Tutorial 6 Solution Emagnet

TUTORIAL 6

(SOLUTION)

EEE 3133 ELECTROMAGNETIC FIELDS AND WAVES DEPARTMENT OF ELECTRICAL AND ELECTRONIC ENGINEERING

1. Briefly define uniqueness theorem. Uniqueness theorem has defined that any solution is unique if a solution to Laplace’s equation can be found that satisfy the boundary condition. 2. List THREE (3) criteria uniquely describe a problem in order to solve boundary-values problems. a. The appropriate differential equation b. The solution region c. The prescribed boundary conditions 3. Show that the following potentials satisfy Laplace’s equation. a.
      

1

b.




c.








EEE 3133 Sem 1 2009/2010 prepared by Mohd Taufik

4. Find  and  at point P(1,2,-5) if the potential field of !" #  is placed in free space.

5. The potential field
 " #  $   exist in a dielectric medium having %  % . a. Does the potential field satisfy Laplace’s equation?

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EEE 3133 Sem 1 2009/2010 prepared by Mohd Taufik

b. Calculate the total charge within the unit cube & ' "( (  ' meter. )#
 $

* +

 $  $ $$,    % 4

-  .  /0  12 %3 /"/ /  % -  565789:



6. Two infinitely large conducting plates are located at "  and "  8. The space between them is free space with charge distribution

and
283  7&
.



;<

n C/m3. Find
at "   if
2 3  $7&


"

&>  /#
!= $   $!"

&> /" # % != /
 $" # ? @ $ $$,
 $"  ? @" ? A /"

$7&  $ ? @ ? A $ $$, @ ? A  $8B 7&  $!8 ? 8@ ? A $ $$, 8@ ? A 

8 Thus : @  786 C/ A  $ &6
 $"  ? 786" $ &6 Substituting "  
23  $6!!D


7. Find the equation for potential field,
using Laplace’s equation in homogeneous region between two concentric spheres with radius of C and E, where E F C. Calculate the capacitance between the two conducting spheres if given
 & at G  E and

 at G  C.

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EEE 3133 Sem 1 2009/2010 prepared by Mohd Taufik

Then apply the boundary conditions, we gain

Finally;

Finding the electric field intensity: Assume permittivity as %;

The inner sphere, the charge density will be

Then the capacitance

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EEE 3133 Sem 1 2009/2010 prepared by Mohd Taufik

8. Two coaxial conducting cylinders are located at   C and   E, where E F C. The region between the cylinders is filled with a homogeneous perfect dielectric. a. Determine the potential field equation, V using Laplace’s equation if the inner cylinder has

 and the outer cylinder has zero potential value.

By using prescribed boundary condition, we gain  HI JEK

 HICJEK

b. Find the location of the &
equipotential surface in centimeter (cm). Let inner cylinder 2  &67L3 is at &&
and the outer 2  6L3 is at &
.

Substituting the boundary potential values we gain,   HI JEK HM J&6& N  &&

 HI&6&&7J&6& K HICJEK

If
23  &
O P  HM J&6& N HR
 &
 && Q $867BDB  HR HI&6&&7J&6& K HR  $867BDB S HRQ   &T6>U>SVW#  &6&8 B  86 BL c. Maximum value of electric field intensity, XYZ .

[
/



&& $ $ $ [ / HICJEK HI&6&&7J&6& K Since the inner cylinder has higher value of potential, thus it also has high value of electric field intensity, hence the XYZ occur at   &67L

X  $

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EEE 3133 Sem 1 2009/2010 prepared by Mohd Taufik

Therefore, XYZ  $

&&  658\
]L &6&&7HI&6&&7J&6& K

d. Calculate the relative permittivity, % if the charge per meter length on inner cylinder is & S &>C/meter. #< b
 =a
 ^  . / . /  $% -  . % _ /`  $%  HICJE K  HICJEK =a =a  $%  $% %
 HICJEK HICJEK C - HI JEK =a  Q %  $ $% %
 % =a HICJEK
 &6&&7J & S &> HI &6& K S  6 8D %  $

&& % =2 3 9. If
 " #   and relative permittivity, %  , determine the following at point P(1,2,1). a. Potential, V Substituting the coordinate into V, we have

b. Electric field intensity, E

c. Volume charge density, 

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EEE 3133 Sem 1 2009/2010 prepared by Mohd Taufik

d. Does
satisfy Laplace’s equation? No, since the charge density is not ZERO

10. The cylindrical capacitor whose cross section in Figure 1 has inner and outer radius of 5 mm and 15mm, respectively. If
2  7LL3  &&
and
2  7LL3  &
, calculate the following at   &LL; (Assume %  ) a. Potential field, V

From Laplace’s equation solution above, we gain

 Then by doing the minus gradient of V to find E;

 VcI JZK VcMdJZN




ef  HIEJCK %h %
 g  %  $

ef  HIEJC K   $)
 $

i  jc  k l

%h %
 l Q   C( E  HIEJC K

In this case
  &&
, E  7LL, C  7LL, %   Hence at ,   &LL

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EEE 3133 Sem 1 2009/2010 prepared by Mohd Taufik

 HM JCN

  !6B
HIEJCK b. Electric field intensity, E In this case
  &&
, E  7LL, C  7LL, %   Hence at ,   &LL   $)
 $

mn

 VcMdJZN

ef  $B &6Bef V/m

c. Electric field density, D g  %  $

%h %


ef  $ ! ef :]L# E  HI JC K

And; d.  on each plate i  jc  k l

%h %
 l Q   C( E  HIEJC K

For   7LL,

i   :]L#

For   7LL,

i  $ &D6 :]L#

Figure 1

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EEE 3133 Sem 1 2009/2010 prepared by Mohd Taufik