Field Plotting Using Teledeltos Paper
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1
2
3
4δ
δ
2
2
neglecting terms in
y
and
x
. Since we are considering
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where
x and y
are the unit vectors in the x and y
a steady state there must be no net rate of loss or
direction respectively. The equation is sometimes written
Department of Electronic Engineering and Applied Physics
accumulation of charge in the element: thus the expression
E
V or E
grad V
= −∇
= −
given above must be equal to zero. Divide by the area of
the element and
†
Then the continuity equation, 1, can then be written
Field Plotting Using Teledeltos
∂
∂
∂
∂
2
2
J
J
V
V
y
x
Paper
+
=
0
(1)
0
+
=
∂
∂
∂
∂
2
2
x
y
x
y
Sometimes this is written for short as
which is sometimes written
Introduction
2
∇ ⋅
J
=
0 or as div J
=
0
∇
V
=
0 or divgrad V
=
0
A sheet of electrically conducting, but still fairly resistive,
The equation is commonly referred to as Laplace's
∂
material called Teledeltos paper is used. An electrical
J
δ
δ
y
equation.
J
y +
y
x
current is passed through the sheet from one edge to
∂
y
Note that the theory can be applied in three dimensions
another . The edges can be arbitrarily shaped. A current can
with only modest increase in complexity.
also be driven from an electrode attached to the paper to
δ
another electrode. The electrodes can be formed from
x
δ
J
y
Even the two dimensional Laplace equation at first
electrically conducting paint.
x
δ
encounter looks fiendish to solve. It is a partial differential
y
∂
J
δ
δ
x
equation. There are in fact many known solutions. The
J
x
y
x +
The current flow is essentially in two dimensions, i.e.
∂
x
difficulties arise when trying to find a solution that fits the
across the surface of the paper but not through its thickness
given boundary conditions. Approximate methods are now
from one face to the other.
available on computers including the finite element method
δ
and the finite diference, or relaxation method.
-1
J
x
Suppose that the vector J (A m
) represents the current
y
density per unit width at some point on the sheet. Then the
The relationship is called the equation of continuity and
In earlier days Teledeltos paper and other analogue methods
electric field (or voltage gradient) at the same point in the
applies to the flow of any fluid like material whose volume
were used to find solutions of the Laplace equation and to
sheet is
ρ
or quantity does not change flowing across a region in
calculate flows, voltage gradients and related phenomena.
-1
E = J
(V m
)
steady conditions. It applies to the flow of heat, fluids,
ρ
Ω
electricity, mass particles: see later. In this case it shows
Current flow is perpendicular to the equipotentials. The
where
(
per square) is the specific resistance of the
ρ
that the current flows clear through the element without
flow lines and the equipotentials form a net of curvilinear
sheet. Note that
represents the resistance between
shedding any charge.
squares
opposite sides of a square: you should check that this is so
Complete the table on the last page of these notes as far as
in the experiment.
you are able to show what are the corresponding quantities
Applications
in the different systems all obeying the equation of
Theory
continuity.
Typical problems that are addressed are
Consider a square element of the surface as illustrated in
Potential
(i)
given two fixed electrodes at fixed voltage
the figure.
(equipotentials) what is the total current flow
The net outflow of current from the element is
The electric field, E, across the surface of the sheet may be
between them. This is of interest in calculating
∂
∂
expressed in terms of the potential, V
resistance or capacitance.
J
(
J
)
(
)
δ
δ
δ
δ
δ
δ
y
x
(
)
y
J
x +
x
+
x
J
y +
y
−
yJ
x +
xJ
∂
∂
y
∂
∂
x
y
V
V
^
(2)
^
E
x
y
(ii)
what is the field strength at different points in the
= −
+
∂
∂
∂
∂
J
J
(
)
x
y
(
)
δ
δ
δ
δ
y
x
space between the electrodes?
=
y
x
+
x
y
∂
∂
x
y
1
Teledeltos WTN/CGP 20/10/94
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