Engineering Electromagnetism and DriversLab 1 Electrostatics Field Plotting

Author: Ruimin Zhao 1302509Ruochen Fu 1302509

Module: EEE 108

Lecturer: Dr.Gray

Date: March/30/2013

Abstract

• Experiment Purpose

– Electric filed of conductor system is an essential physical property in electronics,

which can be measured using analog methodology. This lab examined the electric

field of conductor systems composed of two parallel plates and two concentric cylin-

ders. In addition, the obtained data was comprehensively analysed thus the features of

electric field of two cases, such as the changing mode of the electric strength with the

increase of radius in the two concentric cylinder system, were learned.

• Experimental Procedure

– First, apparatus and construct circuit were set up and built. A circular grided sheet

which can conduct electricity was used to measure each point of a certain voltage.

Then, conductive water was poured into the set to finish the preparation stage of the

experiment. Following that, the voltage probe was placed to various points upon the

sheet to find the locations of points where the voltage value is 1V,2V,3V,4V,and 5V. So

that, the equipotential level lines can be drawn out and corresponding analysis based

on the obtained experimental results and theoretical results.

• Main Conclusion

– The electric field lines distribution of the two parallel plates is parallel and the electric

strength is a constant. While for the two concentric cylinders situation, the electric

field lines are concentric circles whose line density decrease with the increase of ra-

dius. However, the experimental results had errors, to improve the experimental accu-

racy, the experimental circumstance should be more professional - experimenters are

supposed to be separated with a safe distance so that the walking around people will

not interfere others’ experiments by shaving the experimental set table without even

knowing it. Also, more points should be measured so that the analog results can be

more convincing.

i

Contents

Abstract i

Contents ii

1 Introduction 11.1 Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Apparatus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

2 Methodology 22.1 Experimental Set Up and Procedure . . . . . . . . . . . . . . . . . . . . . . . . 2

2.1.1 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2.1.2 Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2.2 Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.2.1 Experimental Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.2.2 Theoretical Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

3 Error Analysis and Discussion 103.1 Error Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3.2 Task complement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3.3 Problem - solving Experience . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

4 Conclusion 154.1 Conclusion for Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

4.2 Suggestion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

A Questions 16

B Matlab Code 17

Reference 19

ii

Section 1

Introduction

A fairly useful means of visually representing the vector nature of an electric field is to make use

of electric field lines of force, and in this way, a pattern of several lines that extend between source

charge.[1] The fundamental relationship between electric field and potential is E = −dVdx [2].

1.1 Objective

This lab aims at gaining a more comprehensive understanding of the concept of Electric Field by

examining the Electric filed lines distribution patterns among different conductor systems. In this

case, the conductor systems specially refer to the systems composed of two parallel plates and two

concentric cylinders respectively.[2]

During the observation of the electric fields, the electric charges’ roles of sources or sinks of lines

should also be paid attention to. Moreover, the physical fact that ”the electric field is the negative

gradient of potential” should be verified with the aid of obtained data and reasonable analysis.

1.2 Apparatus

• High-impedance digital Voltmeter: Accurate magnitude of voltage among two polar of it

can be measured.

• Voltage Supply: Voltage and frequency can be set according to requirement.

• Voltage probe: The potential difference between the point pointed by voltage probe and the

negative electrode can be measured.

• Circular Experiment Set: A piece of coordinate paper is placed in it with the same size.

A grid of reference marks are printed on that sheet making it easier to identify the points

location between the conductors.

• Electrode(parallel plate and concentric cylinder): conductor system are composed of by

them.

1

Section 2

Methodology

2.1 Experimental Set Up and Procedure

2.1.1 Theory

• Analog Experiment

– Analog experiment method has extremely broad application in scientific experiments.

In order to overcome the difficulty of direct measurement of electrostatic field, we

modeled on fully consistent current field, so that it is easy to directly measure the

electric current field to simulate electrostatic field.

• Gauss’s Law

– The integral of electric field (normal component) can be worked using Gauss’s Law as

shown in equation below. Where S represents the corresponding closed surface, Q is

the closed charge, and parameter ε permittivity.∫E • dS = Q

ε [2]

2.1.2 Procedure

• 1)Preparation: set up apparatus and construct circuit

– Place the circular experiment set

∗ The piece of grided sheet was neatly placed on the circular set and properly ori-

ented with its axis parallel with the central horizontal conducting bar so that the

raw and column can effectively represents the coordinate of points.

