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7/22/2019 Charges and Fields

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Determination of the Magnetic Field of Earth and the Elementary Charge

Damian Boh8thDecember 2012

Abstract

The horizontal and vertical components of the Earths magnetic field were measured. This was done using themaximum deflection ymax of a current carrying wire in the field. Experimental data suggest a positive linear

relationship between the horizontal ymaxand currentIwhile an additional quadratic relationship was observed for the

verticalymax andI. These effects were analysed and the total magnitude of the Earths magnetic field was calculated tobe 3436T.

Millikans oil droplet experiment wasalso reproduced to determine the value of the elementary electric charge. The

terminal velocities of charged oil droplets in known electric fields between a parallel plate capacitor were measured.These were analysed using fluid dynamic forces to give the radius aand charge qof the droplets. The deviation of the

data from Stokes law was also investigated and this was found to be little. The most accurate value of q wasdetermined to be (1.80.2)10

-19C and this was compared to the known value.

1. Introduction

A current carrying wire will experience a force and deflect in a magnetic field. The aim of the first experiment is to

determine the vertical magnetic fieldBzof the Earth and horizontal fieldBxusing the maximum horizontal and vertical

deflections of a fine, current carrying wire .respectively. Experimental data suggest a positive linear relationshipbetween the horizontalymaxand currentIwhile an additional quadratic relationship was observed for the vertical ymaxand I. The linear relationship was expected and will be discussed in Section 2. The quadratic relationship wasinterpreted to be due to the sagging of the wire as it expands under the power dissipated as heat in the wire. Furthercomplications arose when the tension caused by sagging of the wire built up and overcame the tension of a pulleyconnected to one of the wires ends. These effects were analysed and factored out before the Earths magnetic field

was calculated to be 3436T.

The measurement of the elementary charge on an electron was achieved by Robert Millikan in 1909 in his famous oildroplet experiment[1]. This was reproduced in the second experiment. The motions of charged oil droplets within anelectric field between a parallel plate capacitor were analysed. The terminal velocity vgat free fall under gravity and

the terminal velocity vEas it rises under both the influence of gravity and an oppositely directed electric field Ewere

measured. These were analysed using fluid dynamic forces to give the radius aand charge qof the droplets. Resultswere plotted on scatter diagrams of charge against radius and charge against vg

-1/2. Experimental data suggest a general

positive correlation between the magnitude of charges and radii of the droplets. Clustering of charges were alsoobserved around different radii suggesting that the charge is a discrete quantity. Stokes law was used in thisexperiment and the deviation of the observed q from its prediction was studied and found to be little. The mostaccurate value of qwas determined to be (1.80.2)10

-19C and this was compared to the known value.

2.1 Experimental Method (Earths Magnetic Field)In the first experiment a tungsten wire was made to pass over fixed pulleys mounted onto an aluminium optical rail atboth ends (Figure 4, page 62). It was held under tension by a small weight T0at one end of the wire below the leftpulley while the right end of the wire was fixed to the insulating right pulley stand. An ammeter and power source wasconnected to the wire as shown in Figure 5 of page 63 to produce a current flowing towards the right direction. Thepositions of the left and right ends of the wire were labeled as x=-L/2andL/2respectively, with that of its centre asx=0. A microscope with a graticule scale was mounted at the centre of the wire to observe for its deflection as the

deflection is very minute. The apparatus was arranged in an East-West direction to maximize the vertical deflectionsof the wire. The wire was very sensitive to movement in the lab and the wind in that it vibrated very vigorously whenseen under the microscope when there was movement and wind. Hence such movement and air flow were minimizedand the deflection reading was only recorded when the wire vibration was minimal.

According to theory it should deflect in an arc with the maximum deflection ymaxat its centre when a current passesthrough it[2]. Figures 2 and 3 on page 61 show how different forces are balanced at a position x along the wire.Clearly, for the wire to be in equilibrium, the tension T(x)should balance the magnetic force Fx in the negative x-direction and T0 in the negative y-direction. For small deflections the total magnetic force from x=0 to x is just Fx

=IBzx.Balancing the forces gives T(x)sin= IBzx [Equation 1]andT(x)cos=T0[Equation 2].Diving equation 3 byequation 4 gives

0

tanT

xIB

dx

dy z [Equation 3] and integrating gives I

T

xBy z

0

2

max2

[Equation 4] where ymax is the

maximum horizontal deflection at the middle of the wire. Equation 4 predicts a positive linear trend betweenymaxandI.

