A practical problem from the engineering field requires the use of calculus tools. Beyond that, problems might be either well-defined or open-ended.

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Cover Page

MATHEMATIC-ENGINEERING PROJECT

PROJECT TITLE

Electrical Felds Due to a Charged Rod

Problem Statement

How to compute different kinds of electric fields such as the one due to a charged

rod, on the axis of a ring of charge and of a disk with uniform charge density?

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Table of Contents

1. Abstract

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2. Motivation

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3. Mathematical Description and Solution Approach

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4. Discussion

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5. Conclusions and Recommendations

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6. Nomenclature

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7. References

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Appendix (calculations, graphs, pictures, spreadsheet information …)

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Abstract

Throughout this project report, I use calculus especially integration to calculate

different types of electric field. An E-field is a vector field incorporating an electric

charge that applies oblige on various charges, pulling in or spurning them. The electric

field is radially outward from a positive charge and radially in toward a negative point

charge. Scientifically, the electric field is a vector field that accomplices to each point in

space the power, called the Coulomb drive, that would be experienced per unit of charge,

by a modest test charge by then. The units of the electric field in the SI system are

newtons per coulomb (N/C), or volts per meter (V/m). Electric fields are made by electric

charges, and by time-moving alluring fields. Electric fields are basic in various zones of

material science and are abused in every practical sense of electrical development. The

electric field is accountable for the engaging force between the atomic centre and

electrons that holds particles together, and the forces between particles that reason

invention holding. Electric fields and appealing fields are the two appearances of the

electromagnetic power, one of the four noteworthy forces of nature.

Motivation

Electric field primarily centres on applying force and fascination measures from

material science to make significant devices and materials. The electric field around a

charged framework unveils to us the power a unit positive test charge would

understanding whether set by at that point. The electric field is typical for the plan of

charges and is free of the test charge that we put at a point to choose the field. The term

field in material science can allude to a sum that is portrayed at each point in space and

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may move from point to point. There is much of the time an impressive proportion of

cover between electrical planning and material science, especially in the district of solid

state physical science. Material science is basically the examination of how stuff

capacities. In some sense, it is the most significant science; that is, it finds and portrays

the most fundamental laws that manage the universe (Cengage Learning, 2011). For the

situation that we take a gander at material science in that way, by then it relates to

electrical structuring and added to each other science. In order to genuinely perceive how

the particles participate with each other, it is imperative to consider the individual

particles making up the particles and how the atoms security with each other. The

strategy that the electrons demonstration is additionally depicted by an electric field, and

this is considered by physicists.

Mathematical Description and Solution Approach

The E-field vector E due to a charged rod is given by

E = Ex + E y

where

Ey = −

Ex =

k

( cos2 − cos1 )

yp

k

( sin 2 − sin 1 )

yp

For the E-field due to a ring of charge, we have calculated that

Ex =

(x

kxQ

2

+ a2 )

3/2

For the E-field due to a disk of uniform charge density, we have

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x

E = 2 k 1 −

x2 + R2

Firstly, suppose that we want to calculate the E-field due to a charged rod, shown

in the following figure:

The E-fields in both directions are calculated to be:

k

k 2

dE y = dE sin =

sin d E y =

sin d

yp

y p 1

Ey = −

k

( cos 2 − cos 1 )

yp

k

k 2

dE y = dE cos =

cos d Ex =

cos d

yp

y p 1

Ex =

k

( sin 2 − sin 1 )

yp

where is the charge per unit length, Q is the total charge length, and L is the length of

the rod.

Suppose that we want to have the E-field calculated on the axis of a ring of

charge, as shown in the below figure:

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We determine the field at point P on the axis of the ring, as the differential xcomponent of the field is

kdq

cos

r2

kdq x

dEx = 2

r r

kxdq

dEx =

3/2

( x2 + a2 )

dEx =

where r 2 = x 2 + a 2 and cos =

x

=

r

x

x2 + a2

.

We now integrate exploiting the fact that r and x are constant for all points on the ring:

Ex =

(x

Ex =

Ex =

kxdq

2

+ a2 )

3/2

kx

( x2 + a2 )

(x

3/2

dq

kxQ

2

+ a2 )

3/2

Finally, suppose that we want to have the E-field computed on the axis of a

uniformly charged disk, as shown below:

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Let the surface charge density be =

Q

, where Q is the total charge, and the

R2

ring of thickness da has an area of dA = 2 ada , which further gives

dq = dA = 2 a da . Th, we have the computation for the field by the ring of change

along the x-axis:

dEx =

x 2 a da

1

4 0

(x

2

+ a2 )

