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Transcript of Chapter 4: Solutions of Electrostatic Problems 4-1 Introduction 4-2 Poisson’s and Laplace’s...
![Page 1: Chapter 4: Solutions of Electrostatic Problems 4-1 Introduction 4-2 Poisson’s and Laplace’s Equations 4-3 Uniqueness of Electrostatic Solutions 4-4 Methods.](https://reader033.fdocuments.in/reader033/viewer/2022061421/56649d6e5503460f94a4f43c/html5/thumbnails/1.jpg)
Chapter 4: Solutions of Electrostatic Problems
4-1 Introduction
4-2 Poisson’s and Laplace’s Equations
4-3 Uniqueness of Electrostatic Solutions
4-4 Methods of Images
4-5 Boundary-Value Problems in Cartesian Coordinates
4-6 Boundary-Value Problems in Cylindrical Coordinates
4-7 Boundary-Value Problems in Spherical Coordinates
![Page 2: Chapter 4: Solutions of Electrostatic Problems 4-1 Introduction 4-2 Poisson’s and Laplace’s Equations 4-3 Uniqueness of Electrostatic Solutions 4-4 Methods.](https://reader033.fdocuments.in/reader033/viewer/2022061421/56649d6e5503460f94a4f43c/html5/thumbnails/2.jpg)
4.2 Poisson’s and Laplace’s Equations
Two fundamental equations for electrostatics
In a linear and isotropic medium
For a homogeneous medium and Poisson’s equation:
2 =V
-
and
( )=-Vand
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2 22
2 2 2
1 1( )
V V VV r
r r r r z
Cylindrical coordinates:
22 2
2 2 2 2 2
1 1 1( ) (sin )
sin sin
V V VV R
R R R R R
Spherical coordinates:
2 0V Laplace’s equation (No free charges):
2 2 2
2 2 2=
V V V
x y z
-Cartesian coordinates:
Charges on the conductor surfaces
Electric field
Once V is known,
![Page 4: Chapter 4: Solutions of Electrostatic Problems 4-1 Introduction 4-2 Poisson’s and Laplace’s Equations 4-3 Uniqueness of Electrostatic Solutions 4-4 Methods.](https://reader033.fdocuments.in/reader033/viewer/2022061421/56649d6e5503460f94a4f43c/html5/thumbnails/4.jpg)
Q: Determine the field both inside and outside a spherical cloud of electrons with a uniform volume charge density ρ=˗ρ0 (where ρ0 is a positive quantity) for 0 ≤ R ≤ b and ρ=0 for R>b by solving Poisson’s and Laplace’s equation for V.
Q: Determine the field both inside and outside a spherical cloud of electrons with a uniform volume charge density ρ=˗ρ0 (where ρ0 is a positive quantity) for 0 ≤ R ≤ b and ρ=0 for R>b by solving Poisson’s and Laplace’s equation for V.
0 R b 0
2 =V
-
2 02
0
1( )idVdR
R dR dR
0 12
0
.3
idV CR
dR R
Electric field inside:
(1) Inside the cloud
Poisson’s equation
ρ=0 ρ=-ρ0
b
Ei cannot be infinite at R=0 C1=0
![Page 5: Chapter 4: Solutions of Electrostatic Problems 4-1 Introduction 4-2 Poisson’s and Laplace’s Equations 4-3 Uniqueness of Electrostatic Solutions 4-4 Methods.](https://reader033.fdocuments.in/reader033/viewer/2022061421/56649d6e5503460f94a4f43c/html5/thumbnails/5.jpg)
0
2 =0V
22
1( ) 0odVdR
R dR dR
Electric field outside:
(2) Outside the cloud
Laplace’s equation
0 22
dV C
dR R
Electric field continuity at R=b
30
4
3Q b
ρ=0 ρ=-ρ0
b
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ρ=0 ρ=-ρ0
b
Potential inside:
Potential outside:
At boundary R = b, Vi = V0, we have
Calculate electric field from potential
Lastly, Vi Ei and Vo Eo.
