Aston University
Department of Electronic Engineering and Applied Physics
Field Plotting Using Teledeltos
Paper
neglecting terms in δy2 and δx2. Since we are considering
a steady state there must be no net rate of loss or
accumulation of charge in the element: thus the expression
given above must be equal to zero. Divide by the area of
the element and
∂ Jx
∂x
A sheet of electrically conducting, but still fairly resistive,
material called Teledeltos paper is used. An electrical
current is passed through the sheet from one edge to
another . The edges can be arbitrarily shaped. A current can
also be driven from an electrode attached to the paper to
another electrode. The electrodes can be formed from
electrically conducting paint.
Suppose that the vector J (A m-1) represents the current
density per unit width at some point on the sheet. Then the
electric field (or voltage gradient) at the same point in the
sheet is
E = Jρ (V m-1)
where ρ (Ω per square) is the specific resistance of the
sheet. Note that ρ represents the resistance between
opposite sides of a square: you should check that this is so
in the experiment.
Theory
Consider a square element of the surface as illustrated in
the figure.
The net outflow of current from the element is
=
∂ Jx
δx
∂x )
∂J
(δy ∂ xx δx)
+
+
δ x ( Jy
+
∂ Jy
(δx ∂ y δy)
Teledeltos WTN/CGP 20/10/94
∂ Jy
δy
∂y )
−
( δyJ x
+
δxJy)
J
∇ ⋅
=
=
0 or as div J
J x δy
= −∇V
or E
= −grad
V
Then the continuity equation, 1, can then be written †
∂ 2V
∂ x2
+
∂ 2V
∂ y2
=
0
which is sometimes written
=
0
∇
∂ Jy 
δy δx
∂y 
δx
δy
E
(1)
0

J y +

The current flow is essentially in two dimensions, i.e.
across the surface of the paper but not through its thickness
from one face to the other.
+
∂ Jy
∂y
Sometimes this is written for short as
Introduction
δy (Jx
+
where x and y are the unit vectors in the x and y
direction respectively. The equation is sometimes written

J x +

∂ Jx 
δx  δy
∂x 
J y δx
2
V
=
0 or divgrad V
=
0
The equation is commonly referred to as Laplace's
equation.
Note that the theory can be applied in three dimensions
with only modest increase in complexity.
Even the two dimensional Laplace equation at first
encounter looks fiendish to solve. It is a partial differential
equation. There are in fact many known solutions. The
difficulties arise when trying to find a solution that fits the
given boundary conditions. Approximate methods are now
available on computers including the finite element method
and the finite diference, or relaxation method.
The relationship is called the equation of continuity and
applies to the flow of any fluid like material whose volume
or quantity does not change flowing across a region in
steady conditions. It applies to the flow of heat, fluids,
electricity, mass particles: see later. In this case it shows
that the current flows clear through the element without
shedding any charge.
Complete the table on the last page of these notes as far as
you are able to show what are the corresponding quantities
in the different systems all obeying the equation of
continuity.
In earlier days Teledeltos paper and other analogue methods
were used to find solutions of the Laplace equation and to
calculate flows, voltage gradients and related phenomena.
Potential
(i)
given two fixed electrodes at fixed voltage
(equipotentials) what is the total current flow
between them. This is of interest in calculating
resistance or capacitance.
(ii)
what is the field strength at different points in the
space between the electrodes?
Current flow is perpendicular to the equipotentials. The
flow lines and the equipotentials form a net of curvilinear
squares
Applications
Typical problems that are addressed are
The electric field, E, across the surface of the sheet may be
expressed in terms of the potential, V
E

