PC1143 Physics III Electric Field Mapping 1 Objectives Map the electric fields and potentials resulting from three different configurations of charged electrodes rectangular, concentric, and circular. 2 Equipment List Electric Field Mapping apparatus Power supply Digital Multimeter (DMM), crocodile clips and wires Polycarbonate plate, papers, copy papers and conductive carbon papers Vernier caliper and ruler 3 Theory The electric field at a point in space shows the force that a unit charge would feel if it were placed at that point: E = F (1) E is a vector quantity: it has both magnitude and direction, and has units of newtons per coulomb. The field is set up by electric charges somewhere in the surrounding space. For continuous charge distributions, it is much easier to analyze electric forces using the field concept than using Coulomb s law for the forces between point charges. If we measure the field (magnitude and direction) at enough points around the charge distribution, we could make a map of the electric field lines. These lines show the direction of the electric force at each point. However, instead of measuring the electric field directly, we will measure and map the electric potential around the charge distribution. The electric fields can be determined from this. To place a positive test charge in an electric field, we must do work against the field since the field tries to push the charge away. Since the electric force is conservative, the work we do in placing the charge in the field is stored as electric potential energy. Page 1 of 5
Electric Field Mapping Page 2 of 5 Electric potential is defined as the work done per unit charge to move the charge into an electric field. Like electric potential energy, electric potential is measured relative to some reference position, so that we define it as V = W (2) where V is the change in electric potential in going from the reference point to the point in question, W is the work done (or charge in electric potential energy) and is the charge. The unit of electric potential is volt, which is equivalent to a joule per coulomb. We usually use the Earth itself as our reference point and take the electric potential of the Earth to be zero. Thus, a point in an electric field or in an electric circuit can be characterized by the electric potential, or simply the voltage, there. A quantity of charge placed at that point has electric potential energy equal to the voltage there times the amount of charge. We can measure electric potential directly using a voltmeter. We can then map the potential field by connecting points that are at the same electric potential. Lines between such points are called equipotential lines. Moving a charge along an equipotential line requires no work since the energy of the charge does not change. To find the direction of the electric field, we make use of the fact that the equipotential lines must always be perpendicular to the electric field lines. This is because the electric field lines show the direction of maximum decrease in the electric potential. To determine the electric field from a series of voltage measurements, we note that the electric potential difference between two points is the integral of the electric field taken along the path between the two points: V = W = F d s = q0e d s = E d s (3) where ds is an infinitesimal element of the path. The minus sign arises because the external agency must exert a force that is equal and opposite to the force exerted by the field E. Conversely, the above expression tells us that E is the negative (vector) derivative of the electric potential it gives the rate and direction of maximum decrease in electric potential. In situations of symmetry, the vector notation simplifies. For instance, suppose E is constant is magnitude and direction, as between two opposite charged parallel electrodes. Then the change in electric potential in moving a distance x away from the positive plate is V = W = E d s = E x (4) Conversely, the magnitude of the electric field can be determined from E = dv dx = V x The minus sign signifies that E points in the direction that V decreases. Note also that the electric field can be expressed as volts per meter. (5)
Electric Field Mapping Page 3 of 5 In more general case, where the field is not symmetric, or when the symmetry involves more than one coordinate, the derivative must be taken with respect to each spatial coordinate. However, over small distances, one can treat the field as approximately linear and can approximate E(x) V (6) x where x is the relevant coordinate. 4 Laboratory Work Equipment Setup Figure 1: Equipment setup. 1. Assemble the Electric Field Mapping apparatus as shown in Figure 1. 2. Mount the universal holder on the mounting plate. The universal holder has three poles on which electrodes of different shapes can be connected. This is done by lowering the poles until they come into firm contact with the electrodes. 3. Place a polycarbonate plate on top of the mounting plate as a solid insulating base. 4. To record the equipotential points, place a sheet of white paper, a copy paper and a special conductive carbon paper in sequential order on top of the insulating base. 5. Connect the DMM, as a voltmeter, to adjust the dc power supply to a voltage output of 12 V. Switch off the power supply and disconnect the DMM from the power supply. Do not change this voltage setting for the rest of the experiment.
