Electric Field in Planar Geometry PDF

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Electric Field in Planar Geometry

James Van Cleave August 27th, 2020 Team Members: Nick Schulten Robert Pettas Coby Hipp

Physics 224 Section 03 Introduction Concerning electrical phenomena, electrical fields (  E ¿,

which is a vector, and electric

potentials (V), which are scalar, are two descriptive physical properties. When examining the electric force of a specific point in an electrical field, the force is directly proportional the charge at that specific point in the field, as described by the equation:

 E F =q  The work done by moving the charge from point (a) the point (b) is equal to the charge (q) multiplied by the change in electric potentials which are perpendicular to the electrical field. W = -q∆V Since the electrical field and electrical potential are related, if you were to map out a uniform electrical field, one could calculate the electrical potentials at points within that field. One could also solve for the electrical field if one were to know the electrical potentials within that specific field. In the following equations

( E ) is the electrical field, (V) is the electrical potential as a specific point, (d) is

the distance between two points.

|V a−V b|  E= d

and

∆ V =− E ∆ d

Objectives The objective of this experiment was to study the properties of a uniform electric field generated between two parallel metal bars and to map the equipotential surfaces within that field, all while practicing proper laboratory techniques and procedures.

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Physics 224 Section 03 Experimental Setup and Procedure

A piece of graph paper was placed under a clear plastic tray which was then filled with water. Two metallic L-shaped bars were placed in the water about 6 inches away from each other in a parallel position. A probe connected to a power source was placed on each metal bar. These metal bars were referred to as P1 and P2. P1 was connected to the positive terminal of the power source and P2 was connected to the negative terminal. A digital voltmeter labeled B was connecter from P1 to its positive terminal and the negative terminal was connected to a handheld probe labeled P3. A second voltmeter labeled A was connected the circuit with the negative being connected to another stationary probe labeled P4 and the positive terminal of voltmeter A was connected to P3. The sensitivity of both voltmeters were turned to 20V. Once the set-up was completed, the power source was turned on and the voltage output was set to 12V. With voltmeter B turned off and voltmeter A turned on, P4 was placed at x=2, y=0. P3 was then placed at points where y=0 and x= 2.2, 2.4, 2.6, 2.8, 3.0, 3.2, 3.4, 3.6, 3.8, 4.0, and a voltage reading was taken at each point. Next voltmeter B was turned on and voltmeter A was turned off. P3 was placed at x=2, y=0 to find the potential for that point. Once a potential had been recorded, voltmeter B was turned off and voltmeter A was turned on. Next P3 was placed on x=2, y=+1. The voltage was 0V at this point. P3 was moved along the y axis at x=2 and y= +1, -1, +2, -2 and the exact x coordinate was recorded where the voltage read 0V. This step was repeated with the tip of P4 being moved to (x=2.5, y=0), (x=3.0, y=0), (x=3.5, y=0), and (x=4.0, y=0). The resulting points where P3 read 0V were graphed out the field and 3

Physics 224 Section 03 vertical lines were drawn to show the locations of the measured electrical potentials. This graph is listed below. Analysis The values from Table 1 show that the electrical potentials are different and change in a uniform way as you move from one side of the electrical field to the other. This would indicate that there is a planar electrical field present. Since only one line of data points along the x axis was tested, one cannot definitively say that the electrical field is uniform based off this data. Table 2 offers more information as to the uniformity of the electrical field. From the mapped-out data points that were collected in part 2 of the experiment, the electrical potentials appear to be equidistant from each other at all points along the potential, indicating that they are parallel to each other. We know that electrical fields are perpendicular to potentials and since the potentials are parallel to each other, we can assume that the electrical field is uniform. This can be proved mathematically. The data from table 2, when entered into the equation for electrical fields, yielded an average value of 71.69. The linear trendline of the plot of (x values vs. potentials) had a slope of 71.834. 71.69 is only 0.195% different from the theoretical value of the electrical field. Conclusions The data collected in this experiment showed a very small amount of error. The minor variations in the x values when mapping the electrical potentials could have possible been caused by the length of the metal bars. Since the space between the metal bars was so close to the length of the bars, the electrical field was not 100% uniform. Had the bars been longer or the distance between the bars been shorter, the electrical field would have appeared more uniform. Other possible sources of error in the experiment could have been user error by the person measuring the x coordinate of P3. This could have been avoided by using more detailed graph paper or magnified the entire set up to minimize the inaccurate readings of the coordinates. Even though there was some error present, this experiment was conducted in a way and

