PPT: Poisson’s & Laplace Equations | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) PDF Download

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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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FAQs on PPT: Poisson’s & Laplace Equations - Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)

1. What is Poisson's equation and how is it used in physics?
Ans. Poisson's equation is a partial differential equation that relates the distribution of a scalar field to its sources in physics. It is commonly used to describe phenomena such as the electric potential, gravitational potential, or fluid flow. By solving Poisson's equation, we can determine the potential or field distribution caused by the given sources.
2. What is Laplace's equation and why is it important in electrostatics?
Ans. Laplace's equation is a second-order partial differential equation that describes the behavior of scalar fields in physics. In electrostatics, Laplace's equation is used to model the electric potential in regions where there are no charges or external sources. Solving Laplace's equation allows us to determine the electric potential in these regions, which is crucial for understanding the behavior of electric fields and charges.
3. How are Poisson's and Laplace's equations related?
Ans. Poisson's equation is derived from Laplace's equation by introducing a source term. While Laplace's equation describes the behavior of scalar fields in regions without sources, Poisson's equation accounts for the presence of sources. Thus, Poisson's equation can be seen as an extension of Laplace's equation that incorporates the effect of sources on the field distribution.
4. Can Poisson's and Laplace's equations be solved analytically?
Ans. Yes, Poisson's and Laplace's equations can be solved analytically in certain cases. If the boundary conditions and sources are simple enough, it is possible to find analytical solutions using techniques such as separation of variables or Fourier series. However, in more complex situations, numerical methods like finite difference or finite element methods are often employed to approximate the solutions.
5. What are some applications of Poisson's and Laplace's equations in engineering and science?
Ans. Poisson's and Laplace's equations have numerous applications in various fields of engineering and science. Some examples include the analysis of electric and magnetic fields, fluid dynamics, heat conduction, and quantum mechanics. These equations are fundamental tools for understanding and solving a wide range of physical problems, making them essential in fields such as electrical engineering, physics, and computational science.
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