Screening Electric Fields using a Tube of Water:The Transition from Conductive to Dielectric Screening

J.S. Bobowski∗ and Jaklyn De Vos†

Department of Physics, University of British Columbia Okanagan,3333 University Way, Kelowna, BC, V1V 1V7, Canada

(Dated: August 24, 2015)

Abstract: An experiment that investigates the transition from conductive to dielectric screeningof electric fields by a tube of water has been designed for senior physics undergraduates. A parallel-plate capacitor is used to generate a uniform electric field. Two concentric acrylic plexiglass tubespass perpendicularly through the electric field generated between the plates. The region betweenthe tubes can be filled with air or water. An electrode, suspended within the inner plexiglass tube,is used to sense the electric potential at its location. The sensor is designed so that it can be rotatedto measure the potential at a second symmetric position. From the difference in the two potentials,the frequency dependence of the magnitude and phase of the electric field can be determined.With deionized water between the tubes, the magnitude and phase of the interior electric field wasmeasured from 100 Hz to 300 kHz. The high-pass filter frequency response expected for a dielectrictube with non-negligible conductivity was observed. Fits to the data yielded a very reasonableexperimental value for the ratio of the water’s conductivity to its dielectric constant.

Keywords: Electromagnetics, Screening, Dielectric, Conductivity, Voltage sensor, Lock-in detection

PACS numbers: 01.50.Pa, 07.50.Hp, 77.22.-d

I. INTRODUCTION

It is common and convenient to use a conducting enclo-sure, or Faraday cage, to exclude external electric fieldsfrom a region of space. In a Faraday cage, the conduc-tion electrons redistribute in response to the applied fieldsuch that they generate an electric field inside the cagethat exactly cancels the applied field. It is, however, alsopossible to attenuate external electric fields using dielec-tric enclosures. In the case of a dielectric shield, polarmolecules within the dielectric material reorient in re-sponse to the applied field. This alignment of electricdipoles generates an electric field that aligns with theapplied electric field within the dielectric material andpartially cancels the applied field inside the enclosure1,2.

Faraday cages and dielectric enclosures can both beused to suppress static electric fields. Oscillating elec-tromagnetic (EM) fields are also effectively screened byFaraday cages provided that any holes in the cage aremuch smaller than the wavelength of the incident EMradiation and that the cage walls are much thicker thanthe frequency-dependent EM skin depth. Dielectric en-closures will also suppress EM radiation provided that,at the frequency of the radiation, |εr(ω)| ≫ 1 where εr(ω)is the frequency-dependent complex relative permittivityof the shield1,2.

In this paper, an experiment designed to study thetransition from conductive to dielectric screening by atube of water is described. This experiment is suitablefor senior undergraduate physics students that have com-pleted a course in introductory electrodynamics.

FIG. 1. A dielectric tube of inner radius a and outer ra-dius b immersed in a uniform electric field E0 applied perpen-dicularly to its axis. Cross-section view (left) and side view(right).

II. THEORY

At the interface of two different uncharged dielectricmedia, Maxwell’s equations can be used to show thatthe perpendicular and parallel components of the electricfield E must satisfy the boundary conditions:

ε1E1⊥ = ε2E2⊥ (1)

E1∥ = E2∥ (2)

where ε1 and ε2 are the dielectric constants of the twomedia2. The geometry of interest in this experiment isshown in Fig. 1. A dielectric tube is immersed in a uni-form electric field applied perpendicularly to its axis. Inthis case, the boundary conditions in terms of the electricpotential V become:

ε1∂V1

∂r= ε2

∂V2

∂r(3)

V1 = V2 (4)

One way to determine the electric potential in all threeregions (r < a, a < r < b, and r > b) is to use separationof variables to solve Laplace’s equation ∇2V (r, ϕ) = 0

edited by Eblen-Zayas, Behringer, and Kozminski; Peer-reviewed, doi:10.1119/bfy.2015.pr.004 Published by the American Association of Physics Teachers under a Creative Commons Attribution 3.0 license. Further distribution must maintain attribution to the article’s authors, title, proceedings citation, and DOI.

