PHYS1156 Experiment #16 Lab Report PDF

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Report for Experiment #16 Electric Field and Electric Potential

Alec Braverman Lab Partner: Your Partner’s Name TA: Nathaniel Avish 9/29/20

Introduction The experiment that we conducted was meant to investigate the electric potential that is produced within an electric field. We observed both electric potential and electric fields between two electrodes. The primary purpose of performing the investigations was to better understand the relationship between electric potential and electric fields in different scenarios; we know that an electric field is a vector field that is related to the forces between electric charges. Conversely, we know that the electric potential field is scalar and is instead related to the energy between electric charges. The following equation demonstrates this relationship between the electric field E and the electric potential V:

E x=

In this formula, x is distance and

−∆ V ∆x

∆ x is the change in distance. We were able to study and

visualize the electric field produced in the experiments due to the equipotential lines that connect points of equal potential. We also know that the electric field is always perpendicular to these lines. In Investigation 1, we examined the electric field that was produced by parallel electrodes; two parallel electrodes were both attached to positive and negative ends of a power supply, which created an electric field. A voltmeter was then used to trace and help draw the equipotential lines at varying electric potentials. We recorded the values for the electric field and electric potential and compared them to theoretical values. In Investigation 2, the electric field produced between concentric electrodes was examined. Two circular electrodes with different radii were used to create an electric field by connecting the negative terminal of the power supply to an outer brass

ring and the positive terminal to the electrode in the center. We collected measurements for the inner radius a, outer radius b, the electric field, electric potential, and radial distances between equipotential lines. We used this data to calculate theoretical values that we then compared to our measured values. There may have been slight deviations from error due to inexact voltages or faults in the equipment, which could have contributed to deviations in the results reflected in this report. Investigation 1 In this investigation, we examined the electric field that was produced by parallel electrodes; two electrodes were placed 10 ± 0.1 cm apart (parallel to each other) on a sheet of conducting paper. The conducting paper was placed on a rubber pad in order to isolate the currents being produced. A grease pencil was used to mark the edges of the electrodes to help with drawing the equipotential lines. Next, the negative terminal attached to the power supply was hooked up to the left electrode (at x=0 cm) and the positive terminal was attached to the electrode on the right (at x=10 ± 0.1 cm). The power supply was set to 10V, and the negative terminal was grounded and attached to the voltmeter. This is because the voltage was equal to 0 at x=0 cm; finally, the voltmeter precision level was set to 20V max. Once the voltmeter and power source were set up and connected, we started using the probetipped wire to trace out the equipotential lines. This was done in 1V increments, starting with 5V and then moving to the 4V and 6V lines, the 3V and 7V lines, and concluding with the 2V and 8V lines. Additionally, we traced the lines on one end of the electrode past the end and into the fringe field (this also helped show the curvature of some of the lines). The drawing of all lines

was done with equal pressure to ensure accurate voltage readings. After recording all measurements and drawing all the lines, the power source was disconnected. During the investigation, electric field lines were drawn perpendicularly in the center and roughly every inch along the inside of the electrodes. The distances between adjacent lines,

∆ x , were also recorded. Using these measurements, we used the equation

E x=

−∆ V ∆x

to

calculate the electric field for each pair of equipotential lines, where Ex is the electric field for a particular line. The following graphs show the plots for E vs. x and V vs. x (E was plotted against the average distance xav, which was obtained by taking the two distance measurements for a pair

of lines, xa and xb, and taking their average (

xa + xb )). For uncertainties in the positions of the 2

lines, we used half the distance of the line thickness drawn by the pencil, which is approximately 0.15 cm. This value is very close to the one obtained through error propagation; multiplying 0.1 cm (the standard error determined for the ruler) by

√ 2 , we get a value of δ x=0.142 .

Electric Potential v. Position 12

Electric potential, V (V)

10

f(x) = x R² ==11.1 x − 0.28 f(x) R² = 0.97

8

X1 Linear (X1) Linear (X1) Theoretical X1 Linear (Theoretical X1)

6 4 2 0

1

2

3

4

5

6

7

8

Position, x (cm)

9 10 11

Figure 1: Plot of Electric Potential vs. Position

Electric field v. Position 1.4

Electric field, E (V/cm)

1.2 f(x) = 0.02 x + 0.95 R² = 0.12

1 0.8 0.6 0.4 0.2 0

1

2

3

4

5

6

7

8

9

10

Position, x (cm)

Figure 2: Electric field vs. Position The slope of the line in Figure 1, 1.1022, was the calculated electric field value. This was calculated using the electric potential equation, V (x)=Ex Where V is the electric potential, E is the electric field, and x is the distance. This value can be compared to the expected value, which was 1.0. All slopes were calculated using IPL’s straightline fit calculator. This value was expected considering voltage was set at a constant 10V at x=10 cm. The observed value of E=0.954 ± 0.066 was very close to the expected value (error was calculated using an online error calculator) but is still not quite equal within experimental uncertainty. Even though E was expected to equal 1.0 V/cm at all points, there was still some variance. This is likely due to systematic error with the lab equipment, or inconsistent measurement techniques.

