Download hw3 - UCF EECS

Department of Electrical Engineering and Computer Science
EEL 3470 Electromagnetic Fields
Homework Set No. 3
Please turn in your answers, MatLab files and results.
Each group should turn in ONLY one set of homework.
Print this page and write your answers. After that, attach your MatLab files and results.
Names: __________________
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Problem 1. A line charge is distributed uniformly along a straight line from A(3,2,1) to
B(1,2,6). The total charge of this line is 3 nC (nano Coulomb). Find the
potential at point P(3,4,5).
Answer:
Problem 2. A sheet charge plate is on yoz plane. The charge density is given by
(1  y 2 )(1  z 2 ) for  1  y  1,  1  z  1
0
elsewhere

Find the potential at (2, 3, -4).
s  
Answer:
Problem 3. Electric potential V in free space is given as:
𝑉 = 3𝜌2 cos⁡(𝜙)𝑧
Find:
(1) Electric field E and displacement D;
(2) Volume charge density 𝜌𝑣 ;
(3) Total charge within spherical volume defined as:
1 ≤ 𝑟 ≤ 6,
25𝑜 ≤ 𝜃 ≤ 80𝑜 ,
10𝑜 ≤ 𝜙 ≤ 85𝑜
Based on Gauss’s law, solve the problem by:
(a) Evaluating the closed form integral ∯𝑆 𝐃 ∙ 𝑑𝐬 over the closed region.
(b) Calculating ∭𝑉𝑜𝑙 𝜌𝑣 𝑑𝑣 .
Answer: (1)
(2)
(3)
Problem 4. A parallel plate capacitor is filled with a nonuniform dielectric characterized
by  r  3  3x , where x is the distance from one plate. If the area of each
plate S and the separation between the two plates is d, find the capacitance C.
Answer :
Problem 5. (1) A coaxial line is 20 cm long the radius of the inner conductor is 5 cm and
the outer conductor is 10 cm, Calculate the electrostatic energy that is
stored in the structure if the intermediate space is filled with a dielectric
whose relative dielectric constant is 10 and a voltage is applied between
the conductors creating an electric field
106
E
V/m .

Compare this with the value that would be obtained if the intermediate
space is a vacuum.
(2) Determine the capacitance of the coaxial structure described in (1)
assuming that a vacuum exists between the two conductors. In addition,
determine the value of the applied voltage
Answer: (1)
(2)
Problem 6. Assume that there are two concentric spheres with radius 𝑎 < 𝑟 < 𝑏⁡. The
dielectric distribution is given as: 𝜖𝑟 (𝑟) = 3 + 3𝑟 . Find the capacitance
between these two spheres.
Answer:
Problem 7.
(1) Find the potential distribution V(r) by solving Laplace’s equation analytically
for the region between two concentric hollow spheres (spherical capacitor).
Apply a spherical coordinate system with the following boundary conditions:
V= V0 at r=a and V=0 at r=b. Simplify the calculation with symmetry
argument.
z
y
a
b
x
(2)
(3)
(4)
(5)
Find the electric field Find the electric field E between two conductors;
Find the displacement vector D between two conductors;
Find the total charge on inner conductor r  a ;
Find the capacitance C of the spherical capacitor.
Answer: (1)
(2)
(3)
(4)
(5)
Problem 8. For a coaxial line with inner conductor radius a and outer conductor radius b,
𝑉 = 𝑉0 ⁡at 𝜌 = 𝑎 and 𝑉 = 0⁡at 𝜌 = 𝑏.
(1) Find the potential distribution by solving Laplace’s equation.
(2) Find the electric field E between two conductors;
(3) Find the displacement vector D between two conductors;
(4) Find the total charge per unit length on inner conductor   a ;
(5) Find the capacitance per unit length between two conductors.
Answer: (1)
(2)
(3)
(4)
(5)
Problem 9. Two radial planes with an interior angle  as shown in the figure
below.
(1) Find the potential distribution assuming the planes are infinite;
(2) Find the electric field E between two planes;
(3) Find the displacement vector D between two planes;
(4) If the planes have length L in z direction and radius a in 𝜌 direction, find
the total charge on the bottom plane. (Assume a tiny gap of 106 m
between these two planes.)
(5) Find the capacitance between two planes
V=0

V=V0
a
Answer: (1)
(2)
(3)
(4)
(5)
L

0


0
z