2015 BFY Proceedings,

16

in cylindrical coordinates. Then, one must apply theboundary conditions given by Eqs. 3 and 4 at r = aand b1,3. The electric field inside the dielectric tube canthen be found using E = −∇V . The result for a tube ofrelative permittivity εr immersed in air is:

Ein =4εrE0

(εr + 1)2 −

(ab

)2

(εr − 1)2

(5)

and the shielding effectiveness is defined to beSE = E0/Ein

1,3. The experiment discussed in the nextsection investigates the shielding effectiveness of a tubeof water. The dielectric constant of room-temperaturewater is 79± 1 such that εr ≫ 1 and4:

SE ≈[1−

(ab

)2]εr4. (6)

All water contains a finite concentration of dissolvedions and therefore has a finite conductivity σ. The chargeflow through a cross-sectional area A and length ℓ of wa-ter can be modelled as parallel conduction through ca-pacitive and resistive impedances:

1

Zeff=

1

ZC+

1

ZR= jωε0εr

A

ℓ+ σ

A

ℓ(7)

= jωε0

(εr − j

σ

ωε0

)A

ℓ(8)

which allows one to characterize the charge dynamics us-ing a capacitive impedance Zeff with an effective complexand frequency-dependent dielectric constant5–7:

εeff (ω) ≡ εr

(1− j

σ

ωε0εr

). (9)

The shielding effectiveness of a tube of water is given byEq. 6 with εr replaced by εeff (ω) such that:

|SE(ω)|−1 ≈ 4/εr

1−(ab

)2

1√1 +

(σ

ωε0εr

)2(10)

tanβ ≈ σ

ωε0εr(11)

where β is the phase difference between the applied elec-tric field E0 and the field inside the shield Ein. Notethat |SE(ω)|−1

has the frequency dependence of a high-pass filter. At high frequencies ω ≫ σ/(ε0εr), the shield-ing effectiveness reduces to the form given in Eq. 6 andthe screening is determined by the dielectric propertiesof the water. On the other hand, at low frequenciesω ≪ σ/(ε0εr), |SE(ω)|−1 ∝ ωε0εr/σ and goes to zero atdc as would be expected for a Faraday cage.Finally, as will be discussed in the following section, the

tube of water used in the experiment is formed by fillingthe space between two concentric plexiglass tubes. Thepresence of the plexiglass modifies the geometry shownin Fig. 1 as now five regions and matching conditions at

four boundaries must be considered (air-plexiglass-water-plexiglass-air). The method used to calculate the shield-ing effectiveness of this geometry is exactly as describedabove but the solution is very complicated8. However,tanβ and the frequency dependence of the |SE|−1

re-main unchanged provided that the dielectric constant ofplexiglass (≈ 3.4) is much less than that of water (≈ 80).

III. EXPERIMENTAL DESIGN

Figure 2 shows a schematic of the experimental appa-ratus and Fig. 3 shows a photograph of the actual appa-ratus built in-house9. A pair of 1/2 in.-thick aluminumplates (16 in. × 16 in.) separated by 6 in. were used toconstruct a parallel-plate capacitor that creates uniformelectric field. The plates were driven by a center-tappedtransformer so as to establish a ground plane midwaybetween the capacitor plates. This allowed a groundedcopper tube to extend partially into the region betweenthe plates without resulting in a substantial redistribu-tion of the electric field lines.

The 5/8 in. copper tube supports an electrode sensorand houses a buffer circuit. The sensor was made from arectangular piece of 0.01 in.-thick of aluminum foil 3.6 in.long and 0.1875 in. wide. With the electrode placed toone side of the ground plane, it acquires a voltage thatis proportional to the capacitor plate voltage and is de-pendent on the input impedance of the buffer circuit10.A simple circuit model that can be used to estimate thevoltage acquired by the suspended electrode is shown inFig. 4. The capacitance C ′

0 represents the net capaci-tance between the electrode and the ground plane whichmust include the input capacitance of the buffer circuit.The voltage at the buffer output is given by10:

V0 = VC1 − C2

C ′0 + C1 + C2

. (12)

To ensure that the largest possible buffer output is mea-sured, it is important to minimize C ′

0 and hence the bufferinput capacitance. The op-amp used was an Analog De-vices AD549L which has a nominal input capacitance of0.8 pF (and input resistance 1015 Ω)11. It is also im-portant to consider the layout of the buffer circuit so asto minimize stray capacitance contributions to C ′

0 that

1

2 3

4

56

7

1 Function generator 2 Center-Tapped Transformer3 Capacitor Plates

4 Water/air-filled tubes5 Foil Sensor6 Shielded AD549 Op-Amp

7 Output

(to Lock-in Amplifier)

FIG. 2. A parallel-plate capacitor generates a uniform electricfield that is perpendicular to the axis of a tube of water. Anelectrode, used to sense the electric potential, is suspendedwithin the tube of water and feeds into a buffer.