∗ Water was poured into the set till the upper surface of water level reached the line

marked around the set.

– Set up the power source

∗ The voltage was set to be 8V and frequency was around 200Hz.

∗ The negative terminal of power supply was connected to one side of the electrode

(the edge of the circular set)

2

Two Parallel Plates

• 2) Measurement: conduct concrete measurements as required

– The voltage probe was placed to various points upon the sheet to find the locations of

points where the voltage value is 1V,2V,3V,4V,and 5V.

– According to previous physics knowledge, it was expected that the equipotential sur-

faces of two parallel plates should be close to ”paralleled pattern”, therefore in the

concrete measurement process, once one point of a certain voltage was found, the rest

of series of points of this voltage were found approximately along the line parallelled

to the plate and cross that point. In this way, the work efficiency was improved.

Two Concentric Cylinders

• 3) Measurement: conduct concrete measurements as required

– The voltage probe was placed to various points upon the sheet to find the locations of

points where the voltage value is 1V,2.5V,4V and 5V.

– Again, according to previous physics knowledge, it was expected that the equipotential

surfaces of two concentric cylinders should be close to ”circular pattern”, therefore in

the concrete measurement process, once one point of a certain voltage was found,

the rest of series of points of this voltage were found approximately along the circle

crossing that point. In this way, the work efficiency was improved.

The detailed cable connection achieved by three cables are presented below:

Figure 2.1: Cable Connection

3

2.2 Result

2.2.1 Experimental Result

Two Parallel Plates

The coordinates of measure points matching different potential are presented is Table 2.1, and also

marked in a 2 dimensional graph as scattering graph as shown in Figure 2.4(a).

Table 2.1: Coordinates of points obtained at each voltageVoltage

1V (-5.6,10) (-4.9,8) (-4.3,6) (-4.3,4) (-4.2,2) (-4.2,0) (-4.3,-2) (-4.3,-4) (-4.7,-6) (-5.2,-8) (-6.2,-10) (-7,-12)2V (-1.6,0) (-1.2,-2) (-1.2,-4) (-1.1,-6) (-1.1,-8) (-1.4,-10) (-1.2,2) (-1.1,4) (-0.9,6) (-0.9,8) (-0.6,10) (-0.1,12)3V (2.5,0) (2.3,2) (2.4,4) (2.5,6) (2.8,8) (3.3,10) (4.5,12) (2.5,-2) (2.3,-4) (2.8,-6) (3.2,-8) (4.5,-10)4V (5,0) (5,2) (5,4) (5,6) (5.3,8) (6.3,10) (5.3,-2) (5.4,-4) (5.6,-6) (6.1,-7)5V (6.5,0) (6.5,2) (6.5,4) (6.5,6) (6.5,8) (6.8,9) (6.7,-2) (6.7,-4) (6.7,-6) (7,-7)

Additionally, to visualize the result more directly, contour map and even 3 dimensional graph con-

necting the relationship among position and potential are simulated as shown in Figure 2.2(b) and

2.2(c). to illustrate the equipotential pattern in this case, where the conductor system is consisting

of two parallel plates.

(a) (b)

(c)

Figure 2.2: contour map and 3 dimensional graph

4

Two Concentric Cylinders

The coordinates of measure points matching different potential are presented is Table 2.2, and also

marked in a 2 dimensional graph as scattering graph as shown in Figure 2.5(a).

Table 2.2: Coordinates of points obtained at each voltageVoltage

1V (0,9.5) (6.6,6) (9.2,0) (6.5,-6.4) (-6.4,5.9) (-8,0) (0,-7.9) (-5.2,-6.2)2.5V (0,3.8) (2.7,2.7) (3.6,0) (2.4,-2.5) (0,-3.4) (-2.1,-2.7) (-3.5,0) (-2.2,2.5)4V (0,1.6) (1.2,1.2) (1.6,0) (1.2,-1.1) (0,-1.7) (-1.1,-1.3) (-1.8,0) (-1.2,1.2)5V (0,1.1) (0.7,0.7) (1.1,0) (0.8,0.7) (0,-1.2) (-0.7,-0.8) (-1.2,0) (-0.8,0.8)

Moreover, again, corresponding contour map and even 3 dimensional graph connecting the re-

lationship among position and potential are simulated as shown in Figure 2.3(b) and 2.3(c). to

illustrate the equipotential pattern in this case, where the conductor system is consisting of two

concentric cylinders.

