Select Page

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

project_guidelines.pdf

Unformatted Attachment Preview

1
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?
2
1. Abstract
3
2. Motivation
4
3. Mathematical Description and Solution Approach
6
4. Discussion
8
5. Conclusions and Recommendations
9
6. Nomenclature
11
7. References
12
Appendix (calculations, graphs, pictures, spreadsheet information …)
13
3
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
4
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
( cos2 − cos1 )
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
5

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:
6
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:
7
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-
8
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
9
field has a predictable size and is guided inverse to the plates from the positive to the
negative plate.
10
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
11
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.
12
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
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.
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
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.
Page 1

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

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.
Project Checklist
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
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
MATHEMATICS – ENGINEERING PROJECT *
(e.g., MAC2282.902)
PROJECT TITLE
Student:
First Name
Last Name
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.
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 …)
Page 3
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.
Page 4