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In this experiment, I explored concepts relating to electric fields, such as fields generated by dipoles, between evenly spaced bars, and in the presence of conductors. The potential within the fields was calculated, along with the magnitude of the fields themselves. Although partial derivatives were not used, the magnitude was estimated by determining the changes in voltage over the changes in distance. This was accomplished by using a voltmeter and taking measurements at different points of the electric fields. Although our results did not follow the expected results in our dipole setup, as experimental calculations revealed our dipoles to be 7.44 ±0.17 cm, rather than 7.00 cm, our other results had greater accuracy. In the electric field between two bars, a linear fit with a high r2 value was observed to calculate an electric field of magnitude of 98 ± 7 V/m. Finally, in the presence of a conductor, a small electric field of magnitude 11 ± 1 V/m was observed, which, although not within the expected 0 V/m of the area contained within a conductor, was relatively precise in measure.
Introduction and Theory:
The electrostatic field is the force exerted on a positive test charge at any point in space. Coloumb's Law defines the electrostatic force exerted by two charges as
q1 and q2 refer to the charges, r is the distance between the charges, and k refers to Coloumb's constant. To determine the force of the electrostatic field on the test charge, or the force per charge exerted by the presence of another charge, equation 1 is divided by the test charge, yielding
Where E is the electrostatic field and Q is the charge generating the field. Consequently, the force on the test charge itself is given as follows:
Electrostatic fields can be depicted by electric field lines, which can be visualized as the joining of the electric field vectors. The field lines are drawn proportional to the strength of the charge and perpendicular to equipotential surfaces.
Voltage is a measure of electric potential. The difference in electric potential between two points in space, a and b, is the negative of the work down per charge by an electric field when a test charge is moved between the two points, as given
The electric potential at a point r is the negative work done as a test charge is moved from a distance infinitely far from a charged point to some distance r. Based on equation 4, the following equation gives the absolute potential at some distance r from a single point charge, Q.
In the presence of a single charge, an equipotential surface is the same distance from the charge at all points. Equipotential surfaces never intersect.
Electric fields can be measured by determining the electrostatic potential with a voltmeter. The relationship between the electric field and its electrostatic potential is given by
Equation 6 can be simplified if it is known that the field points in one direction only, for example, along the x axis:
Thus, the strength of the electric field is proportional to the negative rate of change of the electric potential
An electric field produced by equal and opposite charges, otherwise known as dipoles, separated by a certain distance, can be calculated through vector addition. Similarly, the potential can be calculated by scalar addition of the individual potentials. By equation 5, the potential along the x-axis is
By Equation 2, the electric field along the x-axis is as follows:
A uniform electric field can be created by two infinitely long conducting sheets with opposite charge densities. This field's potential will increase directly as the distance between the sheets changes, while the potential outside the sheets will remain constant.
A conductor is a material in which the charge carriers can move freely. When in the presence of an applied electric field, the charge carriers will redistribute themselves so that the field is cancelled and the electric forces on the charge carriers equals zero. Consequently, no static electric field exists within a conductor.
In this experiment I plotted the electric field generated under several conditions by determining the placement of the equipotential in a quadrant. Additionally, I quantified the magnitude of the electric field by finding ΔV between adjacent contours and dividing that value by the separation of the contours. The direction of the field was then determined as field lines are perpendicular to equipotential surfaces. The magnitude of the field was graphed along the x-axis between two dipoles and along the y-axis in between parallel conductors.
A water tray was set up with an AC transformer and voltmeter as shown in Figure 5. Graph paper was placed under the water tray to divide the tray into quadrants. The probe was held vertical when measurements were made.
Figure 5: Electric Potential Field setup. Courtesy Driscoll
Figure 2 shows the three different field geometries that were tested in this lab. Brass cylinders, blocks, and a hollow brass tube were used, respectively. The anode and cathode were attached to the AC transfer, while the common input of the DMM was connected to the 'a' anode and the probe was plugged into the V/Ω input. The DMM was set to read at 20 V, given that the actual voltage was 12 V. Because the DMM was being used to record potentials, error uncertainties were ±0.5% of the actual readings.
Figure 6: Field geometries. Courtesy Driscoll
The electrodes were placed as shown in Figure 6 7 cm from the y axis along the x axis. The potentials for each electrode, 'a' and 'b', along with the potential in between the two electrodes was determined. Measurements were taken along the entire x axis along values of ΔV=1.5 volts. These values were plotted in Origin and fit to the theoretical model of Equation 8. The Origin plot is attached and labeled Figure 1: Voltage vs. Distance. The chi-squared per degree of freedom was also determined. Measurements were taken along the y-axis every 5 cm, and also along equipotential lines of values ΔV=1.5, 3.0, and 4.5 volts compared to the reference y-axis in the fourth quadrant only. As before, measurements from the DMM had uncertainties of ±0.5% of the actual readings, while the error in determining the coordinate values were ±0.1 cm, as the graph paper gave readings in cm.