(a) (b)

(c)

Figure 2.3: contour map and 3 dimensional graph

5

2.2.2 Theoretical Result

Condition: The field in parallel plate case is predicted to be directed perpendicular to the plates,

and directed along the radial direction in the case of the concentric cylinders.

Parallel Plate

Figure 2.4: Conceptual graph for parallel plate system in mathematical model

According to E = −dVdx , the integration derived is V = −Ex+C where C is a constant. Then

consider the boundary conditions we obtain that: V=0 when x=0; V=Vd when x=d.

Therefore we can substitute the parameters worked out and express the integrated equation as:

V (x) = Vdd x

Corresponding distances from 0 potential plate worked out theoretically are presented below:

Voltage(V) x(cm)

1 74

2 72

3 214

4 7

5 354

6

The coordinates of points derived by the calculated values recorded in above Table are pre-

sented in Table 2.3, and also marked in a 2 dimensional graph as scattering graph as shown in

Figure 2.5(a).

Table 2.3: Coordinates of points obtained at each voltageVoltage

1V (-5.25,0,1) (-5.25,2,1) (-5.25,4,1) (-5.25,6,1) (-5.25,8,1) (-5.25,-2,1) (-5.25,-4,1) (-5.25,-6,1) (-5.25,-8,1)2V (-3.5,0,2) (-3.5,2,2) (-3.5,4,2) (-3.5,6,2) (-3.5,8,2) (-3.5,-2,2) (-3.5,-4,2) (-3.5,-6,2) (-3.5,-8,2)3V (-1.75,0,3) (-1.75,2,3) (-1.75,4,3) (-1.75,6,3) (-1.75,8,3) (-1.75,-2,3) (-1.75,-4,3) (-1.75,-6,3) (-1.75,-8,3)4V (0,0,4) (0,2,4) (0,4,4) (0,6,4) (0,8,4) (0,-2,4) (0,-4,4) (0,-6,4) (0,-8,4)5V (1.75,0,5) (1.75,2,5) (1.75,4,5) (1.75,6,5) (1.75,8,5) (1.75,-2,5) (1.75,-4,5) (1.75,-6,5) (1.75,-8,5)

Additionally, to visualize the result more directly, contour map and even 3 dimensional graph con-

necting the relationship among position and potential are simulated as shown in Figure 2.5(b) and

2.5(c). to illustrate the equipotential pattern in this case, where the conductor system is consisting

of two parallel plates.

(a) (b)

(c)

Figure 2.5: contour map and 3 dimensional graph

7

Concentric Cylinder

Figure 2.6: Conceptual graph for concentric cylinder system in mathematical model

In this case, applying Gauss’s Law, we obtain E(r) = −dVdr = Q

2πrε (a < r < b). Rearranging,

we obtain Vrb =Q2πε ln(

br ). Then, consider the boundary conditions: V=0 when r=b; V=Va when

r=a. Substituting the parameters and rearranging, we obtain Vrb =ln(b/r)ln(b/a)Va and Q = 2πεVa

ln(b/a) .

Subsequently, there are two method to obtain E using the derived two equations of Vrb and Q

respectively with the aid of original expression E(r) = −dVdr = Q

2πrε .

Corresponding radius with the central cylinder to be the center worked out theoretically are p-

resented below:

Voltage(V) r(cm)

1 8ln(14)

2.5 165ln(14)

4 2ln(14)

5 85ln(14)

8

The coordinates of points derived by the calculated values recorded in above Table are pre-

sented in Table 2.4, and also marked in a 2 dimensional graph as scattering graph as shown in

Figure 2.7(a).

Table 2.4: Coordinates of points obtained at each voltage

Voltage

1V (3.03,0,1) (-3.03,0,1) (0,3.03,1) (0,-3.03,1)2.5V (1.212,0,2.5) (-1.212,0,2.5) (0,1.212,2.5) (0,-1.212,2.5)4V (0.7575,0,4) (-0.7575,0,4) (0,0.7575,4) (0,-0.7575,4)5V (0.606,0,5) (-0.606,0,5) (0,0.606,5) (0,-0.606,5)

Moreover, again, corresponding contour map and even 3 dimensional graph connecting the re-

lationship among position and potential are simulated as shown in Figure 2.7(b) and 2.7(c). to

illustrate the equipotential pattern in this case, where the conductor system is consisting of two

concentric cylinders.

