Laboratory #4
EE Science II Laboratory #04
Charge Distributions and Electric Field Visualization in 3D
Pre-Laboratory Assignment
Your name and group number: _____________
1.
Consider a charge q placed at (x, y) = (0, 0). The magnitude of the charge depends on your group number:
Groups 1 and 2 → q = 3×10-3
Groups 3 and 4 → q = 4×10-3
Groups 5 and 6 → q = 5×10-3
Groups 7 and 8 → q = 6×10-3
C
C
C
C
From Coulomb’s Law, the electric field produced by this charge at any point on the x-y plane is given by the
following equation in the cylindrical coordinate system:
𝐸⃗ =
a.
𝑞
𝑟̂
(4𝜋𝜖)𝑅2
Transform this expression into the Cartesian coordinate system and write down the x and y components of
the field below: [1 Point]
Ex = ___________________________
Ey = ___________________________
b.
Complete Table 1 by calculating the x and y components of the E field for the given (R, ϕ) values. [1 Point]
Table 1: Calculate the electric field in Cartesian coordinates for the given values of (R, ϕ)
R
Observation point (x, y)
𝝓 (deg)
𝑬𝒙
0.5
0
(
,
)
1.0
0
(
,
)
1.5
0
(
,
)
0.5
45
(
,
)
0.5
90
(
,
)
0.5
180
(
,
)
1.0
270
(
,
)
1.5
135
(
,
)
2.
𝑬𝒚
c.
Plot the electric field calculated in Table 1 using Grid #1 attached at the bottom of this document. [1 Point]
d.
How will the electric field change if the charge was located at (x, y) = (1, 2) instead of (0, 0)? [0.5 Point]
A linear charge distribution of length l and charge density λ is located along the x axis as shown in Figure 1, where
l and λ depend on your group number –
Groups 1 and 2 → l = 20 m, 𝜆 = 3 × 10−3
Groups 3 and 4 → l = 10 m, 𝜆 = 4 ×
Groups 5 and 6 → l = 40 m, 𝜆 = 5 × 10
University of South Florida
C
m
C
10−3
m
−3 C
m
1
EE204-pre-lab.docx
Laboratory #4
Groups 7 and 8 → l = 30 m. 𝜆 = 6 × 10−3
C
m
Calculate the electric field at any point on the y axis by following the steps listed below –
a.
Consider an infinitesimally small charge dQ = λ dx’ as shown in Figure 1.
b.
If (x’, y’) is used to denote any point on the charge distribution and (x,y) is used to denote any observation
point, then the net electric field at (x,y) due to an infinitesimally small charge dQ placed at (x’, y’) is:
𝑑𝐸⃗ =
𝑟̂
1 𝑑𝑄
1 𝑑𝑄 ((𝑥 − 𝑥 ′ )𝑥̂ + (𝑦 − 𝑦 ′ )𝑦̂)
𝑟̂
=
4𝜋𝜀 𝑅2
4𝜋𝜀 ((𝑥 − 𝑥 ′ )2 + (𝑦 − 𝑦 ′ )2 )3/2
((𝑥−𝑥 ′ )𝑥̂+(𝑦−𝑦 ′ )𝑦̂)
c.
Show how
d.
In this problem, dQ = λ dx’, y’ = 0 (line charge distribution is on the x axis) and x = 0 (observation point is
on the y axis). Since the charge distribution is uniform, the x components of the electric field cancel at any
point on the y axis. Therefore, the net electric field is due to the y component alone. Calculate the electric
field by integrating the y component in the above expression for x’ varying from -l/2 to l/2. Show your
work clearly for full credit. [3 Points]
𝑅2
equals ((𝑥−𝑥 ′)2
+(𝑦−𝑦 ′ )2 )3/2
in the above step. [1 Point]
Hint:
∫
𝑑𝑥
𝑥
=
(𝑎2 + 𝑥 2 )3/2 𝑎2 √𝑎2 + 𝑥 2
Figure 1: Problem #2 – line charge distribution placed on the x – axis.
3.
You will work with MATLAB in this lab. Please review the MATLAB tutorial and answer the following
questions:
3.1. Write a MATLAB code line to implement the following equation [1.5 Points]
1
𝑌=∫
(𝑥 + 5) 𝑑𝑥
𝑥=−1
3.2. Explain the output of the following MATLAB code in your own words: [1 Point]
x = 1:4;
y = 1:6;
[X,Y] = meshgrid(x,y)
University of South Florida
2
EE204-pre-lab.docx
Laboratory #4
Grid #1: Cartesian coordinate Grid for Problem #1
Y
X
University of South Florida
3
EE204-pre-lab.docx
Laboratory #04
EE Science II Laboratory #04
Charge Distributions and Electric Field Visualization in 3D
Summary
In this laboratory experience, you will explore the concepts of charge distributions and electric field distributions
in a 3-dimensional space. You will compute the electric field of various charge geometries (ie, single charge, dipole
and line distribution). You will use MATLAB as a tool for carrying out the calculations leamed during the lab
experience.
During this lab, you will:
a) Understand the use of coordinate systems to represent 2D Electric Fields,
b) Study the most fundamental charge distribution densities (point, dipole and linear),
c) Develop basic skills in the use of MATLAB to compute the electric field produced by a single point charge
confined in an arbitrary volume and compare to the results obtained by two-point charges of opposite charge sign
enclosed in an arbitrary volume (i.e. electric dipole).
d) Develop basic skills in the use of MATLAB to compute the electric field produced by a line with constant charge
along the length (i.e line charge distribution) and compare to the results obtained in the pre-lab experience,
e) Gain knowledge about uniform electric fields and their applications,
By the end of this experience, you should be able to understand the graphical representation of fields produced
by different electric charge distributions, comprehend the concept of distributed charge in two-dimensional and three-
dimensic spaces and get a sense of the physical meaning of the electric field.
Objectives
Gain an understanding of electric field produced by different charge distributions
Gain proficiency in the use of MATLAB to solve vector field problems
Gain understanding about the concepts of charge distribution, electric field, coulombs law by visualizing
2D and 3D representations of electric fields
.
.
Equipment and Software
Advanced Visualization Center
.
MATLAB

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