Brass bars were placed as shown in Figure 6 with their inner edges 6cm from the x-axis. Data was taken along the y-axis every 1.5 volts from the reference value at the origin. This data was plotted in Origin. A linear fit of data between the plates was performed. The Origin plot is attached and labeled Figure 2. Parallel Plates. Data was taken along the x-axis every 5.0 cm, and data was also collected along equipotential lines in the same manner as before in the fourth quadrant.
A hollow conducting cylinder was placed midway between the two plates as shown in Figure 6. Readings were taken every 0.5 cm along the y-axis between the bars. This data was also plotted in Origin, and is attached as Figure 3: Y-axis Data of Hollow Cylinder between Two Plates. Equipotential lines were also determined as before.
Results and Analysis:
Electric potentials and field maps of the three setups were charted and are attached as Figures 7, 8, and 9 for the dipole, parallel plates, and hollow cylinder, respectively. Figure 1 shows the voltage plot along the x-axis for the dipole setup. The actual data was plotted against Equation 8.
A χ2 value of 2799.2 was obtained, and Origin calculated 10 degrees of freedom. Thus, the χ2 per degree of freedom value was 280.0. Additionally, the value of 'a' or the distance between the origin and each electrode was experimentally determined to be 7.44 ±0.17cm. Uncertainties in the values of x were ±0.1 cm, as the graph paper was marked every 1 cm, while uncertainties in the values of V, or the potential recorded by the DMM was ±0.5% of the actual reading. Based on this information, Origin calculated the uncertainty and the χ2 values. Error bars were automatically calculated and inserted into the Origin graph.
Figure 2 shows the linear fit of the parallel plate setup as compared to the actual potentials recorded along the y-axis.
Figure 3 shows a plot along the y-axis of the parallel plate with a hollow cylinder setup. As before, uncertainties in the voltage and distance remained the same, which were plotted by Origin.
Figure 4 is a plot of the field as a function of position for the dipole setup. Since Equation 6 gives the magnitude of the field using partial derivatives, a similar equation can be used to estimate the magnitude of the electric field:
Where Δr can be reduced to Δx since I only considered the charges along the x-axis. The calculation was done automatically by Origin, and fitted to Equation 9:
Because Equation 8 from Figure 1 is concerned with finding the potential, it takes on a form of Equation 5:
While Equation 9 takes on a form of Equation 2:
For this reason, Equation 9 differs from equation 8 in that I am concerned with r2, not |r|. In addition, because I am dealing with an electric field or the force per charge at a certain point, Equation 9 finds the negative sum of the change in r2. Thus, the electric field is calculated by vector addition. In contrast, Equation 8 finds the actual potential and therefore finds the difference in the change in |r|. A χ2 value of 41.78 was obtained, while the 7 degrees of freedom were calculated, giving a χ2 per degree of freedom value of 6.0.
The field along the y-axis for the parallel conductor setup was calculated by Origin to be 0.98 ±0.01 V/cm, or 98 ± 7 V/m as shown in Figure 2. This was calculated in the same manner by following Equation 10, and finding the slope of the best fit line.
Figure 3 shows the Origin calculated field within the hollow cylinder. The slope of the line gives the electrical field to be 0.11 ±0.01 V/cm, or 11 ±1 V/m.
Our data did not always support our conclusions. For example, Figure 1 shows the expected model given the voltage between the dipoles. However, our experimental data showed a high χ2 value, indicating that error was present in our measurements. This could have been due to shifting of the dipole. As measurements were taken, both the movement of the probe and the contact between the wires was observed to have shifted the dipoles which would explain the high degree of inaccuracy observed. This conclusion is justified by the fact that although our dipoles were originally spaced 7 cm from the origin, the experimental data returned by Origin analysis shows that our dipoles were spaced 7.44 ±0.17 cm from the origin.
Figure 2 shows that our data was precise - a constant change in voltage was observed as distance increased, as shown by the high r2 value. This linear fit follows the prediction given by Equation 10, where the magnitude of the electric field is directly proportional equal changes in voltage over changes in distance. Additionally, Figure 7, our field map, shows approximately evenly spaced equipotential surfaces in between the two brass bars. The field along the y-axis for the parallel conductor setup was calculated by Origin to be 0.98 ±0.01 V/cm, or 98 ± 7 V/m.
In Figure 3, we expected a slope of zero inside the conducting sphere. As previously discussed,within a conductor, the electric forces on the charge carriers equals zero. Thus, static electric field should exist within a conductor; however an electric field was observed with a magnitude of 11 ± 1 V/m. This minute electric field is most likely the result of the brass cylinder not being a perfect conductor, and as a result, electric forces built up on the charge carriers within the conductor. This minute electric field was observed in Figure 8, where the area contained within the cylinder had a roughly constant charge of 8.0 V.
The electric field analysis in Figure 4 once again shows that our data contained a high degree of imprecision and inaccuracy, most likely due to the shifting of the dipoles. Thus error could be reduced by ensuring that the dipoles did not shift. In addition, having a lab partner confirm that the probe was vertical would also result in higher precision. A high χ2 value is evidence of this. However, we did observe a nonlinear relationship between the electric field and the distance, which is in line with our expectations given Equation 9.
I would like to thank Sandra Kikano and the Case Departments of Physics for help in obtaining the experimental data and preparing and providing the figures.