(a) (b)

(c)

Figure 2.7: contour map and 3 dimensional graph

9

Section 3

Error Analysis and Discussion

3.1 Error Estimation

Distinctive Error

• Experimental Error

– The contour graph for parallel plates shown in Figure 2.2(a) has excessively large

difference with the theoretical one and even with its scatter graph.

– The cause is that the number of points measured was not huge and virtually all the

measured points laid in range between -7 to 7 in x axis. However, when i simulated the

graph, the range was set to be -14 to 14 in x axis as the real range of the experimental

set. Therefore, for the left side of the graph, the results are far from theoretical situation

as there was a lack of data in that domain.

– To solve this problem, the scatter graph in Figure 2.2(c) should be paid more attention

to replace the Figure 2.2(a) because the data are real and the situation presented in

scatter graph was more real.

Slight Errors

By observing the electric potential distribution graphs of both the experimental results and the

theoretical results, it can be clearly seen that they are reasonably similar in the micro perspective.

However, the exact patterns are still slightly different: the theoretical graphs are apparently more

smooth and have more regular shape:

The one for parallel plate has parallel lines with generally the same space between each pair of

adjacent lines; while the one for concentric cylinder has a series of circular equipotential planes

with different space between adjacent circles.

10

Possible causes for the slight errors

• Accidental Error

– The connection parts among terminals or electrodes might not be stable all the time.

– The experimental set may not be kept still because of the occasional chaos in lab.

– The relative position of the sheet to the circular set may be slightly changed after

dozens of times of pointing by voltage probe.

• Systematic Error

– The accuracy of the apparatus cannot be completely correct, for example the high-

impedance voltmeter may have slightly inaccurate voltage supply.

– The coordinates of each points were read by experimenters, which cannot be totally

right because the estimation of each number is only two or three digits after the decimal

point.

3.2 Task complement

Electric Field of concentric circles

The features of the E field of the case of two concentric cylinders are that the equipotential level

lines distributed as several concentric circles whose center is the central cylinder. Also, in terms

of the density of electric field, it was found that the potential drop slower and slower with the

increase of the radius, which indicates that the electric field also has smaller density when the

radius increase.

Electric Field Expression Using Gauss’s Law

The expression is E(r) = −dVdr = Q

2πrε , which was derived in previous section.

11

Electric Field Distribution

The direction of electric field is vertical to the equipotential level lines. The electric field distribu-

tions of two cases are presented below.

(a) (b) (c)

Figure 3.1: Electric Field for parallel plates

(a) (b) (c)

Figure 3.2: Electric Field for concentric cylinders

• Parallel Plates: The direction of electric field is from high positive potential to low nega-

tive potential, and are virtually parallel with horizontal axis in this case. Theoretically, the

electric field density should be the same between two long parallel plates. However, in this

case, the length of plate is limited thus the experimental result is not completely the same as

the theoretical result.

• Concentric Cylinders: The direction of electric field is virtually along the radius of the

circular equipotential level lines. Theoretically, the circular equipotential level lines should

be strict concentric circles and the electric density is decrease with the increase of radius.

12

Electric Field Function and Graph

The graphs electrostatic potential as a function of the distance from the negative electrode for

the parallel plates and from the center to the position of the outer electrode for the cylindrical

electrodes are presented in Figure 3.3(a) and Figure 3.3(b). In each graph, the more smooth one is

the theoretical one while the less smooth one is the experimental one.

Function: V = ln(14/r)14 × 8 for parallel plates, and V = 8

14 × x for concentric cylinders.

(a) (b)

Figure 3.3: Voltage

The deduced plots of electric strength as a function os distance are presented in Figure 3.4.

Function:E = − 8ln14 × 1/x for parallel plates, and E = 8

14 for concentric cylinders.

(a) (b)

Figure 3.4: Electric Field Strength

Comments:It can be clearly seen that the general trend of experimental graph and theoretical graph are basi-

cally the same, while the error is likely to be caused by the conductance of the water within the

experimental set (In the experiment, it was found that the highest voltage point that could be found

in set was 5 rather than 8, which suggested that the conductance of the water is fairly limited).

For the parallel plates case, the electric field theoretically should be the same, while that of con-

centric cylinders should decrease with the radius.

