- •Advanced chapters of theoretical electroengineering.
- •Lecture 4
- •Image method for the flat boundary between dielectrics.
- •Image method for the flat boundary between dielectrics.
- •Image method for the flat boundary between dielectrics.
- •Image method for the flat boundary between dielectrics.
- •Image method for the flat boundary between dielectrics.
- •Equivalent charge density.
- •Method of images for cylindrical boundaries between dielectrics.
- •Problem formulation
- •The inverse point
- •Normal component of the field intensity
- •Normal component of the field intensity
- •Geometrical relations
- •Angles
- •Geometrical relations
- •Geometrical relations
- •Trigonometric relations
- •Trigonometric relations
- •Geometrical relations
- •Field induced by the line sources
- •Geometrical relations
- •The field sources for the external domain
- •The field sources for the internal domain
- •Application of the Images Method for calculating magnetic fields in the presence of
- •Image method for the flat boundary between magnetic media.
- •Equivalent magnetic charge density.
- •The field in the presence of a cylindrical magnetic object
- •The field sources for the magnetic field intensity in the external domain
- •The field sources for the magnetic field intensity in the internal domain
- •Images of a two-wire transmission line (external domain)
- •Dependence of the field intensity on the coordinate
- •Inductance of the two-wire transmission line per unit length
- •External fluxes
- •Total inductance
- •Forces. The first line.
- •Forces. The second line.
Advanced chapters of theoretical electroengineering.
1
Lecture 4
Method of images
(метод зеркальных изображений)
after: K. Binns, P. Lawrenson. Analysis and computation of electric and magnetic field problems
2
Image method for the flat boundary between dielectrics.
y
1 h r
x
2
E
2 r
Application of the image method is not so evident.
Here the electric field exists in both half-spaces.
Each half-space requires separate solution
- field induced by the charged line source
3
Image method for the flat boundary between dielectrics.
|
1-st region. |
Dielectric constant is the same in both half-spaces. |
||||||||||||||||||||||||||||||||
|
|
|
y |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
1 |
|
|
|
|
|
|
|
|
|
|
|
|||||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
||
|
Let us place the image into the point of |
|
y=-h. |
|
||||||||||||||||||||||||||||||
|
|
|
|
|||||||||||||||||||||||||||||||
1 |
|
h |
r1 |
|
|
|||||||||||||||||||||||||||||
|
|
|
||||||||||||||||||||||||||||||||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
||||
|
|
Charge density of the image is |
|
|
τ1 |
|
|
|||||||||||||||||||||||||||
|
|
|
x |
|
|
|
|
|
|
|||||||||||||||||||||||||
1 |
|
h |
r2 |
Field intensity at the boundaries |
|
|
|
|
|
|
|
|||||||||||||||||||||||
|
|
|
|
|
|
|
|
|||||||||||||||||||||||||||
|
|
|
|
|
|
x |
|
|
1 |
|
|
x |
|
|
x |
|
1 |
|||||||||||||||||
|
|
|
Ex |
|
|
|
|
|
|
|
|
|
|
|||||||||||||||||||||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|||||||||||||||||
|
|
|
1 |
2 1 |
r2 |
2 1 |
r2 |
2 1r2 |
||||||||||||||||||||||||||
r1 r2 r |
|
Ey |
|
|
|
|
|
h |
|
1 |
|
|
|
|
h |
|
h |
|
|
|
1 |
|
||||||||||||
|
2 |
1 |
|
|
r2 |
2 |
1 |
|
|
r2 |
2 |
r2 |
||||||||||||||||||||||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
4 |
||||||||||||||||||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
1 |
|
|
|
|
|
Image method for the flat boundary between dielectrics.
2-nd region. |
Dielectric constant is the same in both half-spaces. |
|
2 |
y |
2 |
h |
r |
x
2 No charges here!
2
Charge density of the image is |
τ2 |
|
|
Field intensity at the boundaries
Ex |
|
2 |
|
x |
|
||||
2 2 |
r2 |
||||||||
|
|||||||||
Ey |
|
2 |
|
|
h |
||||
|
2 2 r2 |
||||||||
|
|
|
5
Image method for the flat boundary between dielectrics.