= −

∂V ^
x
∂x
+
∂ V ^
y
∂ y 
(2)
1
This is of importance in determining hot spots in
resistance heating or places where the voltage gradient
(sometimes called the electric stress) is high and might lead
to dielectric breakdown.
2.
Experimental Procedure
silver paint to form electrodes along the outer lines as
shown in the figure. Wire fastened with drawing pins is
used to form an electrical contact with the paint.
1.2 Prepare both strips at the same time and allow to dry
hastening the process with lamps or heaters. While waiting,
prepare the next experiment.
15 mm radius
Electrodes
60 mm between
centres
Objective
To use Teledeltos paper to establish some of the
characteristics of potential fields.
Apparatus
1.
2.
3.
4.
5.
Teledeltos paper
Board, connectors, wire, silver conducting
paint, drying equipment
Graph paper and carbon paper for
measuring the positionof the equipotentials
DC power supply
Voltmeter and ammeter: probe electrode(s)
drawing pin to hold wire
Wire for
electric contact
conducting
paint strip
1.3 Measure the resistance of each square. Pass a
current between the electrodes and measure the potential
between the two inner lines using the probes. Make several
measurements placing the probe at different points along
the lines.
1.4
Compare the resistance of the two squares. Look at the
scatter in the resistances that you calculated. How would
you express the confidence you have in your results in a
quantitative way?
convenient for the next part to arrange the voltage between
the circles to be an integral number of volts.
Why have we suggested measuring the resistance in the
manner suggested? Why not simply paint the two
electrodes along the edges of the square, apply the current
and measure the voltage between the two electrodes? Why
not use an ohmmeter?
2.4 Trace out 3 or 5 equipotentials at equally spaced
voltages between the two electrode voltages. The probe
attached to the voltmeter can be used for this purpose.
Record a number of positions on each equipotentials by
pressing on the probe to make a mark on the plain paper
beneath the Teledeltos paper.
2.
Pencil lines
Repeat with two different currents.
Calculate the resistance.
Trace equipotentials between two cylinders
2.1 Take an 'infinite' sheet of Teledeltos paper about 150
mm on a side. Carefully paint two circles about 30 mm
diameter with centres separated by 60 mm. The pair of
circles should be central on the sheet as in the figure.
2.5 Remove the plain paper and fill in by hand each
equipotential
2.6 Sketch in by hand flow lines perpendicular to the
equipotentials trying to form 'curvilinear' squares.
Procedure
1.
To show the resistance between opposite sides of a
square is independent of the size of square.
2.2 Fasten the Teledeltos paper to the board with sheets
of plain and carbon paper underneath so that pressure on
the Teledeltos paper will make a mark on the plain paper.
1.1 Carefully cut two parallel sided strips of Teledeltos
paper 25 mm and 50 mm wide. Draw two lines across each
strip on the dark conducting side so as to form a square.
Draw two further lines outside the first pair of lines spaced
away rather more than half the sides of the paper. Use the
2.3 Pass a current from one circle to another and
measure its value. One of the electrodes should be earthed.
Measure the voltage between the circular electrodes using
the probes and doing your best to measure the voltages at
the outside of the circular electrodes. You may find it
Teledeltos WTN/CGP 20/10/94
(Note: this diagram is schematic and definitely not to scale:
yours will be different in important respects. Nonetheless
in former times engineers would analyse field by hand,
sketching potential and flow lines to produce what seemed
to them good curvilinear squares; this led to goodish
estimates of resistance. Note also that some of the squares
at the outside will have five sides: of such stuff are
engineering approximations made).
2
Equipotentials
3.2 Plot the potential along the line AB, between the tip
of the sharp electrode and the flat electrode. In your log
book draw a graph of potential versus distance along the
line.
3.3
4.
Parallel Plate Electrodes with Tube
4.1 Take a piece of Teledeltos paper 150 mm on a side.
Paint two straight parallel electrodes 120 mm long and
about 100 mm apart, and a hollow circular electrode of
Measure the resistance as best you can
10cm
Flow
lines
3.4 Now add paint to make the tip of the electrode 15
mm radius. Repeat the potential plot along the line AB.
How do the potential plots compare? Why the difference?
A
A'
3cm
3.5 Measure the resistance again. Again compare.
Comment
2.7 Count the number of 'squares' parallel to the
equipotentials and the number parallel to the flow lines
(this will be the number of equipotentials plus one). The
curvilinear net can be 'transformed' (in the minds eye) to a
rectangular net with the same number of squares in each
direction as the curved set. Since the resistance of each
square is independent of its size the resistance of the
rectangular set is the same as that of the original. Since you
know the resistance of a square you can estimate the
resistance of the array of squares. Compare this with the
measured value.
There is an extensive mathematical theory of such
conformal transformations arising from theory of complex
numbers. You may be allowed to enjoy this in later years.
2.8 Keep the results not only to write up a report but also
to compare with the results of a computer simulation to be
made in a later experiment.
3.
3.6 Measure the resistance between the electrodes.
Make a cut in the Teledeltos sheet about 20 mm long and
parallel to the current flow. Measure the resistance. Has it
altered? Why, or why not?
4cm
B'
B
1
2
about 3cm outer radius as shown.
3.7 Make a slit across the current flow and remeasure the
resistance. Compare with the previous measurement and
comment on differences of similarities.
3.8 Make a conducting paint strip about 30 mm long
parallel to the current flow. Again measure the resistance
and compare with the previous results.
3.9 Do you expect a conducting paint strip across the
direction of current flow to alter the resistance? Check
your prediction.
4.2 Set the supply voltage to 5V, and measure the
potential difference between electrode 1 and points along
the line BB'. Repeat for the line AA' and plot graphs of
both sets of results, using error bars to indicate
uncertainties.
4.3 From the p.d. graphs determine values for the electric
field component Ex along the lines, and sketch graphs for
both of them.
4.4 From the results of 4.2 determine Ex midway along
AA', estimating the uncertainty in this result.
Electrodes
Field Near a Sharp Electrode
3.1 Take a new infinite sheet about 150 mm square.
Paint an electrode along one edge and make this the earth
electrode. Make the opposite edge with an electrode
protruding 50 mm inwards, about 15 mm wide and with a
tip of radius 5 mm.
Teledeltos WTN/CGP 20/10/94
A
B
3
Conclusion
1.
2.
Laplace Equation for Different Physical Systems
Present and interpret the results of your
investigations
Electric
current flow
Consider the following:
- is the Teledeltos paper isotropic? If you did not
check, how could you have done so?
Potential
- Justify the resistance of squares being independent
of size by theory.
Force or
Potential
Gradient
- Why distrust measurements of resistance made
with an ohmmeter?
Material
Property
- What happens to the field at the edges of a parallel
plate arrangement?
Flow
Intensity
- Discuss the effects of making cuts in the sheet on
the overall resistance. Can you think how what you
observed might help in making approximate
computation of the resistance of awkward shaped
objects? (Not an easy question until you know the
answer - unless you see it).
Heat
Conduction
Gas or
chemical
diffusion
Fluid
Flow
Electrostatic
Fields
Magnetostatic
Fields
Force or potential gradient = (∆ potential / ∆ distance)
Flow intensity = (material property) × potential gradient
div (flow intensity) = 0
except at sources or sinks
This is the continuity equation.
† Appendix1
∂J
∂x
∂J
∂y
=
0
and since J
=
σE
+
and E
then
J
∇ ⋅
= −∇V
=
σ∇ ⋅ E
= −
i .e.
∇
2
V
=
Teledeltos WTN/CGP 20/10/94
σ∇
⋅ ∇V = −
σ∇2V
=
0
0
4