13

Comparison Graph

By comparing Figure 2.2(a) with Figure 2.5(a), Figure 2.3(a) with Figure 2.6(a) respectively, it can

be seen clearly that the basic pattern were the same. Detailed explanation can be seen in ErrorEstimation section.

(a) (b) (c)

Figure 3.5: Two Parallel Plates

(a) (b) (c)

Figure 3.6: Two Concentric Cylinders

3.3 Problem - solving Experience

• Failure of finding the point of voltage 5

– At the beginning, when find the coordinate of the point where voltage is 5V, it was

failed because the highest voltage is merely 3V. Then, after a few minutes of reflection

and communication with the teacher, we solved this problem by replacing the purified

water with usual water obtained from drag directly. The reason why purified water

cannot satisfy our need is that it is noe=t conductive enough because of its purified

features of lacing conducting materials in water.

– This experience taught us that it is important to ensure that every step in an experiment

should be reasonable with enough scientific support. It is always too careless to ran-

domly choose a material to use in experiments. Strict and scientific attitude should be

held when conducting experiments.

14

Section 4

Conclusion

4.1 Conclusion for Objective

• Achievement: The electric field lines distribution and corresponding potential level of this

two type of conductor systems - two parallel plates and two concentric cylinders - have been

throughout examined and discussed both in analog experimental prospective and mathemat-

ical theoretical prospective.

• Limitation: The accuracy of apparatus used in this lab might not be extremely high. Also,

the stability of the experimental set during the experiment might not have been kept in

perfect circumstance due to the time limit and large number of experimenters.

4.2 Suggestion

• Improve Experimental Accuracy

– The experimental circumstance should be more professional as well. For example,

experimenters are supposed to be separated with a safe distance so that the walking

around people will not interfere others’ experiments by shaving the experimental set

table without even knowing it.

• Broaden The Study

– More various composition of conductors are suggested to be studied in the similar

procedure as conducted in this lab, so that a more comprehensive understanding of

electric field distribution of different geometrical conductor system might be obtained.

– Also, more electric method of solving electric field might be worth learning. For

example: superposition method, filling the vacancy method for conductor system with

special features.

15

Appendix A

Questions

• Pre-lab Q1: Can you think of other fields of force that you have encountered in yourscience studies?

– Answer:Gravitational field: This terminology refers to a field of force which surrounds a phys-

ical body of finite mass

Magnetic field: This terminology refers to a field containing lines of force surrounding

a permanent magnet or a moving charged particle which have magnetic feature.

• Pre-lab Q2: In terms of the information already cited in the introduction, why is it thenegative and not the positive of the gradient of potential that is involved here?

– Answer:In brief, this negative sign is because that in terms of magnitude, the direction of

gradient is from low to high, while that of E is from high to low. E is a vector with

a direction pointing from a positively charged source to negative charges. However,

along the direction of E, the potential decreases because the further from the positive

source, the lower the potential is. The E direction is opposite to the direction of the

gradient increase of potential.

• Pre-lab Q3: What are the units of electric flux density and electric field? What is therelationship between them?

– Answer:The unit is V

m .

The relationship is: They equivalent terminology.

16

Appendix B

Matlab Code

Parallel Plates

a.Experimental Result1 A=[ −5.6 ,10 ,1 ; −5.1 ,9 ,1 ; −4.9 ,8 ,1 ; −4.7 ,7 ,1 ; −4.3 ,6 ,1 ; −4.25 ,5 ,1 ; −4.3 ,4 ,1 ; −4.2 ,3 ,1 ; −4.2 ,2 ,1 ; −4.2 ,1 ,1 ; −4.2 ,0 ,1 ; −4 .25 , −1 ,1 ;23 −4.3 ,−2 ,1;−4.3 ,−3 ,1;−4.3 ,−4 ,1;−4.5 ,−5 ,1;−4.7 ,−6 ,1;−4.9 ,−7 ,1;−5.2 ,−8 ,1;−5.6 ,−9 ,1;−6.2 ,−10 ,1;−6.8 ,−10 ,1;−6.8 ,−11 ,1;45 −7 ,−12 ,1;−1.6 ,0 ,2;−1.2 ,−2 ,2;−1.2 ,−4 ,2;−1.1 ,−6 ,2;−1.1 ,−8 ,2;−1.4 ,−10 ,2;−1.2 ,−12 ,2;−0.4 ,−14 ,2;−1.2 ,2 ,2;−1.1 ,4 ,2;−0.9 ,6 ,2;