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Physics 224 Section 03 achieved results that adequately demonstrated the characteristics of and the relationship between electrical field and electrical potentials. Appendix Table 1 x' (inch) 2.2 2.4 2.6 2.8 3 3.2 3.4 3.6 3.8 4

Tip of P3 (x=2.0 y=0) (x=2.5 y=0) (x=3.0 y=0) (x=3.5 y=0) (x=4.0 y=0)

x' (m) 0.05588 0.06096 0.06604 0.07112 0.0762 0.08128 0.08636 0.09144 0.09652 0.1016

x (m) 0.0254 0.0254 0.0254 0.0254 0.0254 0.0254 0.0254 0.0254 0.0254 0.0254

Potential Difference ∆V (V) 0.3 0.6 0.92 1.2 1.53 1.9 2.25 2.62 2.99 3.33 Table 2

∆V/(x’-x) (V/m) 9.843 16.873 22.638 26.247 30.118 34.001 36.909 39.673 42.042 43.701

Potential (V)

x for (y=0)

x for (y=+1)

x for (y=-1)

x for (y=+2)

x for (y=-2)

Average x (in)

-7.75

2.0

1.99

1.99

2.01

2.00

2.00

-6.88

2.5

2.50

2.51

2.52

2.49

2.51

-6.00

3.0

3.00

3.01

2.99

2.99

3.00

-5.05

3.5

3.51

3.50

3.49

3.50

3.50

-4.09

4.0

4.01

4.01

4.01

4.00

4.01

Between st 1 and 2nd equipotential 2nd and 3rd equipotential

 E

(V/m) 67.49 70.35

rd

th

74.43

th

th

74.47

3 and 4 equipotential 4 and 5 equipotential

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Average  E (V/m) 71.69

Average x (m) 0.0507 0.0636 0.0761 0.0889 0.1018

Physics 224 Section 03

Conversion of (inch) to (m) = 2.2 inches

V/m for (x=2.2in) =

1∈¿ 0.0254 m × ¿

= 0.0508m

∆V 0.3 V =9.834 = ' 0.0508 m−0.0254 m ( x −x )

1.99 + 1.99 + 2.01 + 2.00 ¿ Average x value = in ¿ ¿

 E =

∆V ∆ x´

=

2.51∈− 2.00 ∈¿ × 0.0254 m ¿ ¿ (−6.88 V − ( −7.75 V ) ) ¿

theoretical− experimental ¿ ¿ % Error = ¿ ¿ ¿

71. 83−71.69 ¿ ¿ ¿ ¿ ×100 %=¿

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V/m

Physics 224 Section 03

Average x value vs. Potential -3.50 -4.00 -4.50

Potental (V)

-5.00

f(x) = 71.83 x − 11.43 R² = 1

-5.50 -6.00 -6.50 -7.00 -7.50 -8.00 0.0400

0.0500

0.0600

0.0700

0.0800

0.0900

0.1000

0.1100

Average x value (m)

Questions 1. For the measurement by step 2, is the measured ∆V linearly proportional to (x’-x)? Yes. By looking at the graph of the average x value vs. the potential at that point, we can see that a linear trendline has been established. With an R2 value of 0.9995, we can say with confidence that the data is precise and follows a definite trend. The slope of 71.83 stands for the average change in potential as you increase the value of x by 1 meter which is the value of the electrical field (V/m). 2. Within a 10%-error margin, do the measured data infer a uniform electric field between the metal bars? Yes. The percent error for this data set was 0.195%. One can be confident inferring that the data would suggest the presence of a uniform electrical field between the metal bars. 3. Does the result change when you place the tip of P2 at different points on the 12V bar? Explain. No. The charge is distributed evenly throughout the metallic bar. Changing the placement of the tip of P2 on the same metallic bar would not change the electric field as long as the metallic bars remain parallel and the same distance apart from each other as before.

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