Shering Bridge

8/18/2019 Shering Bridge

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DIELECTRIC CONSTANTS

References: - Bleaney and Bleaney Electricity and Magnetism, 2nd Ed., Chapter 17.Duffin, Electricity and Magnetism, 4th Ed. p 298-9 (or 3rd Ed p 325-6).

Aims

The aim of this experiment is to measure the dielectric constants of a number of liquidsat high (optical) and low (audio) frequencies. These results are then used to obtainvalues for the dipole moments of the molecules of the liquid. The experiment alsointroduces the use of phase sensitive detectors.

The relative permittivity ε r of an insulating medium, which is also referred to as thedielectric constant, may be defined by C m /C 0 = ε r where C 0 is the capacitance of acapacitor in vacuum and C m is that of the same capacitor with the medium filling the

space between the capacitor plates. The fact that ε r is not unity is due to thepolarisation of the medium by the electric field. This polarisation can arise in a numberof ways. The electric field can induce dipoles by distortion of the electronic or ionicdistribution of an initial non-polar medium. With a polar medium it can orientate pre-existing permanent molecular dipoles along the field direction. The time taken for suchre-orientation is much longer than that needed to induce dipoles. This can lead to ε r being frequency dependent and this dependence can be used to measure thepermanent dipole moments of materials.

In this experiment the dielectric properties of toluene (C7H8) and chlorobenzene(C6H5Cl) will be compared by measurement of ε r at a low (audio) frequency and very

high (optical) frequency. The two parts of the experiment can be done in any order.There are only two sets of equipment for the optical measurements (which should takeless than one hour) and so students should co-operate in the timing of the use of theequipment.

Audio Measurements

The low frequency measurements are made using parallel plate capacitors, which aremounted in containers, which can be filled with liquid. The capacitance in air, C A, andin the liquid, C L, is measured using a Schering capacitance bridge operating at 1 kHz(see Appendix II). For safety reasons the chlorobenzene filled capacitor is in a sealedcontainer. This has been chosen to have exactly the same capacitance in air as theother capacitors. The ratio C L /C A gives the ratio of the dielectric constants. ε r for air is1.00058.

Optical Measurements

The refractive index of a substance at a particular frequency is given by n = ( µ r ε r ) ½ .

For the liquids used in this experiment one can assume that the permeability µ r = 1. ε r may therefore be obtained from a measurement of n . The refractive index of Tolueneis to be measured at an optical frequency (corresponding to λ= 5893Å) using the Abbé

Refractometer. (See Appendix I). That of chlorobenzene, which again cannot bemeasured for safety reasons is n = 1.525 at 20oC.


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Calculating Dipole Moments

If it is assumed that any difference between the high and low frequency values of ε r isdue solely to the presence of permanent dipoles, which are unable to contribute to thepolarisation of the liquid at optical frequencies, one can calculate the values of the

dipole moments of the liquids, p , from the formula below:-

where ε s = static (low frequency) dielectric constant. (This contains both induced and

permanent dipolar contributions to the polarisation). ε i = induced (high frequency)dielectric constant, (i.e. the high frequency value where permanent dipolar

contributions are zero). N 0 is Avogadro's number, ρ = density, M = molecular weight,

ε 0 = permittivity of free space, and k B is Boltzmann's constant. The value of p shouldbe expressed in Debye units. (1 Debye unit is 3.336 x 10-30 coulomb metre).

Additional considerations

Look up the dielectric constant for water and hence calculate its dipole moment. Why

are we unable to make a satisfactory measurement of ε r for water at low frequencies?If water is such a good dielectric (optically clear), why do microwave ovens work sowell? Mathematically, what is the lock-in detector doing?

( )( )

( ) T k M

p N

Bis

isis

0

2

0

292

2

ε

ρ

ε ε

ε ε ε ε =

+

+−


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APPENDIX I

The Abbé Refractometer

NOTE Great care must be taken not to damage the polished surfaces of the prismsand test pieces used in this experiment. Surfaces are to be cleaned only with the

cotton buds or lens tissue provided. The Abbé Refractometer is an instrumentdesigned to permit rapid measurement of the refractive index, n , and the dispersion ofliquids and solids. The instruments used in this experiment can measure in the range1.30 < n


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specimens that may have the same value of n but different dispersions. The value ofthe dispersion is not needed in this experiment.

The temperature at which each measurement is made should be noted, as therefractive indices of some liquids are strongly temperature dependent.

Calibration

Check the scale calibration using the glass test pieces. The arrangement to be usedis that of Figure 1 (a); a fluorescent lamp will act as the light source. To ensure goodoptical contact between the test-piece and the prism, a very small drop ofmonobromonapthalene (n = 1.658) should be placed on the prism before the test-piece is placed upon the prism. The absolute minimum amount of the contact liquidshould be used to evenly cover the interface. (Note the contact liquid used mustalways have a larger refractive index than that of the solid being investigated so thatthe critical angle is determined by the value of n for the solid). If the scale error is

greater than ½ of a division consult a demonstrator.

As additional experiments in the use of the Refractometer: -

(1) Measure the refractive index of distilled water. This is frequently used as a testsubstance to check the calibration of the instrument.

(2) Examine the quartz test-piece and account for the appearance of twoborderlines in this case.


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Appendix IISchering Capacitance Bridge

Air capacitors have low values of capacitance and liquid filled capacitors may havesignificant losses, which appear as series resistances. These two factors make the

accurate measurement of capacitance a little difficult. Using the Schering capacitancebridge circuit of Fig 2 however, it is possible to achieve quite good sensitivity and toindependently determine both the capacitance and series loss resistance of the aircapacitors. The out-of-balance signals in the bridge are measured using a "Lock-inAmplifier" (or "Phase Sensitive Detector"). This measures the AC voltage only at theoscillator frequency and is also able to determine the phase of the out of balancevoltage.