17

Concentric Tubes

Sensor

Capacitor Plates

Water-Fill Line

Op-Amp Output

Grounded Copper Tube

AD549 Op-AmpSensor

FIG. 3. Photographs of the apparatus. Top: Two concentricplexiglass tubes pass through the capacitor. The sensor issuspended within the inner tube. Bottom: The buffer circuitis contained within an electrically grounded copper tube. Thecopper tube fits inside the inner plexiglass tube and allows thesensor to be rotated into two symmetric positions.

FIG. 4. A circuit model showing the electrode suspendedjust above the ground plane (dotted line) between oppositely-charged capacitor plates.

can occur between the electrode and the buffer input10.Many useful guidelines, such as tying the shield of thebuffer input to the output, are given in Refs. 11 and 12.As shown in Fig. 3, a pair of concentric plexiglass tubes

are suspended midway between the capacitor plates. Theouter tube has a inner diameter (id) of 1 in. and an outerdiameter (od) of 1.25 in. The copper tube slides freelyinside the inner tube (id = 5/8 in., od = 3/4 in.) andallows the electrode to be rotated into two symmetricpositions. In this experiment, the space between theplexiglass tubes is filled with either air or water. Theentire assembly is tilted at a slight angle so that, whenthe tubes are filled with water, there are no air pocketsin the region between the capacitor plates.A General Radio 1310-B function generator was used

to drive the transformer which stepped down the volt-age by a factor of eight at 1 kHz. The AD549L op-ampwas powered using two 9 V batteries and its output wasmeasured using a Princeton Applied Research 5204 lock-in amplifier. The lock-in amplifier was referenced to thevoltage applied to the top capacitor plate.To probe the electric field within the inner plexiglass

tube, and therefore test Eqs. 10 and 11, the potentialsat two symmetric positions were measured. The firstposition placed the electrode closer to the top plate asshown in Fig. 2 and the second position placed it nearerto the bottom plate. The difference of these two readingsis proportional to the electric field E ∝ V2−V1 such that:

EA cos (ωt+ γ) ∝A2 cos (ωt+ θ2)−A1 cos (ωt+ θ1) (13)

where EA and A are the amplitudes of the electric fieldand potentials respectively and γ and θ are phases mea-sured relative to the phase of the top-plate voltage. Ap-plication of some trigonometric identities leads to:

EA ∝√A2

1 +A22 − 2A1A2 cos∆θ (14)

tan γ =A2 sin θ2 −A1 sin θ1A2 cos θ2 −A1 cos θ1

(15)

where ∆θ ≡ θ2 − θ1. As a result, γ and a quantity pro-portional to the electric field amplitude can be extractedfrom the V1(A1, θ1) and V2(A2, θ2) measurements.

IV. EXPERIMENTAL RESULTS & SUMMARY

With air between the plexiglass tubes, the magnitudeand phase of the electric field were measured as a func-tion of frequency from 100 Hz to 300 kHz. This mea-surement was then repeated with water placed betweenthe tubes. The water used was filtered by a BarnsteadE-Pure water purification system. It is important to usehigh-purity water with a low conductivity so that the cut-off frequency given by σ/ (ε0εr) is as low as possible. Theresults are shown in Fig. 5. As expected, Figs. 5(a) and(b) show that the air measurement has a magnitude thatis approximately flat and a phase that is close to zeroat low frequencies. However, above 10 kHz the magni-tude and phase both start to decrease with increasingfrequency. These effects are due to the high-frequencylimitations of the op-amp and lock-in amplifier.

