The inverse point
The inverse point is located inside a circle
r
A B C
A position of inverse point is defined by a relation
AC BC r2
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Normal component of the field intensity
The normal component of |
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the external field intensity |
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cos( ) |
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2 0 |
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surface charges: |
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(x) 1 |
2 2D(ext) |
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C
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Field displacement induced by surface charges:
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Normal component of the field intensity
Field intensity induced by |
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the surface charges: |
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cos( ) |
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Normal component of the field intensity outside the cylinders:
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En En |
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To find a proper solution of the problem outside the cylinder it is |
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enough to ensure the right values of the normal component of the |
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field intensity along the interface! |
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Geometrical relations
Wire with the linear charge
density of τ is located in the point A
P is an arbitrary point at the circle
B is the inverse point
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A |
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Consider triangles APC and PCB
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PC r |
1. |
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BC |
2.Angle C is common
The triangles are similar !
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Angles
APQ 180 APB |
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ABP 180 APB |
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APQ ABP
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Geometrical relations
Wire with the linear charge
density of τ is located in the point A
P is an arbitrary point at the circle
B is the inverse point
b
A
Evidently |
AB b cos( ) a cos( ) |
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From the red triangle: |
AB sin( ) AO |
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OO 90
a
C
B
b sin( ) AO
So we get |
b cos( ) a cos( ) b sin( ) |
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sin( ) |
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Geometrical relations
PQ a sin( )
PQ r sin( )
Inside a triangle a sum of angles is 180º:
90 180
90 180
So we have:
P
ra
C
QB
Combining all these relations: |
a sin( ) r sin( ) |
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Trigonometric relations
We just have got two relations: |
b cos( ) a cos( ) b sin( ) |
a sin( ) r sin( ) |
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sin( ) |
Let us combine them: |
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b cos( ) a cos( ) a b sin( ) r sin( )
Taking into account: sin( ) sin( ) sin( ) sin( )
2
sin( ) ar sin( )
sin( ) sin( ) 2 sin( ) cos( ) sin( ) 2ar sin( ) cos( )
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Trigonometric relations
b cos( ) a cos( ) a b sin( ) r sin( )
sin( ) sin( ) 2 sin( ) cos( ) sin( ) 2ar sin( ) cos( )
b cos( ) a cos( ) |
a b sin( ) |
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r sin( ) |
a b 2r cos( ) sin( ) r a sin( )
b cos( ) a cos( ) |
a b |
b 2 cos( ) |
: ab |
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We shall get: |
cos( ) |
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Geometrical relations
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b |
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cos( ) |
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cos( ) |
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cos( ) |
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2 0 |
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The normal component of the field intensity induced by the surface charges may be expressed as
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2 |
1 |
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2 |
1 |
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cos( ) |
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2 0 |
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