Electric Field Mapping Page 4 of 5 6. With the power supply off, connect one electrode to the ground terminal of the DC output on the power supply. Connect the other electrode to the positive terminal on the power supply. Once the power supply is switched on, an electric field is established between the electrodes. Note: In a conductor, such as the silver electrodes, even small electric fields will cause large amounts of charge to move in such a way as to tend to cancel out the field. Thus charges will distribute themselves on the electrodes so as to make them equipotential surfaces. The field between the electrodes will be that appropriate to the resulting distribution of charges. 7. Connect the negative terminal (COM socket) of the digital voltmeter (DMM) to the negative terminal (0 V) of the power supply. Recall that one of the electrodes is also connected to the same negative terminal of the power supply. Thus, the electrode will be considered at zero potential (0 V). 8. Connect the moveable test probe, which consists of a knitting needle held with a crocodile clip, to the positive terminal of the DMM. Touch the test probe to a point on the carbon paper to measure the voltage there. Note: The surface of the carbon paper has a much higher resistance than the electrodes. Small currents will tend to flow along the electric field lines (i.e. between the potential differences) set up by the charged electrodes but these will not be sufficient to significantly disturb the field pattern. CAUTION: At the end of each part of the experiment, always switch off the power supply before removing the crocodile clips. Part A: Parallel Electrodes In this part of the experiment, you will map out the equipotential lines between two oppositely parallel electrodes. A-1. Lower the two side poles on the universal holder until it is in contact with two bar electrodes on the carbon paper. Make sure the bar electrodes are parallel to each other. A-2. Measure the distance between two parallel electrodes and record it as d in the laboratory worksheet. Outline the shape of electrodes on the paper using the test probe. A-3. Starting at the negative electrode (this should be at 0.0 volts) find the equipotential lines at every 2 volts. That is, find the 2.0 V, 4.0 V, 6.0 V, 8.0 V and 10.0 V equipotential lines. The edge of the positive electrode should be at 12.0 V. To find a line, slowly drag the test probe away from the negative electrode until the DMM displays the voltage you seek. Keep the probe vertical and allow the DMM enough time to settle down before you take the reading. When you find the exact voltage, pressure the probe to leave a mark on the white paper. Repeat this process for the same potential for about 9 evenly spaced points, including the region beyond the ends of the electrodes.
Electric Field Mapping Page 5 of 5 Part B: Concentric Electrodes In this part of the experiment, you will map out the equipotential lines in the region between concentric oppositely charged cylindrical electrodes. B-1. Lower the central pole of the universal holder until it is in contact with the carbon paper. Replace the bar electrodes with a ring electrode and make it contact firmly with the two sides poles. B-2. Connect the central pole to the positive terminal on the power supply and the outer ring electrode to ground. B-3. Measure the radius of the central cylindrical pole and record it as a in laboratory worksheet. And, measure the inner radius of the ring electrode and record it as b in laboratory worksheet. Outline the shape of electrodes on the paper using the test probe. B-4. Find the 2.0 V, 4.0 V, 6.0 V, 8.0 V and 10.0 V equipotential lines. Find 8 10 points for each line in the region between the electrodes. Part C: Circular Electrodes In this part of the experiment, you will map out the equipotential lines around a pair of oppositely charged cylindrical electrodes. C-1. Without using any specially shaped electrodes, just lower that two side poles of the universal holder until they are in firm contact with the carbon paper. C-2. Connect one side pole to the positive terminal on the power supply and the side pole to ground. C-3. Measure the distance between two side poles from center to center and record it as L in laboratory worksheet. Outline the shape of electrodes on the paper using the test probe. C-4. Find the 2.0 V, 4.0 V, 6.0 V, 8.0 V and 10.0 V equipotential lines take readings both between and on the far side of the electrodes. Find enough points (10 15) for each line to clearly map the field. REMEMBER: Before leaving the laboratory, turn the voltage dial to minimum and turn off the power supply. Last updated: Wednesday 14 th January, 2009 12:40pm (KHCM)