Boundary conditions are:
|
|
y |
Ex(1) Ex(2) |
|
Dy(1) Dy(2) 1Ey(1) 2 Ey(2) |
||||||||||||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
||
1 |
|
h |
r |
|
In the upper half-space |
Ex(1) |
|
|
x |
|
|
1 |
|||||
|
|
|
|
|
|||||||||||||
|
|
|
|
|
|
||||||||||||
|
x |
2 1r |
2 |
||||||||||||||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
2 |
|
|
|
|
|
Ex |
|
|
2 |
|
|
|
x |
|
|
|
|
|
|
In the lower half-space |
(2) |
|
|
|
|
|||||||
|
|
|
|
|
|
|
|
2 2 |
|
r2 |
|
||||||
|
|
|
|
1 |
2 |
|
2 1 |
|
|
|
|
|
|
|
|
|
|
|
First relation |
|
|
|
or |
1 2 |
|
|
|
|
|
|
|
||||
|
1 |
2 |
|
|
|
|
|
|
|
||||||||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
6
Image method for the flat boundary between dielectrics.
Boundary conditions for vertical |
1Ey(1) |
2 Ey(2) |
component of the field intensity |
|
|
y
1 h r
x
2
Combining with the 1-st
2 1 1 1
In the upper half-space
In the lower half-space
|
Second relation |
||||
|
|
|
|
|
|
or |
|
1 |
1 |
2 |
|
|
|
|
1 |
2 |
|
Ey(1) |
|
|
h |
|
|
1 |
|||
|
|
|
|
||||||
2 1r2 |
|||||||||
|
|
|
|
h |
|||||
(2) |
|
|
|
2 |
|
|
|||
Ey |
|
|
|
|
|
|
|
||
2 2 |
|
|
r2 |
||||||
|
|
|
|
|
1 2
2 2 2
1 2
7
Equivalent charge density.
Assuming the same dielectric constant in the whole space: |
|
|
1 |
|
|
||||||||||||||||||||||
|
|
|
y |
|
|
Substituting the expressions for the derived |
linear |
|
|
||||||||||||||||||
|
|
|
|
charge densities we shall get |
|
|
|
|
|
|
|
|
|
|
|
||||||||||||
1 |
|
|
|
r |
|
(1) |
|
|
2 |
|
|
h |
|
|
(2) |
|
|
|
|
|
h |
|
|||||
|
|
|
|
|
|
|
|
||||||||||||||||||||
|
h |
|
|
Ey |
|
|
|
|
|
|
|
|
|
Ey |
|
|
1 2 r |
2 |
|
||||||||
|
|
x |
1 1 |
2 |
r |
2 |
|
|
|
||||||||||||||||||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|||||||||||||
|
|
|
1 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
(1) |
|
2 |
|
|
|
|
|
|
|
|
|
|
1 |
2 h |
|
|
|
|
|
|||||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|||||||||
Charge density |
Ey |
|
Ey 1 |
|
|
|
|
|
|
(x) |
1 |
|
|
|
|
|
|
||||||||||
|
|
|
|
|
|
2 |
r2 |
|
|
|
|
|
|
||||||||||||||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
||||
There are no free charges at the interface! |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|||||||||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
||||||||||
|
|
|
|
(x) 1 |
|
2 2D(ext) |
|
|
|
|
|
||||||||||||||||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|||||||||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
1 |
2 |
|
n |
|
|
|
|
|
|||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
8 |
Method of images for cylindrical boundaries between dielectrics.
9
Problem formulation
Primary field source – charged line |
|
|
|
Dielectric cylinder |
1 |
|
2 |
|
r |
||
|
|
|
|
C |
|
0
The goal is to replace a cylinder by thin charged wires
E(1) |
E(2) |
|
0 |
E(1) |
E( |
2) |
|
|
|
n |
n |
|
10