Hence in this experiment, measurements ofymaxat differentIwere taken and a graph ofymax againstIwas plotted. The


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vertical magnetic fieldBz was determined from its slope. The horizontal magnetic fieldBxcauses a vertical deflectionzmaxin the same way. Hence a graph ofzmax againstIwas plotted and analyzed as well.

2.2 Experimental Method (Elementary Electrical Charge)In the second experiment Milikans oil droplet experiment was reproduced with an experimental set-up shown inFigure 13 on page 88. A power source whose voltage can be varied was connected to a circular parallel plate capacitorsurrounded by a Perspex air shield. The electric field Ebetween the parallel plates can be adjusted by varying the

power source voltage. This field was directed downwards so that it created a force opposite to the weight of the oil

droplets. A nebulizer containing oil was connected to the set-up via holes in the air shield.

Oil droplets negatively charged by friction were introduced in between the parallel plates when the nebulizer bulb issqueezed. Their motions were observed using a microscope connected to the front of the Perspex air shield via a hole.

The terminal velocities vgand vEas mentioned in Section 1 was determined by measuring the time taken for the oil totravel a distance of a few deflections on the microscope graticule scale under different E. The voltage V0required to

exactly cancel gravity and bring the droplet to rest can also be recorded to serve as a cross check for the charge oneach droplet q.

From theory, the terminal velocity depends on the viscous drag of the air given by Stokes Law [3] to be F=6av[Equation 5] where is the viscosity of air and ais the radius of a spherical droplet. As vEis the terminal velocity

where viscous drag balances the combined influences of the electric force and weight of droplet, qE-mg=6avE

[Equation 6] where mis the mass of a droplet and gis the gravitational constant. Similarly vgis measured whenE=0,hence the weight of droplet is mg=6avg[Equation 7]. The mass mcan be determined from the droplets radius aanddensity by m=(4/3)a

3 [Equation 8]. Further manipulations to these equations will give the droplet radius as

g

va

g

2

9 [Equation 9] and the observed charge on each droplet as

E

vvv

gq

Egg

obs

)(

2

96

2/12/3

[Equation 10]. V0

can be used to cross check this value of qusing the equation mg=qE[Equation 11] when the electric force exactlybalances the weight to bring the droplet to rest. In this caseEis V0/dwhere dis the distance between the parallel platesof capacitor.

Stokes law is actually based on a continuous medium and it breaks down when the molecular mean free pathlof airis comparable with the droplet radius a. Hence it is important not to use the results of droplets that have too small a

values. This will be further discussed in Section 3. On the other hand, droplets that have high awill fall quicklyresulting in vgbeing more difficult to measure accurately. Thus it is important reach a compromise between both ofthe effects where the avalues of droplets measured are not too high or low.

3.1 Results and Discussion (Earths Magnetic Field)

A graph of the measured ymaxagainst Iwas plotted by Orgin in Graph 2 of page 71. As predicted by Equation 4,experimental data suggest a positive linear trend between the 2 quantities. Equation 4 suggests that the slope of thisgraph should be Bzx

2/2T0where x=L/2 as in Section 2. x=L/2was measured to be 0.970.02 mwhile T0 (weight of

mass)was measured to be 0.1330.001N. The value of the vertical component of Earths magnetic field Bz was hence

calculated as Tx

slopeTBz 132

22

0

. The errors in this result include errors from the measurement ofL/2with

a metre rule and also the measurement of T0with a mass balance. The main source of error comes from the vibration

of the wire as described in Section 2.