6 −0 . 9 , 8 , 2 ; −0 . 6 , 1 0 , 2 ; −0 . 1 , 1 2 , 2 ; 2 . 5 , 0 , 3 ; 2 . 3 , 2 , 3 ; 2 . 4 , 4 , 3 ; 2 . 5 , 6 , 3 ; 2 . 8 , 8 , 3 ; 3 . 3 , 1 0 , 3 ; 4 . 5 , 1 2 , 3 ; 2 . 5 , −2 , 3 ; 2 . 3 , −4 , 3 ; 2 . 8 , −6 , 3 ; 3 . 2 ,

7 −8 , 3 ; 4 . 5 , −1 0 , 3 ; 5 . 7 , −1 1 , 3 ; 5 , 0 , 4 ; 5 , 2 , 4 ; 5 , 4 , 4 ; 5 , 6 , 4 ; 5 . 3 , 8 , 4 ; 6 . 3 , 1 0 , 4 ; 5 . 3 , −2 , 4 ; 5 . 4 , −4 , 4 ; 5 . 6 , −6 , 4 ; 6 . 1 , −7 , 4 ; 6 . 5 , 0 , 5 ; 6 . 5 , 2 , 5 ;

8 6 . 5 , 4 , 5 ; 6 . 5 , 6 , 5 ; 6 . 5 , 8 , 5 ; 6 . 8 , 9 , 5 ; 6 . 7 , −2 , 5 ; 6 . 7 , −4 , 5 ; 6 . 7 , −6 , 5 ; 7 , −7 , 5 ] ;9

1011 x=A( : , 1 ) ; y=A( : , 2 ) ; z=A( : , 3 ) ; s c a t t e r ( x , y , 5 , z )%A s c a t t e r d iagram1213 [X, Y, Z]= g r i d d a t a ( x , y , z , l i n s p a c e (−14 ,14) ’ , l i n s p a c e (−14 ,14) , ’ v4 ’ ) ;%The i n t e r p o l a t i o n o f v a l u e s o f maximum and minimum1415 p c o l o r (X, Y, Z ) ; s h a d i n g i n t e r p%Pseudo c o l o r c h a r t1617 f i g u r e , c o n t o u r f (X, Y, Z ) %Contour map1819 f i g u r e , s u r f (X, Y, Z )%3 d i m e n s i o n a l s u r f a c e

b.Theoretical Result1 A= [ 3 . 0 3 , 0 , 1 ; −3 . 0 3 , 0 , 1 ; 0 , 3 . 0 3 , 1 ; 0 , −3 . 0 3 , 1 ; 1 . 2 1 2 , 0 , 2 . 5 ; −1 . 2 1 2 , 0 , 2 . 5 ; 0 , 1 . 2 1 2 , 2 . 5 ; 0 , −1 . 2 1 2 , 2 . 5 ; 0 . 7 5 7 5 , 0 , 4 ;23 −0 . 7 5 7 5 , 0 , 4 ; 0 , 0 . 7 5 7 5 , 4 ; 0 , −0 . 7 5 7 5 , 4 ; 0 . 6 0 6 , 0 , 5 ; −0 . 6 0 6 , 0 , 5 ; 0 , 0 . 6 0 6 , 5 ; 0 , −0 . 6 0 6 , 5 ] ;456 x=A( : , 1 ) ; y=A( : , 2 ) ; z=A( : , 3 ) ; s c a t t e r ( x , y , 5 , z )%A s c a t t e r d iagram78 [X, Y, Z]= g r i d d a t a ( x , y , z , l i n s p a c e (−14 ,14) ’ , l i n s p a c e (−14 ,14) , ’ v4 ’ ) ;%The i n t e r p o l a t i o n o f v a l u e s o f maximum and minimum9

10 p c o l o r (X, Y, Z ) ; s h a d i n g i n t e r p%Pseudo c o l o r c h a r t1112 f i g u r e , c o n t o u r f (X, Y, Z ) %Contour map1314 f i g u r e , s u r f (X, Y, Z )%3 d i m e n s i o n a l s u r f a c e