Theoretical values for the electric potential V were calculated by taking the values obtained from the above equation for electric potential and multiplying the theoretical value

E=V /x

by the measured distance. Therefore, every value for V corresponded with the appropriate measured distance. The error quantity for electric potential, δV, was obtained by estimating the variance in voltage when recording the data and using it to perform error propagation. Due to the fact that our theoretical values did not exactly match up with the observed values, it was clear we underestimated the amount of error present. The likely cause for this was probably inconsistent pressure on the probe when tracing the equipotential lines and taking measurements. Any other experimental error could have also contributed to deviations in results. Investigation 2 In this investigation, we examined the electric field produced between concentric electrodes. A pair of perpendicular lines the length of the radius of the outer brass ring were drawn in the center of the same conducting paper from Investigation 1. A smaller brass ring was placed inside the outer one, and these two rings acted as the concentric electrodes. The radius of the smaller, inner ring (a) was measured at a=1.5 ± 0.1 cm, and the larger, outer radius (b) was measured at b=11.4 ± 0.1 cm. The grease pencil was used again to trace an outline of the electrodes. The grounded negative terminal of the power supply was attached to the outer ring, while the positive terminal was connected to the inner electrode in the center. Once again, voltage was set to 10V, and the probe was calibrated to ensure that it was reading 10V. In a similar manner to the first investigation, we started by tracing the 5V equipotential line. Then, every 45˚, subsequent lines

were traced. Following the 5V line, we traced the 4V, 3V, and 2V lines, followed by the 6V and 7V lines. The measured points along the circle were connected to form the equipotential lines, and the radial distance r from the origin to each of the lines was measured along the coordinate axis using a ruler. A ravg value of 3.75 and an error δravg value of 0.15 were obtained by averaging all the radial distances and performing error propagation. Using this data, we were able to produce a plot of electric potential V vs. radial distance ravg. This data had a logarithmic fit due to nature of the relationship between electric field and electric potential for concentric electrodes, as shown by the equation for the magnitude of the electric field:

(

E (r )=

)

V0 1 ln ( b ) −ln( a ) r

Where V0 is the power supply voltage (10V). The theoretical value for V was calculated using the equation:

V theoretical =V 0∗(

The above data produced the following graph:

ln( b )−ln ( r ) ) ln ( b )− ln ( a )

Electric Potential V v. Average radial distance r 12

Electric Potential (V)

10 8

f(x) = − 6.83 ln(x) + 15.71 R² = 0.97 Observed Logarithmic (Observed) Theoretical Logarithmic (Theoretical)

6 4 2 0

f(x) = − 2.49 ln(x) + 5.22 R² = 0.91 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7

Distance (cm)

Figure 3: Plot of electric potential versus the average radial distance The results were calculated using the data found above, and even though the data followed a somewhat expected trend, the measured data does not match the theoretical predictions within the calculated uncertainty. Considering the same equipment was used for both investigations (minus the brass rings), it is fair to assume that there was also likely error from the probe and from inconsistent pressure on the conducting paper when taking measurements. All of this supports the conclusion that our error value for this investigation should have been larger than what was found. Conclusion Throughout the experiments we conducted, we studied and observed the relationship between electric field and electric potential. Investigation 1 did this by demonstrating the electric field that was produced by parallel electrodes. We were able to measure observed values and calculate theoretical values using the measurements of electric potential, electric field, and distance between points. This led to finding an observed electric field value of E=0.954 ± 0.066 V/cm and

an electric potential value of V=1.1022 V. When compared with the theoretical value of 1.0, we found they were not equal within the calculated uncertainty. This implied additional error that had not been accounted for, likely due to inconsistent measurements. In Investigation 2, we examined the relationship between electric field and electric potential in concentric electrodes. Similar values were obtained, but the deviation from the expected error value was far greater than anticipated. This was likely due to additional systematic errors.

Questions 1. In Investigation 1, if the potential difference ΔV between the plates were doubled, how would the equipotential lines and electric field lines change? The electric field in between the electrodes would point from positive to negative (right to left) 2. Imperfections in the electrodes and poor mechanical contact with the conducting paper can affect the electric fields that are measured in this experiment. What type of error does this introduce into your calculations? This introduced unexpected error into the results, and led to δV, δE, δxavg, and δravg that were larger than expected. For this reason, there was deviation between observed quantities and theoretical calculations. 3. For the parallel electrodes, is the average electric field in the fringe region smaller or larger than in the central region?

The average electric field between the parallel electrodes was smaller in the fringe region than in the center because as you approach the fringe region, the lines begin to curve more; this increases the distance and decreases the magnitude of the electric field. 4. In Investigation 1, how does the magnitude of the electric field change if the distance d between the electrodes is halved, assuming that the power supply stays at 10 V? Assuming the power supply stays at 10V, the magnitude would double V ( x ) =Ex 10 V ( x ) =E ( 0.05 m )

E=200

V V =2.0 cm m

This is double the theoretical value of 1.0 that was found when d = 10 cm 5. In Investigation 2, if everything stays the same, except the diameter of the outer ring is doubled, how does the electric field change? If the diameter of the ring is doubled and everything else remains the same, the electric field would change, but not by double or by half.

(

E (r )=

)

V0 1 ln ( b ) −ln( a) r

In this equation, b is the diameter of the ring. As b increases,

(

V0 ln (b )−ln ( a)

therefore E decreases by an amount that is neither double nor half.

)

decreases. And...