The Experiment 2 of Engineering Electromagnetics:
The Analysis about Electric Distribution of A Line
Charge with Integration and Infinitesimal Method
Zhang Haodong
Department of Electronic and Electrical Engineering
Southern University of Science and Technology
Shenzhen, China
12113010
Abstract—This experiment is to calculate the electric field
distribution of the continuous line charge and plot the relevant
figures with MATLAB. In this experiment, two method are used,
one is to use integration method to calculate the distribution of
electric field, and another is to use infinitesimal method to obtain
the approximate calculation of the distribution. These two method
are compared and analyzed at last. The infinitesimal method is
to divide the continuous line charge into several point charges
and add up the contribution of each charge to the electric field
distribution. There must be some error between this method
and the integral method which is There must be some error
between this method and the integral method which is considered
an accurate method of calculation. This error is related to the
number of segments that are divided from the line charge. And
the more segments divided are, the smaller the error is.
Index Terms—Electric field, Line charge, Integration method,
Infinitesimal method, MATLAB
I. INTROD UCTION
This is a report for the second experiment of Engineering
Electromagentics theory. This experiment is to calculate the
electric field distribution including potential distribution, con-
tours distribution and electric field line of a continuous line
charge using integration method and infinitesimal separately
method with MATLAB and analyze and compare them at last.
In this experiment, a continuous line charge with charge
density of ρ = 1×10
−9
C/m is distributed on the line segment
between points A(-1,0) and B(1,0).
There are many symbols and notations that are used in
this experiment, so the following table lists all the important
symbols and notations that will be used.
TABLE I: List of symbols and notations
Symbol Meaning Unit
k Electromagnetic constant of 1 × 10
9
F/m
E Electric field intensity V/m
V Electric potential V
ρ The charge density of 1 × 10
−9
C/m
Q The electric charge C
∆Q The charge amount of each segment C
V
i
The potential of the i
th
segment V
The basic electrostatic knowledge and relevant formulas
used in this experiment are as follows:
The electric field intensity E generated by point charge in
vacuum is:
E
E
E = k
Q
R
2
a
R
a
R
a
R
(1)
If infinity is the zero potential point, the potential generated
by point charge in vacuum is:
V = k
Q
R
(2)
The strength of the electric field can be obtained by using
the negative gradient of the electric potential, i.e.
E
E
E = −∇V (3)
II. INTEGRATION METHOD
The purpose of this experiment is to obtain the electric
distribution of a continuous line charge. Firstly we use the
integration method to calculate the result that is accurate.
A. Theoretical Analysis
First of all, we should deduce the expression of the potential
of the line charge. Let the coordinates of the point on the plane
is (X
0
, Y
0
). And then divide the line charge into infinitesimal
pieces, each of which is small enough so that the length is
dx. So, the charge of each pieces is Q = ρdx. Because the
potential is scalar so that the total potential in this field is
V = k
Z
1
−1
ρdx
R
(4)
where R is the distance between the point in the field and
the object point being studied.
R =
q
(X
0
− x)
2
+ Y
2
0
(5)
So the equation (4) can be deduced to the expression as
follows:
V = kρ ln(
1 − X
0
+
p
(1 − X
0
)
2
+ Y
2
0
−1 − X
0
+
p
(−1 − X
0
)
2
+ Y
2
0
) (6)