3/2

The total field is hence given by

R

R

0

0

E = dEx = k

x 2 a da

(x

2

+ a2 )

3/2

R

1

E = kx −

2 ( x 2 + a 2 )1/2

0

x

E = 2 k 1 −

x2 + R2

Discussion

We meet the objective of the object as we need to influence we to have effectively

registered the different electric fields. We get the expected result rather than the counter-

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intuitive one because the problem was planned to underline a numerical relationship with

electric fields. When all is said in done, the suggestions my outcomes have to the current

issue and to the field are that we have plotted the use of analytics with a couple of cases,

in which we expect the charge is reliably appropriated on a line, on a surface, or all

through a volume. While theoretical conjectures of electric fields for direct charge

scatterings are quickly open, their exploratory confirmation can be dismal. One procedure

is to choose the equipotential surfaces or lines and after that build up the electric lines

wherever at right edges to the equipotential. If two are on an equipotential surface, by

then, the electric stream between the two must be zero.

Conclusions and Recommendations

Overall, we have processed the required electric fields, and one approach to

managing nonstop charge conveyances is to describe electric movement and make use of

Gauss’ law to relate the electric field at a surface to the total charge encased inside the

surface. There are things that a man doing likewise extend may do any other way or

thoughts for another investigation that is recommended by our outcomes. Determining

different electric fields produced by persistent charge disseminations, for instance,

charged wires or plates is more troublesome than for point charges. The procedures for

analytics are normally connected for reliable charge circulations and general material

science messages, when in doubt, offer estimates to the more run of the mill spread, for

instance, charged parallel plates (Edward, 2011). Consider two oppositely charged

parallel plates whose estimations are generously appeared differently in relation to the

parcel between the plates. The foreseen field resigns from the completions; in any case,

the more crucial desires are that, inside the central region between the plates, the electric

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field has a predictable size and is guided inverse to the plates from the positive to the

negative plate.

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Nomenclature’s

E

Electric field

N/C or V/m

Q

Charge

Coulombs (C)

Linear charge density

C/m

Surface charge density

C / m^2

k

Coulomb’s constant

9.0 109 Nm2/C2

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References

Cengage Learning (2011). pp. 532–533. ISBN 1305142829. Purcell. Electricity and

Magnetism, 2nd Ed., p. 20-21.

Edward, P. (2011). Electricity and Magnetism, 2nd Ed. Cambridge University Press. pp.

8–9, 15–16.

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Appendix

Electric field due to a charged rod:

The electric field on the axis of a ring of charge:

The electric field on the axis of a uniformly charged disk:

Applied Calculus Projects – Guidelines for Students

Nature of the Project

Your project should feature a practical problem from the field you are pursuing and require the

use of calculus tools. Beyond that, problems might be either well-defined or open-ended.

All projects will have at least two advisors – a Subject Area Advisor and a Mathematics

Advisor.

Subject Area Advisor

Your Subject Area Advisor will most likely be the person (e.g., work supervisor, faculty

member, postdoc, etc.) who suggested the project to you. This person might simply hand

you a project and say “Come back when you are done” or schedule meetings with you to

discuss it. How you work with your Subject Area Advisor is between you and her/him.

Math Advisor

Your Calculus course Instructor will be your Mathematics Advisor. You may have more

than one Mathematics Advisor (any Faculty or Graduate student in the Department of

Mathematics & Statistics).

How to Select a Project

The problem for your project can come from a number of different sources. If you have a job or

an internship, your work supervisor might have a problem that is important to the organization

you work for. If you have an undergraduate research position, your research advisor can be a

source of project ideas. Or, you might have already taken a class or two with faculty members in

your major and they might be willing to suggest a problem for you to work on. The best source

of a project might be you though. Consider the things you are interested in and look for an

application of calculus to them. If you can find one, you can probably build a project around

that.

Publication in the Undergraduate Journal of Mathematical Modeling: One + Two

Selected projects will be published in the open access electronic journal UJMM: One + Two

http://scholarcommons.usf.edu/ujmm/ or http://ciim.usf.edu/ujmm

under a Creative Commons Attribution Non-Commercial Share Alike 3.0 license. Submission of

a project report will imply that you are giving the editors of UJMM: One + Two permission to

publish your report in this journal, should it be selected.

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Project Deadlines

•

The deadline for selecting a project will be given to you by your Calculus Instructor.

•

Your subject area advisor might wish to review your report and make suggestions before

you submit it. You should ask her/him if this is the case and, if so, when the deadline for

this is.

•

The official due date of the project (final submission) – the day that it must be uploaded –

will be given to you by your Calculus Instructor.