![Page 7: Chapter 4: Solutions of Electrostatic Problems 4-1 Introduction 4-2 Poisson’s and Laplace’s Equations 4-3 Uniqueness of Electrostatic Solutions 4-4 Methods.](https://reader033.fdocuments.in/reader033/viewer/2022061421/56649d6e5503460f94a4f43c/html5/thumbnails/7.jpg)
4.3 Uniqueness of Electrostatic Solutions
Uniqueness theorem
A solution of Possion’s equation (of Laplace’s equation is a special case) that satisfies the given boundary conditions is a unique solution.
A solution of an electrostatic problem satisfying its boundary conditions is the only possible solution, irrespective of the method by with the solution is obtained.
![Page 8: Chapter 4: Solutions of Electrostatic Problems 4-1 Introduction 4-2 Poisson’s and Laplace’s Equations 4-3 Uniqueness of Electrostatic Solutions 4-4 Methods.](https://reader033.fdocuments.in/reader033/viewer/2022061421/56649d6e5503460f94a4f43c/html5/thumbnails/8.jpg)
4.4 Methods of Images
Methods of Images: replace boundary surfaces by appropriate image charges in lieu of a formal solution of Poisson’s or Laplace’s equation.
Grounded conducting plane
Grounded conducting plane
+
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4.4-1 Point charge and conducting planes
Q: Consider the case of a positive point charge, Q , located at a distance d above a large grounded (zero-potential) conducting planeFind the potential at every point above the conducting plane (y>0).
The V(x,y,z) should satisfy the following conditions:(1) At all points on the grounded conducting plane, V(x,0,z)=0.(2) At points very close to Q, V approaches that of the point charge alone; that is as R→0
04
QV
R
(3) At points very far from , , V →0.(4) V is even with respect to the x and z coordinates:
V(x, y, z) = V(-x, y, z) and V(x, y, z) = V(x, y, -z)
( , , )Q x y z
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Positive charge Q at y=d would induce negative charges on the surface of the conducting plane, resulting in a surface charge density ρs. Hence the potential at points above the conducting plane would be:
2 2 20 10
1( , , ) ,
44 ( )s
s
QV x y z ds
Rx y d z
Trouble: how to determine the surface charge density ρs?
![Page 11: Chapter 4: Solutions of Electrostatic Problems 4-1 Introduction 4-2 Poisson’s and Laplace’s Equations 4-3 Uniqueness of Electrostatic Solutions 4-4 Methods.](https://reader033.fdocuments.in/reader033/viewer/2022061421/56649d6e5503460f94a4f43c/html5/thumbnails/11.jpg)
If we remove the conductor and replace it by an image point charge -Q at y=-d, then the potential at a point P(x,y,z) in the y>0 region is
Note: The image charge should be located outside the region where the field is to be determined
12 2 2 2[ ( ) ]R x y d z
12 2 2 2[ ( ) ]R x y d z
Original charge
Image charge
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Point charge and perpendicular conducting planes
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Q: Consider a line charge ρl (C/m) located at a distance d from the axis of a parallel, conducting, circular cylinder of radius a. Both the line charge and the conducting cylinder are assumed to be infinitely long. Determine the position of the image charge.
Image: (1) the cylindrical surface at r=a an equipotential surface → the image must be a parallel line charge (ρi) inside the cylinder.(2) symmetry with respect to the line OP → the image must lie somewhere along OP. Say at point Pi, which is at a distance di from the axis.
4.4-2 Line charge and parallel conducting cylinder
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Determine: ρi and di
Let us assume that ρi = - ρl, and electric potential at a distance r from a line charge of density ρl is
0 0
0
0 0
1ln
2 2
r r
rr r
rV E dr dr
r r
r0 : zero potential
At any point M on the cylindrical surface, the potential:
0 0
0 0 0
ln ln ln2 2 2
iM
i
r r rV
r r r
r0 for both ρi and ρl
Original charge Image charge
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Triangles OMPi and OPM similar :
The image line charge ρi together with ρl, will make the dashed cylindrical surface equipotential.
Equipotential
2
i
ad
d
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The second equipotential surface
Two wire transmission line
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Electric field and equipotential lines of a transmission line pair
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4.4-3 Point charge and conducting sphere
Q: A point charge Q is at a distance d from the center of a grounded conducting sphere of radius a (a<d). Determined the charge distribution induced on the sphere and the total charge induced on the sphere.
Image charge Qi can be equal to –Q?Triangles OMPi and OPM are similar.