17

Concentric Cylinders

a.Experimental Result1 A= [ 0 , 9 . 5 , 1 ; 6 . 6 , 6 , 1 ; 9 . 2 , 0 , 1 ; 6 . 5 , − 6 . 4 , 1 ; −6 . 4 , 5 . 9 , 1 ; − 8 , 0 , 1 ; −5 . 2 , −6 . 2 , 1 ; 0 , 3 . 8 , 2 . 5 ; 2 . 7 , 2 . 7 , 2 . 5 ; 3 . 6 , 0 , 2 . 5 ; 2 . 4 , − 2 . 5 , 2 . 5 ; 0 ,23 −3 . 4 , 2 . 5 ; −2 . 1 , −2 . 7 , 2 . 5 ; −3 . 5 , 0 , 2 . 5 ; −2 . 2 , 2 . 5 , 2 . 5 ; 0 , 1 . 6 , 4 ; 1 . 2 , 1 . 2 , 4 ; 1 . 6 , 0 , 4 ; 1 . 2 , −1 . 1 , 4 ; 0 , −1 . 7 , 4 ; −1 . 1 , −1 . 3 , 4 ; −1 . 8 , 0 , 4 ;45 −1 . 2 , 1 . 2 , 4 ; 0 , 1 . 1 , 5 ; 0 . 7 , 0 . 7 , 5 ; 1 . 1 , 0 , 5 ; 0 . 8 , 0 . 7 , 5 ; 0 , −1 . 2 , 5 ; −0 . 7 , −0 . 8 , 5 ; −1 . 2 , 0 , 5 ; −0 . 8 , 0 . 8 , 5 ] ;678 x=A( : , 1 ) ; y=A( : , 2 ) ; z=A( : , 3 ) ; s c a t t e r ( x , y , 5 , z )%A s c a t t e r d iagram9

10 [X, Y, Z]= g r i d d a t a ( x , y , z , l i n s p a c e (−14 ,14) ’ , l i n s p a c e (−14 ,14) , ’ v4 ’ ) ;%The i n t e r p o l a t i o n o f v a l u e s o f maximum and minimum1112 p c o l o r (X, Y, Z ) ; s h a d i n g i n t e r p%Pseudo c o l o r c h a r t1314 f i g u r e , c o n t o u r f (X, Y, Z ) %Contour map1516 f i g u r e , s u r f (X, Y, Z )%3 d i m e n s i o n a l s u r f a c e

b.Theoretical Result1 A=[ −5.25 ,0 ,1; −5.25 ,2 ,1; −5.25 ,4 ,1; −5.25 ,6 ,1; −5.25 ,8 ,1; −5.25 , −2 ,1; −5.25 , −4 ,1; −5.25 , −6 ,1; −5.25 , −8 ,1;23 −3.5 ,0 ,2; −3.5 ,2 ,2; −3.5 ,4 ,2; −3.5 ,6 ,2; −3.5 ,8 ,2; −3.5 , −2 ,2; −3.5 , −4 ,2; −3.5 , −6 ,2; −3.5 , −8 ,2; −1.75 ,0 ,3;45 −1.75 ,2 ,3; −1.75 ,4 ,3; −1.75 ,6 ,3; −1.75 ,8 ,3; −1.75 , −2 ,3; −1.75 , −4 ,3; −1.75 , −6 ,3; −1.75 , −8 ,3;0 ,0 ,4;0 ,2 ,4;67 0 , 4 , 4 ; 0 , 6 , 4 ; 0 , 8 , 4 ; 0 , −2 , 4 ; 0 , −4 , 4 ; 0 , −6 , 4 ; 0 , −8 , 4 ; 1 . 7 5 , 0 , 5 ; 1 . 7 5 , 2 , 5 ; 1 . 7 5 , 4 , 5 ; 1 . 7 5 , 6 , 5 ; 1 . 7 5 , 8 , 5 ;89 1 .75 , −2 ,5 ;1 .75 , −4 ,5 ;1 .75 , −6 ,5 ;1 .75 , −8 ,5 ] ;

10111213 x=A( : , 1 ) ; y=A( : , 2 ) ; z=A( : , 3 ) ; s c a t t e r ( x , y , 5 , z )%A s c a t t e r d iagram1415 [X, Y, Z]= g r i d d a t a ( x , y , z , l i n s p a c e (−6 ,6) ’ , l i n s p a c e (−8 ,8) , ’ v4 ’ ) ;%The i n t e r p o l a t i o n o f v a l u e s o f maximum and minimum1617 p c o l o r (X, Y, Z ) ; s h a d i n g i n t e r p%Pseudo c o l o r c h a r t1819 f i g u r e , c o n t o u r f (X, Y, Z ) %Contour map2021 f i g u r e , s u r f (X, Y, Z )%3 d i m e n s i o n a l s u r f a c e