It is interesting to note that the slope of the graph is not entirely uniform. This is due to the power P=I2R being

dissipated as heat in the wire, causing the wire to expand and sag. Though the sagging is in the vertical direction andshould not be observed inymax, this affects the T0which affects the slope of the graph. As currentIwas increased in thewire, the heat dissipated increased and the wire expanded more. This led to an increase in the tension in addition to theoriginal T0 in the wire. From Equation 4, an increase in T0due to an increase inIwill decrease the slope of the graph.

All these are due to the fact that the left pulley is not ideal and frictionless. If it were ideal, any increase in length ofthe wire caused by the expansion would slip off to the left of the pulley and the tension would always remain at T0.Further complications arose when the increased in tension in the wire was large enough to overcome the static frictionat the pulley whenIwas large enough. In this case the pulley slipped (Figure 10, page 73) and the tension became T0again and built up until the additional tension was high enough to overcome the friction yet again. The change in slopeof the graph would thus be inconsistent at these values ofI. Hence the slope in the graph is not uniform.

The maximum vertical deflection zmax was also plotted against I in Graph 4 on page 80 using Origin. The thermalsagging of the wire, being in the vertical direction, affected the results to a huge extent. As mentioned earlier, the heat


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causing the expansion depends on the powerP=I2Rand hence a quadratic effect (dependence on I

2)is expected. This,

in addition to the magnetic deflection of the wire, will lead to a linear and quadraticI dependence in the formzmax=b1I+ b2I

2 [Equation 12] (refer to Figure 12, page 77). The constant b1 should correspond toBxx2/2T0like how the slope of

Graph 2 corresponds toBzx2/2T0.

The graph was observed to be divided into 3 regions of quadratic shapes. The quadratic shapes suggest that thesagging effect outweighs the magnetic effect. Furthermore the fact the graph is divided into 3 regions arises from

exactly the same pulley slipping effect as described earlier. As the pulley slips at high enoughI(for both positive and

negative values), the sagging and hencezmaxwill suddenly decrease and there will be discontinuities for zmaxvalues inthe graph at the values ofIwhere the slipping occurs. Equation 12 was thus fitted onto the middle region of Graph 4using Origin before the slipping occurs (before any discontinuities) and b1was used to determineBx.Bx was calculated

to be Tx

bTBx 713

22

10

. The errors involved were similar to those involved in the determination ofBz, i.e. the

measurements of L/2 and T0. The vibrations of the wire in the vertical plane were much higher than that in thehorizontal plane, leading to a much larger fractional error in theBzvalue.

The overall magnetic field of the earth was calculated to be B = Bx2+ Bz

2= 3436Twith the large error mainly from

Bx. The direction of the field with respect to the horizontal plane was calculated to be = tan-1(Bz/Bx) = 1.20.3 rad

= 7020.The reference value of the magnetic field in London is 48T[4] and this is within 1 from our results.However we would expect the magnetic field in Blackett laboratory to differ from the Earths magnetic field due to

metal structures within the building. The accuracy and precision of the experiment could also be further improvedwith proper shielding of the whole tungsten wire from any air movement in the lab. The convectional currents causedby heating of the air due to the light source being placed closed to the wire could have affected the results of thisexperiment as well.

3.2 Results and Discussion (Elementary Electrical Charge)

The scatter diagram of charge qobs (calculated from Equation 10) of each droplet against the droplet radius a wasplotted in Graph 5 on page 91 by Origin. Clearly, the data suggest a positive correlation between qobs and a. This is notsurprising as bigger droplets are more likely to be more highly charged by friction when squeezed through thenebuliser since they have bigger surface areas. Clustering of qobs around certain qobsvalues and radii can also be seen.This shows that electrical charges have discrete values which corresponds to theory.

It is important to take note of the limitations of Stokes law in this experiment. Equation 1 works on the assumptionthat the air causing viscous drag on the oil droplet is a continuous medium. However, when the radius of the droplet ais comparable to the molecular mean free path lof air, Stokes law breaks down. Hence the value of qobswill be thetrue value of the charge only when l/atends to 0. As shown in Graph 5, the value of a for the droplets used ranges

from 0.4mto more than 1m.The reference value of lis 0.06635mwhich is significantly smaller than the a values.Hence it is reasonable to assume a small value of l/a in this experiment.

