PPT - Poisson’s and Laplace Equations Electrical Engineering (EE) Notes | EduRev

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Electrical Engineering (EE) : PPT - Poisson’s and Laplace Equations Electrical Engineering (EE) Notes | EduRev

 Page 1


Poisson’s and Laplace Equations
A useful approach to the calculation of electric potentials
Relates potential to the charge density.
The electric field is related to the charge density by the divergence relationship
The electric field is related to the electric potential by a gradient relationship
Therefore the potential is related to the charge density by Poisson's equation
In a charge-free region of space, this becomes Laplace's equation
Page 2


Poisson’s and Laplace Equations
A useful approach to the calculation of electric potentials
Relates potential to the charge density.
The electric field is related to the charge density by the divergence relationship
The electric field is related to the electric potential by a gradient relationship
Therefore the potential is related to the charge density by Poisson's equation
In a charge-free region of space, this becomes Laplace's equation
Potential of a Uniform Sphere of Charge
outside
inside
Page 3


Poisson’s and Laplace Equations
A useful approach to the calculation of electric potentials
Relates potential to the charge density.
The electric field is related to the charge density by the divergence relationship
The electric field is related to the electric potential by a gradient relationship
Therefore the potential is related to the charge density by Poisson's equation
In a charge-free region of space, this becomes Laplace's equation
Potential of a Uniform Sphere of Charge
outside
inside
Poisson’s and Laplace Equations
Poisson’s Equation
From the point form of Gaus's Law
Del_dot_ D r
v
Definition D
D eE
and the gradient relationship
E DelV -
Del_D Del_ eE
()
Del_dot_ eDelV
( )
- r
v
Del_DelV
r
v
-
e
Laplace’s Equation
if
r
v
0
Del_dot_D r
v
Del_Del Laplacian
The divergence of the
gradient of a scalar function
is called the Laplacian.
Page 4


Poisson’s and Laplace Equations
A useful approach to the calculation of electric potentials
Relates potential to the charge density.
The electric field is related to the charge density by the divergence relationship
The electric field is related to the electric potential by a gradient relationship
Therefore the potential is related to the charge density by Poisson's equation
In a charge-free region of space, this becomes Laplace's equation
Potential of a Uniform Sphere of Charge
outside
inside
Poisson’s and Laplace Equations
Poisson’s Equation
From the point form of Gaus's Law
Del_dot_ D r
v
Definition D
D eE
and the gradient relationship
E DelV -
Del_D Del_ eE
()
Del_dot_ eDelV
( )
- r
v
Del_DelV
r
v
-
e
Laplace’s Equation
if
r
v
0
Del_dot_D r
v
Del_Del Laplacian
The divergence of the
gradient of a scalar function
is called the Laplacian.
LapR
xx
Vxy , z , ()
d
d
æ
ç
è
ö
÷
ø
d
d yy
Vxy , z , ()
d
d
æ
ç
è
ö
÷
ø
d
d
+
zz
Vxy , z , ()
d
d
æ
ç
è
ö
÷
ø
d
d
+
é
ê
ë
ù
ú
û
:=
LapC
1
r
r
r
r
Vrf , z ,
( )
d
d
×
æ
ç
è
ö
÷
ø
d
d
×
1
r
2
ff
Vrf , z ,
( )
d
d
æ
ç
è
ö
÷
ø
d
d
é
ê
ë
ù
ú
û
× +
zz
Vrf , z ,
( )
d
d
æ
ç
è
ö
÷
ø
d
d
+ :=
LapS
1
r
2
r
r
2
r
Vr q ,f ,
( )
d
d
×
æ
ç
è
ö
÷
ø
d
d
×
é
ê
ë
ù
ú
û
1
r
2
sin q
()
×
q
sin q
()
q
Vr q ,f ,
( )
d
d
×
æ
ç
è
ö
÷
ø
d
d
× +
1
r
2
sin q
()
2
×
ff
Vr q ,f ,
( )
d
d
d
d
× + :=
Poisson’s and Laplace Equations
Page 5


Poisson’s and Laplace Equations
A useful approach to the calculation of electric potentials
Relates potential to the charge density.
The electric field is related to the charge density by the divergence relationship
The electric field is related to the electric potential by a gradient relationship
Therefore the potential is related to the charge density by Poisson's equation
In a charge-free region of space, this becomes Laplace's equation
Potential of a Uniform Sphere of Charge
outside
inside
Poisson’s and Laplace Equations
Poisson’s Equation
From the point form of Gaus's Law
Del_dot_ D r
v
Definition D
D eE
and the gradient relationship
E DelV -
Del_D Del_ eE
()
Del_dot_ eDelV
( )
- r
v
Del_DelV
r
v
-
e
Laplace’s Equation
if
r
v
0
Del_dot_D r
v
Del_Del Laplacian
The divergence of the
gradient of a scalar function
is called the Laplacian.
LapR
xx
Vxy , z , ()
d
d
æ
ç
è
ö
÷
ø
d
d yy
Vxy , z , ()
d
d
æ
ç
è
ö
÷
ø
d
d
+
zz
Vxy , z , ()
d
d
æ
ç
è
ö
÷
ø
d
d
+
é
ê
ë
ù
ú
û
:=
LapC
1
r
r
r
r
Vrf , z ,
( )
d
d
×
æ
ç
è
ö
÷
ø
d
d
×
1
r
2
ff
Vrf , z ,
( )
d
d
æ
ç
è
ö
÷
ø
d
d
é
ê
ë
ù
ú
û
× +
zz
Vrf , z ,
( )
d
d
æ
ç
è
ö
÷
ø
d
d
+ :=
LapS
1
r
2
r
r
2
r
Vr q ,f ,
( )
d
d
×
æ
ç
è
ö
÷
ø
d
d
×
é
ê
ë
ù
ú
û
1
r
2
sin q
()
×
q
sin q
()
q
Vr q ,f ,
( )
d
d
×
æ
ç
è
ö
÷
ø
d
d
× +
1
r
2
sin q
()
2
×
ff
Vr q ,f ,
( )
d
d
d
d
× + :=
Poisson’s and Laplace Equations
Given
Vxy , z , ( )
4y × z ×
x
2
1 +
:=
x
y
z
æ
ç
ç
è
ö
÷
÷
ø
1
2
3
æ
ç
ç
è
ö
÷
÷
ø
:= eo 8.85410
12 -
× :=
Vxy , z , ( ) 12 =
Find: V @ and
r
v at P
LapR
xx
Vxy , z , ( )
d
d
æ
ç
è
ö
÷
ø
d
d yy
Vxy , z , ( )
d
d
æ
ç
è
ö
÷
ø
d
d
+
zz
Vxy , z , ( )
d
d
æ
ç
è
ö
÷
ø
d
d
+
é
ê
ë
ù
ú
û
:=
LapR 12 =
rv LapR eo × := rv 1.062 10
10 -
´ =
Examples of the Solution of Laplace’s Equation
D7.1
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