Current Flux in Half Cell

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Voltage and Current Output from a Stubby
Dipole Immersed in a Vertically Oriented 1000
Volt/meter E Field --- a Simple E Field Sensor
--- A Finite Element Model Solved Using
FlexPDE
Craig E. Nelson Consultant Engineer
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Background A stubby dipole antenna may be
used as an electrostatic field sensor. For a long
time I have been interested in knowing the extent
to which such an antenna sensor will distort the
electric field within which it is immersed.
The following numerical experiment provides
results for one simple physical situation. No
attempt at sensor optimization has been made.
Many further extensions of this experiment are
easily possible.
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Problem Geometry and Physical Layout of
the Solution Domain
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1000 Volts/meter E Field
Copper Rod Length 20 cm Radius 5
cm Conductivity 5.99e7
Vout Plus
Iout
Hi Resistance Rod Length 10 cm Radius 5
cm Conductivity 6.36e-7
Vout
Copper Rod Length 20 cm Radius 5
cm Conductivity 5.99e7
Vout Minus
1000 Volts/meter E Field
3-D Sensor Physical Layout
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1000 Volt/meter E Field
Stubby Dipole
Centerline
Solution Domain (cylindrical Geometry)
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Equations and Boundary Conditions
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The Partial Differential Equation to be Solved
is div ( J ) 0 in cylindrical (r,z)
coordinates div ( J ) (1/r)dr( rJr) dz(
Jz) 0 where JrcondEr JzcondEz
Jvector( Jr,Jz ) Jmmagnitude(J) Jr and Jz
are the current densities in the r and z
directions (amps/meter2) and Er -dr(U)
Ez-dz(U) E-grad(U) Emmagnitude(E) Er
and Ez are the electric field strength in the r
and z directions (volts/meter) and cond
conductivity in the different solution sub
domains (siemens/meter) The Boundary Conditions
are Natural (U) 0 on the centerline and
domain outer wall (Neuman) Value (U)
FieldStrengthHdomain/2 on the top surface
(Dirichlet) Value (U) - FieldStrengthHdomain/2
on the bottom surface (Dirichlet) where U is
the potential (volts) and Fieldstrength and
Hdomain are given parameters note dr(J) d(J)
/ d(r) dz(U) d(U) / d(z) and so on
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Numerical Experiment Results
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Contour Plot of Potential (referenced to the load
resistance vertical axis center)
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Contour Plot of Potential (referenced to the load
resistance vertical axis center)
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Contour Plot of Electric Field Strength
(volts/meter)
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Contour Plot of Electric Field Strength
(volts/meter)
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Contour Plot of log base 10 of Electric Field
Strength (three 1000 volts/meter)
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Contour Plot of log base 10 of Electric Field
Strength
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Plot of Potential along the Solution Domain
Centerline (volts)
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Plot of Electrical Field Strength Magnitude along
the Solution Domain Centerline (volts/meter)
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Plot of log base 10 of Electrical Field Strength
Magnitude along the Solution Domain
Centerline (zero 1 volt/meter)
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Model Parameters and Calculated Results
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Summary and Conclusions A numerical
experiment analysis of a stubby dipole antenna
electric field sensor has been accomplished. The
analysis shows that the despite a moderately high
electric field strength of 1000 volts/meter, the
sensor output voltage and current are rather
small. Apparently only a few tens of micro volts
appear across the resistive load (upper to lower
terminal resistance given as 10 megohm) with a
load current flow of several pico-amps. This is
because the highly conducting copper dipole arms
short the electrical field to near zero in
regions close to the conductors. It would seem
that this particular configuration is far from
optimal. Many other configurations are possible
and could be analyzed by the method presented here