## Field Mills, Electric Fields, and Point Charges

**ABSTRACT**

Electric field was measured over increasing distance to determine the relationship between distance and electric field size (E) using a simulated point charge. Using the expression for

electric field, |?| = ??/?

2 , the data was graphically analyzed to understand the inversely

proportional relationship between the variables. Ultimately, by understanding the direct relationship between these two variables, the expression for electric field was linearized to generate a direct relationship which could then be used to calculate and determine the charge of

the object measured, otherwise known as the objects point charge.

**INTRODUCTION**

An electric field can be characterized by vectors, called electric field lines, that indicate the direction and magnitude of the field. Point charges can be used to determine how the electric field changes given a specific charge and the distance an object is away from that charge. The electric field of a point charge is determined by the equation

|?| = ??/?

2

. This equation was derived from Coulomb’s Law, which is used to

determine how distance effects electric fields.

This experiment studied the relationship between electric field and distance by moving a point charge further away from a field mill, used to read the electric field. By studying the expression for electric field, |?| = ??/? 2, it can be seen that electric field (E) and distance (d) are inversely proportional. Because distance is in the denominator of the equation, as the value gets larger, the value of E will continue to decrease. This equation was “linearized,” meaning that its variables were manipulated so that they would have a linear, or direct relationship with each other. The new linearized equation follows the form ? = ?? + ?. By creating a linear dataset, KQ is not proportional to 1/d2 . The linearized graph, shown in Figure 2 demonstrates the same y-axis, electric field (E), but the x-axis now followed the form 1/d2 . Figure 2 will be described more explicitly in the Results.

By graphing the linearized data, a slope could then be obtained to solve for the charge of the object.

Overall, the study is used to explain how distance from a point charge alters the electric field, by analyzing proportionality of the two variables. Prior to the study, distance was believed to possess a linear relationship to electric fields.

**PROCEDURE**

The experiment was conducted by charging a graphite-covered ball attached to an insulating stick with a rubber rod. The rubber rod was previously charged with a wool cloth. The ball was initially positioned

5.5 cm above the field mill. Then, a reading of the field mill was recorded every 2 cm, moving the ball further away each time.

Initially, only the distance and electric field were recorded. A graph comparing distance (m) and the electric field (V/m) was created

to analyze the relationship between the two. The electric field was initially recorded in Kilovolts per 100 millimeters. In order to maintain consistency in units, the electric field was converted by the following sample

equation:

???????? ????? (

?

?

) =

5.45??

100??

×

1000??

1?

×

1000?

1??

Then, the distance was linearized by converting distance to 1/distance, by using the following unit conversion:

?????????? ???????? (

1

?2

) =

1

(???????? (?))

2

Following the unit conversions, two plots were generated on Logger Pro. One graph, Figure 1 demonstrated the raw, unaltered

distances (m) versus the electric field (V/m).

The second graph was generated using the linearized distances (1/m2 )versus the electric field (V/m).

Using the linearized graph, Figure 2, a slope was generated using the Logger Pro software, by creating a linear fit on the graph. The electric-field equation in pointslope form, is as follows:

|?| = (??) (

1

?

2

) + ?

The variables can be described using the following information: 1) KQ represents the slope, 2) 1/d2 is the equivalent to a typical xvariable,

and 3) b represents the y-intercept. The y-intercept for this experiment was 0. Once the best fit line was generated and the slope was found, the point charge of the ball was determined, by using the value of the slope and setting the equation as follows:

????? ???? ????ℎ = ??

In this equation, K represents Coulomb’s

law constant, 8.99x109N. Thus, Q was

solved using the two conversions:

? =

????? ???? ????ℎ (??)

? (

??2

?

2

)

? =

??

?

Using these two conversions, the new

equation was:

? =

????? ???? ????ℎ (

??2

?

)

? (

??2

? × ?

)

**RESULTS**

The data, displayed on Table 1,

demonstrates how the electric field decreased as distance increased. Every

increment of distance increased by 0.02 meters per recording, but the electric field not display the same linear decrease.

Instead, the distance versus electric field graph, shown by Figure 1 demonstrated that distance and the electric field are, in fact, not

linearly proportional. Based on Figure 1 and the data shown in Table 1, the relationship was determined to be inversely proportional.

In contrast, Table 1 also displayed the linearized distance. After linearization, the distances appeared to have reduced over

time, due to the nature of the relationship between distance and 1/distance2. When placed on a graph, Figure 2, the data presented a linear relationship between linearized distance and the electric field.

Thus, electric field was proportional to 1/distance2

.

The linear fit placed on Figure 2 determined a slope for the equation:

|?| = (??) (

1

?

2

) + ?

The slope was 153.78 Vm, which represented KQ in the equation. Using the

previously discussed relationship, the charge of the ball (Q) was 1.71×10-8C, shown by the following:

? =

153.78 (

??2

?

)

8.99 × 109 (

??2

? × ?

)

= 1.71 × 10−8?

While Figure 2 presented a more linear representation of the relationship between distance and electric field, the data was not perfectly linear. The experiment did not account for charge dissipation from the ball over time. By the end of the experimentation, the ball may have experienced a decrease in charge magnitude, but the data does not demonstrate the effect

of this phenomena. In order to potentially account for charge dissipation, the data would need to be collected quickly and efficiently to limit the amount of charge loss. A second source of error and potential

reason for imperfect data came from using a ruler to measure distance. Rather than use a ruler and hand-measurements, it would have

been more effective to use a laser to detect change in distance.

Table 1. Results from an experiment testing the relationship between distance (m) and converted electric field (V/m).

Distance

(m)

Linearized

Distance

(1/m^2)

Electric

Field (V/m)

0.055 330 5.45E+04

0.075 178 3.82E+04

0.095 111 2.73E+04

0.115 75.6 2.10E+04

0.135 54.9 1.68E+04

0.155 41.6 1.41E+04

0.175 32.7 1.16E+04

0.195 26.3 9.70E+03

0.215 21.6 8.00E+03

0.235 18.1 6.70E+03

Figure 1. Electric field (V/m) versus distance (m). Analyzing the relationship between distance and electric field.

Figure 2. Electric field (V/m) versus linearized distance (1/m2

). Analyzing the relationship between linearized distance and

electric field.

**CONCLUSION**

The expression for electric field, derivedfrom Coulomb’s law, enabled the

determination of the relationship between distance and electric fields. Electric fields are inversely proportional to distance.

However, upon linearization of distance, electric fields are linearly proportional to 1/distance2 . The creation of the linearized equation, allowed for the distance and electric field to be used to determine the

charge of the ball, acting as the point charge, which was determined to be 1.71×10-8C.

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