Electric Strength and Voltage versus displacement (One example)

1 e z p l o t ( ’ ( l o g ( 1 4 / x ) / l o g ( 1 4 ) )∗8 ’ , 0 , 1 4 )2 ho ld on3 x = [ 9 . 5 , 3 . 8 , 1 . 6 , 1 . 1 ] ; y = [ 1 , 2 . 5 , 4 , 5 ] ; p l o t ( x , y )

Electric Field Arrow

1 A=[ −5 .6 ,10 ,1 ; −5 .1 ,9 ,1 ; −4 .9 ,8 ,1 ; −4 .7 ,7 ,1 ; −4 .3 ,6 ,1 ; −4 .25 ,5 ,1 ; −4 .3 ,4 ,1 ; −4 .2 ,3 ,1 ; −4 .2 ,2 ,1 ; −4 .2 ,1 ,1 ; −4 .2 ,0 ,1 ; −4 .25 , −1 ,1 ; −4 .3 , −2 ,1 ; −4 .3 , −3 ,1 ; −4 .3 , −4 ,1 ; −4 .5 , −5 ,1 ; −4 .7 , −6 ,1 ; −4 .9 , −7 ,1 ; −5 .2 , −8 ,1 ; −5 .6 , −9 ,1 ; −6 .2 , −10 ,1 ; −6 .8 , −10 ,1 ; −6 .8 , −11 ,1 ; −7 , −12 ,1 ; −1 .6 ,0 ,2 ; −1 .2 , −2 ,2 ; −1 .2 , −4 ,2 ; −1 .1 , −6 ,2 ; −1 .1 , −8 ,2 ; −1 .4 , −10 ,2 ; −1 .2 , −12 ,2 ; −0 .4 , −14 ,2 ; −1 .2 ,2 ,2 ; −1 .1 ,4 ,2 ; −0 .9 ,6 ,2 ; −0 .9 ,8 ,2 ; −0 .6 ,10 ,2 ; −0 .1 ,12 ,2 ;2 .5 ,0 ,3 ;2 .3 ,2 ,3 ;2 .4 ,4 ,3 ;2 .5 ,6 ,3 ;2 .8 ,8 ,3 ;3 .3 ,10 ,3 ;4 .5 ,12 ,3 ;2 .5 , −2 ,3 ;2 .3 , −4 ,3 ;2 .8 , −6 ,3 ;3 .2 , −8 ,3 ;4 .5 , −10 ,3 ;5 .7 , −11 ,3 ;5 ,0 ,4 ;5 ,2 ,4 ;5 ,4 ,4 ;5 ,6 ,4 ;5 .3 ,8 ,4 ;6 .3 ,10 ,4 ;5 .3 , −2 ,4 ;5 .4 , −4 ,4 ;5 .6 , −6 ,4 ;6 .1 , −7 ,4 ;6 .5 ,0 ,5 ;6 .5 ,2 ,5 ;6 .5 ,4 ,5 ;6 .5 ,6 ,5 ;6 .5 ,8 ,5 ;6 .8 ,9 ,5 ;6 .7 , −2 ,5 ;6 .7 , −4 ,5 ;6 .7 , −6 ,5 ;7 , −7 ,5] ;

2 x=A( : , 1 ) ; y=A( : , 2 ) ; z=A( : , 3 ) ; s c a t t e r ( x , y , 5 , z )%A s c a t t e r d iagram3 [X, Y, Z]= g r i d d a t a ( x , y , z , l i n s p a c e (−14 ,14) ’ , l i n s p a c e (−14 ,14) , ’ v4 ’ ) ;%The i n t e r p o l a t i o n o f v a l u e s o f maximum and minimum4 f i g u r e , c o n t o u r f (X, Y, Z ) %Contour map5 ho ld on6 [DX,DY]= g r a d i e n t ( Z , 0 . 1 , 0 . 1 ) ;7 q u i v e r (X, Y,DX,DY)

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Reference

[1] physics tutor from internet, “Electric field lines,” none, vol. none, no. none, pp. 1–4, 2015.

[2] unknown, “Engineering electromagnetism and drivers lab 1,” none, vol. none, no. none, pp.

1–2, 2015.

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