When the measurements are repeated with water be-tween the tubes, the same high-frequency characteristicsare observed, however, at low frequency there are differ-ences. The magnitude of the electric field is very smallat the lowest frequencies and the measured phase is ap-proximately π/2 radians. The frequency dependence ofthe detection electronics can be removed from the mag-nitude measurements by dividing the water data by theair data as shown in Fig. 5(c). Because only a quan-tity that is proportional to the magnitude has been mea-sured, the ratio of the water-to-air data have been scaledto approach one at high-frequency. The high-pass filterbehaviour predicted by Eq. 10 is clearly exhibited. Like-wise, the contribution of the detection electronics to thephase measurements can be removed by taking the differ-ence of the water phase and the air phase. The tangentof the difference is shown in Fig. 5(d) and exhibits theexpected inverse-frequency behaviour of Eq. 11.

18

(a)

102 103 104 10510-2

10-1

100

101

Air Water

EA (

a.u.

)

Frequency (Hz) (b)

103 104 105-3/4

-1/2

-1/4

0

1/4

1/2

/

Frequency (Hz)

Change to lock-inamplifier filter settings.

(c)

102 103 104 105

10-2

10-1

100

|SE

| -1

Frequency (Hz) (d)

103 104 1050

2

4

6

8

tan

Frequency (Hz)

FIG. 5. (a) Frequency dependence of the electric field with air and water between the plexiglass tubes. (b) The correspondingphase measurements. (c) The frequency dependencies of the buffer and detection electronics are removed from the amplitudemeasurements by dividing the water data by the air data. The ratio was fit to Eq. 10. (d) The difference γwater−γair determinesβ. The tanβ data were fit to Eq. 11. In all four plots, the measurement uncertainties are approximately equal to the size ofthe data points.

The |SE|−1and tanβ data were simultaneously fit

to Eqs. 10 and 11 in order to extract an experi-mental value for σ/ (ε0εr) = 78700± 500 s−1. For wa-ter at room temperature εr = 79 ± 14, such thatσ = 0.550± 0.008 MΩ−1cm−1 which is a reasonablevalue for the E-pure water used in the experiment.

An experiment to measure the frequency dependenceand phase of the shielding effectiveness of a tube of wa-ter has been designed, built, and executed. It is a novelexperiment for senior physics undergraduates that is richin both theoretical physics and measurement techniques.

Investigating the screening of electric fields by a dielec-tric tube is a standard exercise in electrodynamics3 andincluding the conductivity of water is a unique extensionof the problem. Students are challenged to develop acircuit model for an electrode suspended between capac-itor plates and hence understand why a high-impedancebuffer is required. Students are also introduced to thepowerful technique of lock-in detection. The experimen-tal results clearly show the predicted high-pass filter fre-quency dependence of the shielding effectiveness and rea-sonable values for the ratio of the water’s conductivity toits dielectric constant are extracted.

∗ Jake.Bobowski@ubc.ca; https://people.ok.ubc.ca/jbobowsk† Also at UrtheCast, 1055 Canada Place - Suite 33, Vancou-ver, BC V6C 0C3, Canada

1 S. Celozzi, R. Araneo, and G. Lovat, ElectromagneticShielding (John Wiley & Sons, 2008).

2 D. J. Griffiths, Introduction to Electrodynamics, 3rd ed.(Prentice Hall, 1999).

3 J. D. Jackson, Classical Electrodynamics, 3rd ed. (JohnWiley & Sons, 1998).

4 C. G. Malmberg and A. A. Maryott, J. Res. Natl. Stand.56, 2641 (1956).

5 J. S. Bobowski, T. Johnson, and C. Eskicioglu, Prog. Elec-tromagn. Res. Lett. 29, 139–149 (2012).

6 J. S. Bobowski and T. Johnson, Prog. Electromagn. Res.

B 40, 159–183 (2012).7 J. S. Bobowski, Am. J. Phys. 81, 899–906 (2013).8 The four-boundary problem was solved using the mathe-matical analysis software Maple 18 and then verified usingthe finite-element simulation tool COMSOL 4.3b.

9 J. De Vos, B.Sc. thesis, University of British Columbia(2014).

10 J. S. Bobowski, Md. S. Ferdous, and T. Johnson, IEEETrans. Instrum. Meas. 64, 923–934 (2015).

11 AD549 Datasheet: Ultralow input bias current operationalamplifier, Analog Devices, Inc. (2008).

12 AD515A Datasheet: Monolithic precision, low power FET-input electrometer op amp, Analog Devices, Inc. (1997).

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