implies that the electric field is perpendicular to equipotential surfaces. This should make sense intuitively since, by definition of V, there is no work required to move a charge along the equipotential surface. Thus, the mover of the charge along an equipotential surface will not be required to apply any force in the direction of motion. (c) More on Electric Flux Lines The idea of electric flux lines (or electric lines of force as they are sometimes called) was introduced by Michael Faraday ( ) in his experimental investigation as a way of visualizing the electric field. An electric flux line is an imaginary path or line drawn in such a way that its direction at any point is the direction of the electric field at that point. In other words, they are the lines to which the electric field density D is tangential at every point. Electric flux lines have the following properties: 1. The lines always start at positive charges and terminate at negative charges. Positive charges are thus regarded as sources and negative charges as sinks of electric flux lines. 2. No two flux lines can intersect except at singular or equilibrium points; that is, those points at which the resultant D is zero. 3. The electric flux density D is tangential to the flux lines at every point. Figure 1 Flux line geometry 2

3 The analytic expression for flux lines can be obtained as follows. Let D = D x a x + D y a y. Since property (3) above must hold, it is evident from Figure 1 that at any point (x, y): In general, for a three-dimensional field, D = D x a x + D y a y + D z a z, so in Cartesian coordinates, the flux lines are given by (d) More on Equipotential surfaces Any surface on which the potential is the same throughout is known as an equipotential surface. The intersection of an equipotential surface and a plane results in a path or line known as an equipotential line. No work is done in moving a charge from one point to another along an equipotential line or surface (V A - V B = 0) and hence: on the line or surface. From equations above we may conclude that the lines of force or flux lines (or the direction of E) are always normal to equipotential surfaces. Examples of equipotential surfaces for a point charge and a dipole are shown in Figure 2. 3

4 Figure 2 Flux lines and equipotential surfaces for (a) A positive point charge and (b) A dipole. Note that in these examples it is apparent that the direction of E is everywhere normal to the equipotential lines. We shall see the importance of equipotential surfaces when we discuss conducting bodies in electric fields; it will suffice to say at this point that such bodies are equipotential volumes. There are two ways to determine equipotential lines depending on whether V or E is known. If V is known, the equipotential lines or surfaces are given by: V = constant On the other hand, if E is given, equipotential lines may be determined in a manner similar to our determination of the flux lines. From calculus, if a line has slope m, a normal line to it must have slope -1/m. If D = D x a x + D y a y + D z a z, we know from the equations above that the flux lines on the z = constant plane are given by: 4

5 Equipotential surfaces on the same z = constant plane are normal to the flux lines and given analytically by 3. Experiment: Mapping Electric Field Lines and Equipotential Surfaces The lab station is equipped with electrical connectors, conductive paper, corkboard, push-pins, power supply, and a digital multimeter. The semi-conductive paper has different figures painted by conductive ink. These figures will be referred to as electrodes. Locate the paper having electrodes that are a line and a circle. Using a conductive push-pin in each corner, pin this sheet to the corkboard. Stick two other conductive pins into the electrodes for the straight line electrode put the pin in the center of the line. Next, attach the alligator clips to two leads, the leads to the power supply, and then connect an alligator clip onto each of the pins in the electrodes. Connect the low input of the multimeter to the low voltage electrode. Turn on the power supply and adjust its output to 15 volts. Finally, connect a point probe to the high input of the multimeter and turn it on. Make any necessary adjustments to the multimeter so that it will read voltages from 0 20 volts. Using the point probe, you can now measure voltages on the conductive paper. Touch the probe to the low voltage electrode and you should read 0 volts. Touch the probe to the high voltage electrode and you should read 15 volts. If this is not the case, seek assistance. Now, using the probe, find a place on the conductive paper for which the multimeter measures 3 volts (don t worry if it fluctuates a bit). Use a pencil to mark this point on the paper. Repeat this process to find five or six additional locations evenly spaced on the paper. Since these points have the same potential, they are points on the same equipotential surface. Sketch a line on your conductive paper connecting these points to indicate the 3 volt equipotential surface. Repeat this process to locate equipotential surfaces of 5, 8, 11, and 13 volts. Using the surfaces, sketch five or six electric field lines (with direction indicated). Each of these field lines should originate and terminate at an electrode. Make sure that your field lines are perpendicular to the equipotential surfaces. Repeat this process with the second set of electrodes on the other conductive sheet. Turn in next week the conductive sheets with the group notebook. Also try to show characteristic features of flux lines and equipotential surfaces/lines from your data. 5

6 For the final report, in the notebook, also solve analytically the following problem: Problem: For the electric field intensity E = ya x + xa y, (a) (b) Determine the general equation of the flux lines and equipotential lines. Find the lines that pass through the point (1, 4) and E at that point. 6