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The electric potential V at an arbitrary point
0 0
1 1( , ) ( ) ( )
4 4i iQ Q Q Q
aQQ Q adV R
R R R dR
The law of cosines 2 2 1/2[ 2 cos ]QR R d Rd 2 2
2 2 1/2[ ( ) 2 ( )cos ]iQ
a aR R R
d d
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0 0
1 1( , ) ( ) ( )
4 4i iQ Q Q Q
aQQ Q adV R
R R R dR
The R-component of the electric field intensity ER
( , )( , )R
V RE R
R
2
2 2 3/2 2 2 2 2 3/20
cos [ ( / ) cos ]( , ) { }
4 ( 2 cos ) [ ( / ) 2 ( / ) cos ]R
Q R d a R a dE R
R d Rd d R a d R a d
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(2) The total charge induced on the sphere is obtained by integrating ρs over the surface of the sphere.
22
.
0 0
Total-induced-charge sins s i
ads a d d Q Q
d
(1) To find the induced surface charge on the sphere, we set R=a
2 2
0 2 2 3/2
( )( , )
4 ( 2 cos )s R
Q d aE a
a a d ad
Induced surface charge is negative and that its magnitude is maxmium at θ=0 and minimum θ=π.
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4.4-4 Charged sphere and grounded plane
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4.5 Boundary-value problems in Cartesian Coordinates
Method of separation of variablesFor problems consisting of a system of conductors maintained at specified potentials and with no isolated free charges
Three types of boundary-value problems:(1)Dirichlet problems: the potential value V is specified everywhere on the boundaries;(2)Neumann problems: the normal derivative of the potential is specified everywhere on the boundaries;(3)Mixed boundary-value problems: the potential value is specified over some boundaries and the normal derivative of the potential is specified over the remaining ones;
2 0V Potential equation (free source):
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Laplace’s equation for V in Cartesian coordinates
2 2 2
2 2 20
V V V
x y z
Apply method of separation of variables, V(x,y,z) can be expressed as:
( , , ) ( ) ( ) ( )V x y z X x Y y Z z
X(x),Y(y) and Z(z) are functions of only x, y and z, respectively.
2 2 2
2 2 2
( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( ) 0
d X x d Y y d Z zY y Z z X x Z z X x Y y
dx dy dz
2 2 2
2 2 2
1 ( ) 1 ( ) 1 ( )0
( ) ( ) ( )
d X x d Y y d Z z
X x dx Y y dy Z z dz
In order for Equation to be satisfied for all values of x, y, z, each of the three terms must be a constant.
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kx2 is a constant to be determined from the boundary conditions
kx is imaginary, -kx2 is a positive real number
kx is real, -kx2 is a negative real number
2
2
1 ( )[ ] 0
( )
d d X x
dx X x dx
22
2
1 ( )
( ) x
d X xk
X x dx
22
2
( )( ) 0x
d X xk X x
dx
22
2
( )( ) 0y
d Y yk Y y
dy
22
2
( )( ) 0z
d Z zk Z z
dz
2 2 2 0 x y zk k k
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Possible Solutions of
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Q: Two grounded, semi-infinite, parallel-plane electrodes are separated by a distance b. A third electrode perpendicular to and insulated from both is maintained at a constant. Determine the potential distribution in the region enclosed by the electrodes.
With V independent of z , , ,V x y z V x y
In the x direction 00,V y V , 0V y
In the y direction ,0 0V x , 0V x b
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2( ) kxX x D e
1( ) sinY y A ky
2 2 2 0 x y zk k k 2 2 2y xk k k k real number, so kx=jk
0zk 0Z z Bz-independence:
0 2 1, kx kxn nV x y B D A e sinky C e sinky
( , , ) ( ) ( ) ( )V x y z X x Y y Z z
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, 0kxn nV x b C e sinkb
Should be satisfied for all values of x, only if
, 1, 2,3,...n
k nb
( , ) sinn
xb
n n
nV x y C e y
b
01 1
(0, ) (0, ) sinn nn n
nV y V y C y V
b
04,
0,n
Vn odd
C nn even
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Solution of Electrostatic Problems
Method of imagesfree charges near conducting boundaries.
Method of separation of variablesfor problems consisting of a system of conductors maintained at specified potentials and with no isolated free charges.the solution can be expressed as a product of three one-dimensional functions, each depending separately on one coordinate variable only