C1 - test capacitor. This is a variable air capacitor, which should be used with theplates fully overlapping to give maximum capacitance. Being able to open the platesis useful for cleaning and drying the capacitor after immersion in toluene. The tolueneis kept in the fume cupboard. The capacitor plates must NOT be washed in water.R1 - in some applications of the bridge this is just the series loss resistance of thecapacitor under test. In this experiment use a variable resistance box to increase thisresistance. This enables convenient values of other components, especially R3, to bechosen. Note that R1 in the expressions below for the balance conditions is the seriesloss resistance plus the value of dial box resistance.

R1 (ResistorDecade Box)

C2 (FixedCapacitor1500pF)

C1 (TestCapacitor)

Input –

R4 (Resistor

Decade Box)

C4 (CapacitorDecade Box)

R3 (ResistorDecade Box)

LOCK – IN AMPLIFIER

Ref Input Signal Input

Output

Output

FUNCTION

GENERATOR

DIFERENTIAL

PRE- AMPLIFIER Input +

Output

Figure 2. Schering Bridge Schematic


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C2 - this must be a very low (ideally zero) loss capacitor. A mica capacitor of about1500pF is supplied. (Check the value on the Aim LCR Databridge ).

R3 and R4 are variable resistance decade boxes and C4 is a variable capacitordecade box.

Since we have both resistance and reactances in the circuit, the currents flowingthrough the components of the bridge will not generally be in phase with the voltagesupplied by the oscillator. For a pure resistance the current is in phase with thevoltage and for a pure capacitance it is 900 out of phase. It is useful to write thecurrents and voltages as vectors related by complex impedances (see Duffin"Electricity and magnetism" Ch.10 for example)

I Z =V .

Z = R for a resistor andC j

1 Z = for a capacitor, where is the angular frequency.

In a simple Wheatstone bridge (as Fig 2 but with C 2 replaced by R 2 and C 1 and C 4 removed) the condition for balance is R 1 / R 3 = R 2 / R 4 , as the two arms of the bridge actas voltage dividers.

With reactances present this becomes:

For the Schering bridge of Figure 2

Inserting these impedances into equation 1 and rearranging gives two conditions:

and

corresponding to balancing the in-phase and out-of-phase voltages.

From (3): C 1 R1= C 4 R4 (4)

and inserting this into (2) then gives:C 1 R3 = C 2 R4 (5)

Z

Z =

Z

Z

4

3

2

1 (1)

44

4

1

4

4

433

2

2

1

111

1;;

1;

1

RC j

RC j

R Z R Z

C j = Z

C j + R= Z

ω

ω

ω ω +=

+==

−

01

1

42

31

24

24

2

41412

= RC

RC

RC +

R RC C + −

ω

ω (2)

01 24

24

2

4411 =

) RC +(

) RC RC ( j

ω

ω − (3)


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(4) and (5) then give the final balance conditions:

and

The best way to balance the bridge is to choose the values of R 1 and R 3 of a few k and then adjust R 4 and C 4 for balance. Initially, almost all of the voltage across thePSD will be out of phase with the oscillator voltage since the reactances of C 1 and C 2 are very large. Therefore, start by adjusting R 4 to obtain a balance with the PSDmeasuring the signal at 90o to the reference (90o phase button in). Then balance the

in-phase signal (button out) usingC

4. Repeat the balances withR

4 andC

4 until theout-of-phase and in-phase signals are minimised. Note: the bridge is balanced whenthe meter registers zero.

Operation of the Lock-in amplifier (Phase Sensitive Detector (PSD))

A high sensitivity is possible using a phase sensitive detector because noise, atfrequencies other than that at which the bridge is operated, is not detected. Inparticular, harmonics of the driving frequency, generated by non-linear components inthe bridge, are not detected. Furthermore, a PSD can also measure the phase of thesignal relative to that of a reference signal. The EG&G model 5101 Lock-in amplifier

operates by comparing the input signal which is to be measured with a referencesignal. It "locks into" that part of the input signal which is both at the same frequencyas the reference and in phase with the reference. The front panel is sub-divided intothree sections input, reference and output.

Reference: The reference signal in this experiment is taken from the 600 the oscillator, which is in phase with that from the 50 output. The phase of the signalto be detected relative to this reference signal can be adjusted with the 10 turnpotentiometer and the two buttons below this. It is best to set the potentiometer tozero and use the button to select in phase ( = 0o) or out of phase ( = 90o). The F, 2Fbutton should be on F as the 2F selects a signal at twice the reference signal.

Input: The detected input signal at the frequency and phase of the reference isdisplayed on the panel meter. The sensitivity can be varied from 1mV to 250mV full-scale deflection (FSD). A differential amplifier with a gain of x100 first amplifies theoutput from the bridge. This has a very high input impedance and isolates the bridgecircuit from the Lock-in. This pre-amplification increases the sensitivity but canoverload the Lock-in if the voltage from the oscillator is too large (normally about 3-5volts is sufficient).

Output: The time constant knob gives the time over which the measured signal isaveraged. The response of the Lock-in to changing signals is slower for longer time

constants but noise is reduced. A time constant of about 0.1 seconds should be best.The offset and post filter are not used.

(6)

C

C R R

2

43

1 = (7)

3

24

1

R

C RC =

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Shering Bridge