This allows Equation 10 to be expressed as a power series in l/a in which the first order approximation to thecorrection of Stokes law would be enough. This first approximation would hence be linear in the form of

qobs= A(l/a) + B [Equation 13]whereAandBare constants andA(l/a) is the additional term caused by the deviationsfrom Stokes law. The value of Bwould therefore be the true value of qas l/atends to 0and Stokes law becomes

perfectly valid. We can conclude from Equation 9 that l/ais proportional to vg-1/2

. Hence a graph of qobsagainst vg-1/2

could be plotted with its y-intercept corresponding toBin Equation 13.

The graph of qobsagainst vg-1/2

is plotted in Graph 6 of page 9. The clustering of qobsvalues around the lowest valuecorresponds to oil droplets being charged with 1e, where eis the elementary electric charge. Similarly the clusteringthat occurs at the second lowest value corresponds to those charged with 2e. Droplets that have charges higher than 2ewere not used in this experiment for data analysis. A straight line was fitted onto the 1e cluster by Origin and the truevalue of qwas obtained from its intercept to be qa= (1.90.1)10

-19C. This value is higher than the known value of

e = 1.60210-19

C and furthermore the known value does not even lie within 1 of qa. As the random error wouldalready have been minimised by using a lot of data points in the 1ecluster, this suggest some form of systematic errorwhich may be present causing all the measured values of qto be higher than the actual values.

Hence a second method might be more useful. This method concerns the determination of the value of 2e from theintercept of a line fitted to the 2ecluster data points. This value would then be subtracted from qato obtain a value for1e. Such a method of differences is useful in eliminating and checking for any systematic errors and might prove to bemore accurate. The value of the intercept for the line fitted to 2e cluster is shown in Graph 6 to beB = (3.70.2)10

-19C. The value of the elementary charge found from this method is thus obtained to be qb = B qa=


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(1.80.2) 10-19C. This value has a higher error as that of qa as the error from both intercepts are added up in

quadrature. Also, the known value of elies within 1of qband this method is thus concluded to be more accurate.

The fact that this method is more accurate suggests that a systematic error is indeed present. This can be due to any ofthe factors in Equation 10. For example the value of viscosity of air might have been overestimated resulting in toohigh values of qobscalculated for every droplet. Similarly, the density of oil or the electric fieldE might have beenunderestimated. Other sources of systematic errors include that due to an ambientEfield being present in the lab as a

result of the electric power cables in the laboratory.

An even more accurate correction to Stokes law was discovered by Millikan[5] who suggested that Equation 5 should

be replaced byabl

vaF

/1

6

[Equation 14] where bis a constant. This results in the correction qobs

2/3= q

2/3(1+bl/a)

[Equation 15] where qis the true value of the charge. Again noting that l/ais proportional to vg-1/2, a graph of qobs

2/3was plotted against vg

-1/2in Graph 7 of page 96 and its intercept could be used to determine q

2/3and henceq.

Similar methods as those described earlier were used to determine 2 further values for the elementary charge, qcand qd.The intercept from the 1ecluster in this graph was used to determine qc to beqc= (1.90.1)10

-19C.Similarly qdwas

found by the difference from the calculated 2e and 1e to beqd = (1.80.2)10-19C. qcand qd has exactly the same

values and errors as qaand qbdetermined by their corresponding methods respectively. This suggests that the moreaccurate correction to Stokes law is not necessary in the context of this experiment as the first approximation is good

enough. This further confirms the assumption that l/a is small is justified for the less accurate method to work.Furthermore the slopes of the lines in graphs 6 and 7 correspond to the magnitude of the correction terms to Stokeslaw. The fact that these slopes are of the orders of magnitudes 10

-23and 10

-16further suggests that the correction to

Stokes law is very little.