Project Submission

Your project (equations, graphs, diagrams, pictures included) should be presented as a Microsoft

WORD document.

•

•

•

•

•

Clarity of writing is important. At the very least, be sure to use your spell-checking and

grammar-checking facilities.

It is very important that you include the correct first and last names of your project

advisors. Also be certain to include their correct USF Department or Company

Affiliation. This information as well as your own correct first and last name is crucial for

proper identification of your project upon online submission.

You will need to prepare a Project Summary in advance. This is a concise abstract type

description written in the third person. The Project Summary will be posted online so it

should be understandable to a general audience. Therefore it should be focused on the

subject matter rather than mathematical formulas and details.

You should submit your project through the PROJECT SUBMISSION link provided by

your Calculus Instructor. The check list of the required and optional data appears as the

first page in the submission process.

You may be required to provide your advisors with a hard copy of your project.

Project Checklist

1. Find a Subject Area Advisor for your project, by ___________.

2. Meet with your chosen Subject Area Advisor and identify a problem, by ___________.

3. Check back with your Mathematics and Subject Area Advisors concerning your

understanding of the problem and a mathematical approach to solve it. Consult with them

about any difficulties or questions.

4. Show a draft of your report to all advisors no later than _______________.

5. Submit final copy online, by ___________.

Report Format

Project submissions must be in the following format:

(a) Cover page and Problem statement.

The cover page should use the following template, followed by the problem statement (see next

page):

Page 2

Your section

MATHEMATICS – ENGINEERING PROJECT *

(e.g., MAC2282.902)

PROJECT TITLE

Student:

First Name

Last Name

ADVISORS

Mathematics Advisor: First Name Last Name

Affiliation**

Subject Area Advisor: First Name Last Name

Affiliation**

Problem suggested by: First name Last name

Affiliation**

Current semester and year

PROBLEM STATEMENT

Provide an exact statement of the problem as suggested by its author.

* or MATHEMATICS – MEDICINE PROJECT, MATHEMATICS – BIOLOGY PROJECT,

MATHEMATICS-ENVIRONMENTAL SCIENCE PROJECT, etc.

** For instance, Electrical Engineering, University of South Florida, Tampa, FL.

Research and Development, Raytheon Technology. St. Petersburg, FL.

Department of Radiology, Tampa General Hospital, Tampa, FL. , etc.

(b) Table of Contents. Include the following sections in the table and give the page numbers.

Contents

1. Abstract

2. Motivation

3. Mathematical Description and Solution Approach

4. Discussion

5. Conclusions and Recommendations

6. Nomenclature

7. References

Appendix (calculations, graphs, pictures, spreadsheet information …)

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P

A

G

E

#’s

(c) Abstract. The abstract is a short summary of your project report – it should not exceed one

or two paragraphs. It should concisely state what you did, how you did it, and what conclusions

you drew from the results. The abstract will be posted online so it should be well written.

(d) Motivation. In this section you should give some background about why the problem is

important to science or engineering. You should also describe the problem within its

engineering or science context and provide the objective for the project.

(e) Mathematical Description and Solution Approach. In this section, you should formulate

the mathematical approach to solving the problem – providing the relevant equations,

describing the mathematical tools you used and outline the procedure used. Do NOT simply list

the equations – use text between them to provide a clear understanding of them to the reader.

(f) Discussion. Here, you should provide the results and discuss them. Did you meet the

objective of the project? Were they as expected, or were they counter-intuitive? What

implications do your results have to the problem at hand and to the field in general?

(g) Conclusions and Recommendations. Give the basic conclusions of your work. This will

be somewhat similar to what is in the abstract but with a little more detail – for instance,

including a summary of your interpretation of the results. You should also make a few

recommendations – such as things a person doing the same project might do differently or ideas

for a new study that is suggested by your results.

(h) Nomenclature. List the symbols that you use in your report. For each symbol, provide a

description of what it represents and its units. Example:

P

T

v

V

Pressure

Temperature

Velocity

Voltage

kPa

o

C

m/s

V

All units used should belong to the same measuring system: Standard (English) or Metric (SI).

Carefully check whether the units agree and are balanced on both sides of each equation.

(i) References. Any work or ideas that you have taken from someone else should be cited

directly in the text of your report. This includes any figures that you might download from the

web. Do your best to find and cite the original source of information rather than the secondhand

source. At the end of the report should be a list of references that were cited. Book and scientific

journal references are strongly preferable to webpages.

(j) Appendices. You might have detailed calculations, spreadsheets or computer programs that

were used to obtain your results but do not belong in the main report. If so, you should place

these materials in appendices and refer to them as needed in the report.

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