#3 Equipotential Surfaces and Electric Field Lines in 2 Dimensions

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Patrick Clark
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1 #3 Equipotential Surfaces and Electric Field Lines in 2 Dimensions Goals By completing this laboratory you will 1. learn to set up a simple direct current (dc) circuit, 2. learn to use a voltmeter and a galvanometer, 3. improve your understanding of electric fields and equipotential surfaces. Reading Suggestion Young and Freedman, University Physics, 12th ed., Sec Electric Field and Electric Forces, and Sec Equipotential Surfaces. Introduction The concept of field is introduced to describe forces exerted between objects that are separated by space, and not in contact with each other, sometimes referred to as action-at-a-distance. We say that one object creates a field in the surrounding space, and that the second object experiences a force because of the field. The field represents the strength and direction of the force. For example, at any point in space the gravitational field of the earth represents the gravitation force exerted by the earth on a hypothetical unit test mass located at that point. Similarly, the electric field represents the electric force exerted by other charges on a hypothetical unit test charge. In this experiment we will map out the electric field for two different configurations of charged objects. Formally, the electric field E is a vector, equal to the ratio of the electric force F exerted on a small charge q 0, to the charge q 0 : E = F/q 0. Clearly, E and F have the same direction. In general, force fields extend in three spatial directions, but for simplicity we limit our study of electric field lines to a two-dimensional surface. The electric field lines between two charged objects, such as a small point and a ring, can be reproduced by using two charged conductors of those same shapes (point and ring) embedded in a high-resistance conductive sheet. Positive charges on the conductive sheet move in the direction of the force on them, the direction of the electric field. Thus the current is in the direction of the electric field.. The electric potential (or just potential) is the electric potential energy per unit charge. Electric current is the motion of charge, and charges move from high potential energy to low potential energy. That means current flows from high potential to low potential, and each point on the conductive sheet must be at an electric potential V between the potentials of the two conductors. As you will see, the electric potential V varies throughout the conducting surface. (Because the electric potential is measured in units of Volts, using a voltmeter, it is sometimes called the voltage.) An equipotential surface joins all points having the same electric potential energy per unit charge, and charges do not move along the equipotential. This equipotential surface is everywhere perpendicular to the electric field lines that intersect it. (Electric field lines never intersect one another.) This fact allows one to use the equipotential surfaces to sketch the electric field lines -1-

2 Apparatus The apparatus is a mapping board which can have a high resistance plate attached, on which there are two brass conductors. We will investigate a concentric configuration of a point charge at the center of a charged ring, and a dipole configuration consisting of two charged points. The apparatus contains two separate circuits. One circuit consists of a power supply, switch, voltage divider with tap points, and the configuration of conductors. The second circuit consists of a probe attached to a galvanometer and key switch, which can be used to sense the potential at specific points between the conductors. (The galvanometer is a sensitive device for detecting an electrical current.) The probe circuit can be used in two ways: 1. Identify points on the surface that are at the same electric potential i.e. map an equipotential surface. 2. Directly measure the potential difference between a point on the conducting sheet and a reference. The mapping board has a set of convenient tap points that correspond to specific potential differences with respect to a reference. These taps can be used to choose the specific numerical values of the potential for which you might want to map out the equipotential surface. For example, if one side of the galvanometer is connected to a tap at 2.5 V, and the probe is moved to a location on the field mapping board where no current flows through the galvanometer, then you know that the electric potential at that location on the board is 2.5 V. This is a simple example of a null measurement. The resistor on the key switch protects the galvanometer by limiting the current and making the circuit less sensitive. Closing the key switch bypasses the resistor, and maximizes the sensitivity of the circuit. When mapping an equipotential as above, first position the probe, with the key open, so that the galvanometer current is zero. Then, close the key switch to make the circuit more sensitive. Move the probe so that the galvanometer current is again zero. Mark the probe position on the paper. 5-2-

3 Before you begin measurements: Understand the functions of the various components provided to you, such as the power supply, voltmeter, galvanometer and probe. On the underside of the field mapping board, the probe contacts the conductive surface. The point of contact corresponds to the hole in top side of the probe, through which a pencil can be inserted to mark the probe location. Install the conducting sheet with the central circle and concentric ring on the bottom side of the mapping board such that current can flow through the sheet from one brass conductor to the other. Ensure that the screw nuts are in good contact against the conductors. Place a sheet of paper with concentric circles on the upper side of the mapping board. Position the paper using the holes and small metal pins, and hold it in place with masking tape. Connect the power supply through the double-throw switch, and across the voltage divider represented by the tap points. Connect the positive (red) side of the power supply to the silver strip on the bottom of the field mapping board. Before you connect the power supply to the board by closing the double-throw switch, check and recheck the circuit, and convince yourselves that it makes sense. If unsure, check with the instructor before turning anything on. Measurements 1. Use the left side of the power supply to provide a fixed 5.0V across the voltage divider. Measure the voltage between the power supply common and each of the tap points. Do these make sense? 2. For the concentric ring geometry, map out sets of equipotential points for two choices of voltages between 1.5 V and 3.5 V. Each set should contain at least 10 points. What do you get when you connect the dots? 3. When the above measurements are complete, remove the conducting sheet with the concentric ring geometry, and install the dipole geometry, the conducting sheet with two small circular conductors. Also remove the paper, and replace it with a blank sheet, positioned under the rubber tabs, and hold it in place with masking tape. 4. For the dipole configuration, map out a set of equipotential points for each of three voltages: approximately 1.25 V, 2.5 V, and 3.75 V. Each set should contain at least 10 points. Connect the dots. 5. Also for the dipole configuration, rule a straight line on the mapping paper, between the centers of the two poles, the two charged points. Mark off 0.5 cm steps on this line, and, using the digital voltmeter, measure the potential at each of these steps. Plot the potential as a function of distance from the center of one conductor. Use the resulting graph of V vs. x to determine the magnitude of the electric field strength midway between the two poles. [Hint: E x = dv/dx]. As you approach the poles, should the magnitude of the electric field increase, decrease, or remain constant? Is this consistent with your observations? 6. The papers on which you ve plotted the equipotentials should be attached to your reports. -3-

4 #3 Laboratory Report Sheet Equipotential Surfaces and Electric Field Lines in 2 Dimensions Lab Section: Date: Name: Partner: EQUIPOTENTIALS Carefully sketch the 2-dimensional equipotential surfaces, labeling the voltages, including 0 and 5 V. Sketch the corresponding electric field lines. Explain briefly why the field lines and equipotentials have the shapes you ve observed: FIELD LINES -1-

5 Plot the potential measured along the line joining the centers of the two poles of the dipole configuration V x The magnitude of the electric field midway between the two dipoles is V/cm. If you could increase the separation between the two poles, the magnitude of the electric field should increase, decrease, or remain constant. (circle one) Explain your answer: -2-

ELECTRIC FIELD LINES AND EQUIPOTENTIAL SURFACES

ELECTRIC FIELD LINES AND EQUIPOTENTIAL SURFACES The purpose of this lab session is to experimentally investigate the relation between electric field lines of force and equipotential surfaces in two dimensions.