The most accurate measurement of ewas taken to be that of qd, hence e = qd= (1.80.2)10-19C. The theoretical value

of eis within 1of this measured value. qdwas taken to be the most accurate measurement as it was determined by themethod of differences which eliminates systematic errors involved and it was also calculated using the more accuratemethod that Millikan came up with. The errors in this measurement were mainly from the human reaction time inmeasuring the time taken for the oil droplet to travel a particular distance and also the determination of the position ofthe droplet with respect to the microscope graticule scale. The scale has divisions very close together and it wasdifficult determining the positions of the droplets. Moreover the fact that qd is higher than the theoretical valuesuggests that systematic errors were present. These could be due to ambient electric field in the lab as mentionedearlier and also the Brownian motion of air particles and the wind affecting the motion of the air droplets.

Millikans approach to this experiment was somewhat similar in his paper[5]. He, too, selected droplets that were not

too small or big so that Stokes law did not deviate too much and the droplets did not fall too quickly for accuratemeasurements. His result for the elementary charge was e=(4.7740.009)10

-10electrostatic unitswhich corresponds

to (1.5920.004) 10-19

C. This value was far closer to the known value of 1.602 10-19

C. However, Millikansexperiment was conducted over 60 days with 58 droplets. Each droplet was numbered and carefully analysed.Furthermore he considered other factors like pressure of the air around the droplets. He also took great care to keep the

air dust free and to achieve complete stagnancy of the air by absorbing heat rays using a water cell and immersing hiswhole experimental vessel into a constant temperature bath[5]. These precautions could have been adopted in the

experiment to achieve a far greater accuracy comparable to that of his original experiment.

4. Conclusion

The Earths magnetic field was measured using the maximum deflections of a current carrying wire. Also, theelementary charge was measured by subjecting the motion of oil droplets within an electric field to analysis usingfluid dynamical forces. The Earths magnetic field was measured to be 3436T while the elementary charge wasmeasured to be(1.80.2)10

-19C. The known values of the field 48Tand the elementary charge 1.60210-19Cfall

within 1 of both the measured values. In each experiment, complications from other factors such as the pulleyslipping effect and the deviation from Stokes law arose. These had to be analysed carefully before they can befactored out to give an accurate result.

Though not in the main aim of these experiments, they have provided interesting insights into factors like the thermal

sagging effect of a current carrying wire and the accuracy of Stokes law. These suggest that a broad knowledge ofphysics outside the field of work of an experimentalist is vital in handling experiments like these. Finally, the accuracyof the experiments could have been improved by using better apparatus such as those which gives proper shielding ofthe wire from any air movement and those used in the original Millikan experiment.


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5. References

1. Robert MillikanThe Oil-Drop Experiment,http://www.juliantrubin.com/bigten/millikanoildrop.html,lastupdated July 2012

2. Imperial College London Second Year Laboratory, Charges and Fields Experiment Manual,September 2012

3. Stokess Law,http://www.cord.edu/faculty/ulnessd/legacy/fall1998/sonja/stokes.htm,accessed December

2012

4. Magnetometers,http://www.barnardmicrosystems.com/L4E_magnetometer.htm,accessed December 2012

5.

R.A. Millikan, On the Elementary Electrical Charge and the Avogadro Constant, Phys. Rev. 2, 109143 (1913)6. S.G. Jennings, The Mean Free Path in Air, Journal of Aerosol Science, Vol, 19, No. 2, 159-166, 1988
http://www.juliantrubin.com/bigten/millikanoildrop.htmlhttp://www.juliantrubin.com/bigten/millikanoildrop.htmlhttp://www.juliantrubin.com/bigten/millikanoildrop.htmlhttp://www.cord.edu/faculty/ulnessd/legacy/fall1998/sonja/stokes.htmhttp://www.cord.edu/faculty/ulnessd/legacy/fall1998/sonja/stokes.htmhttp://www.cord.edu/faculty/ulnessd/legacy/fall1998/sonja/stokes.htmhttp://www.barnardmicrosystems.com/L4E_magnetometer.htmhttp://www.barnardmicrosystems.com/L4E_magnetometer.htmhttp://www.barnardmicrosystems.com/L4E_magnetometer.htmhttp://www.barnardmicrosystems.com/L4E_magnetometer.htmhttp://www.cord.edu/faculty/ulnessd/legacy/fall1998/sonja/stokes.htmhttp://www.juliantrubin.com/bigten/millikanoildrop.html

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