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.
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1-st region. |
Dielectric constant is the same in both half-spaces. |
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Let us place the image into the point of |
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y=-h. |
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r1 |
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Charge density of the image is |
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τ1 |
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1 |
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h |
r2 |
Field intensity at the boundaries |
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Ex |
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1 |
2 1 |
r2 |
2 1 |
r2 |
2 1r2 |
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r1 r2 r |
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Ey |
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r2 |
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Image method for the flat boundary between dielectrics.
2-nd region. |
Dielectric constant is the same in both half-spaces. |
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2 |
y |
2 |
h |
r |
x
2 No charges here!
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Charge density of the image is |
τ2 |
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Field intensity at the boundaries
Ex |
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2 |
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2 2 |
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Ey |
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h |
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2 2 r2 |
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5
Image method for the flat boundary between dielectrics.
Boundary conditions are:
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y |
Ex(1) Ex(2) |
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Dy(1) Dy(2) 1Ey(1) 2 Ey(2) |
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1 |
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h |
r |
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In the upper half-space |
Ex(1) |
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1 |
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x |
2 1r |
2 |
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Ex |
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In the lower half-space |
(2) |
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2 2 |
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r2 |
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2 1 |
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First relation |
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or |
1 2 |
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2 |
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6
Image method for the flat boundary between dielectrics.
Boundary conditions for vertical |
1Ey(1) |
2 Ey(2) |
component of the field intensity |
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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
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Second relation |
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or |
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2 |
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Ey(1) |
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h |
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1 |
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2 1r2 |
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h |
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(2) |
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2 |
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Ey |
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2 2 |
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r2 |
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1 2
2 2 2
1 2
7
Equivalent charge density.
Assuming the same dielectric constant in the whole space: |
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1 |
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y |
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Substituting the expressions for the derived |
linear |
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charge densities we shall get |
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1 |
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r |
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(1) |
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2 |
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h |
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(2) |
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h |
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h |
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Ey |
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Ey |
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1 2 r |
2 |
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x |
1 1 |
2 |
r |
2 |
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(1) |
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2 |
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1 |
2 h |
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Charge density |
Ey |
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Ey 1 |
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(x) |
1 |
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2 |
r2 |
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There are no free charges at the interface! |
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(x) 1 |
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2 2D(ext) |
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n |
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8 |
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Method of images for cylindrical boundaries between dielectrics.
9
Problem formulation
Primary field source – charged line |
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Dielectric cylinder |
1 |
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2 |
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r |
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C |
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0
The goal is to replace a cylinder by thin charged wires
E(1) |
E(2) |
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0 |
E(1) |
E( |
2) |
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n |
n |
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10