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Electron Charge & Mass Lab

April 1, 2014

Instructor: Prof. Robert Kramer

Dane Mettam

Lab Partners:

1. Abstract:

2. Introduction/Background:

3. Equipment Desc & Procedure:

4. Data:

5. Analysis:

6. Conclusion

Overall Lab Grade:

/ 100

/ 100

Robert Maldonado, Xuan Chen

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1. Abstract The electron charge to mass ratio is observed and found in this experiment. The observed value is then compared to the accepted values for the electrons mass and charge to determine accuracy and show that the mass and charge is fixed. A beam of electrons is subjected to a potential difference inside a set of Helmholtz coils that creates a uniform magnetic field. The magnetic field bends the beam into a circular path which is then measured visually. The changing potential difference and radius of the beams circular path is then used to calculate the electrons charge to mass ratio, which was observed to be 1.34x1011±2.13x1011J/Kg. The accepted value for the electrons charge to mass ratio is 1.76x1011J/Kg which gives this experiment a 23.5% error. This experiment, while not very accurate, was successful since it did strongly correlate with accepted values and helped prove that electrons have a fixed mass and charge.

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2. Introduction/Background The mass and charge of the electron is to be investigated in this lab exercise. It is

hypothesized that the electrons accepted charge value divided by the accepted mass ratio will

be observed. It is predicted that this ratio will strongly correlation with the accepted value. An

accelerating voltage is applied to a beam of electrons which is affected by a magnetic field into a

circular path. The magnetic field is held constant and the accelerating voltage (V) is varied which

in turn increases and decreases the radius of the electron beam. The linear relationship of the

voltage and beam radius is used to calculate the charge to mass ration (e/m). This ratio will be

observed when the voltage radius relationship when applied to a constant is approximately the

same as the accepted e/m value of 1.76x1011J/Kg. This experiment will be considered a success

if a strong e/m correlation to the accepted e/m value is observed. Due to the method of

measuring the radius of the beam; a high error is expected. Despite the high expected error;

strong correlation is expected.

This replication of J.J. Thomson’s 1897

experiment uses a beam of electrons (also known as

cathode rays) that are charged with an accelerating

voltage within a magnetic field. The applied potential

is known at all times. The charged electrons are

within a glass container filled with a fluorescent gas

molecules that emits light when excited (Helium at

10−2mm Hg). The glass container is centered in a pair of Helmholtz coils that bends the beam

into a circular path. The electrons (emitted with a heated cathode) and gas collide inelastically

which excites the gas with the charge of electrons. The gas eventually reverts back to its ground

state and emits light of a certain wavelength and frequency. The above image shows an electron

gun diagram that emits the electrons.

As the voltage is increased, the radius of the beams circular path also increases. The

radius and voltage should be directly proportional.

Image 1: Electron Gun

Image 2: e/m Apparatus Image 3: e/m Tube

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The Helmholtz coils provides a constant

magnetic field. The magnitude of the field can be

calculated using the Biot-Savart Law, given below. The

field has its greatest strength and uniformity at the

center between the coils. The coils, named after

physicist Hermann von Helmholtz is measured in Tesla’s

and is determined by three variables. The three variables

are its number of turns in each coil (N), its current (I),

and the radius of the coils (r). There are two constants

within it as well: μ˳ (4πx10−7), and (4/5)3/2.

𝐵 = (4

5)

32 𝑁𝜇°𝐼

𝑎

The force of the magnetic field (𝐹𝑚) is the only force acting on the electrons and this

force is a cross product of the electrons charge (e) and velocity (S) with the B-field (B). The letter

(S) is chosen for velocity instead of (v) so that it is not confused with voltage later on. The B-field

is providing its force at a 90 degree angle at all times relative to the beam so the values are

simply multiplied normally. This is seen in equations 2 and 3 below. The force can also be

modeled as a centripetal force (𝐹𝑐) since the electrons are moving in a circle. Equation 4 shows

this. The variable (r) is the radius of the beam.

𝐹𝑚 = 𝑒𝑆𝑋𝐵 𝐹𝑚 = 𝑒𝑆𝐵 𝐹𝑐 =𝑚𝑆2

𝑟

When equations 3 and 4 are set equal to each other and solve for e/m, it is clearly seen

in equation 5 that the ratio of e/m is dependent on only the velocity and radius of the beam.

The velocity can be solved by rearranging the nonrelativistic equation for kinetic energy from

equation 6 to equation 7 below.

𝑒

𝑚= 𝑆/𝐵𝑟 eV=𝐾. 𝐸. =

1

2𝑚𝑆2 𝑆 = √(

2𝑒𝑉

𝑚)

Equation 7 can then be inserting into equation 5 and then expanded to the complete formula for e/m. A little algebra is needed to get to this point by first squaring everything then simplifying.

𝑒

𝑚=

𝑆

𝐵𝑟=

2𝑉(54)

3𝑎2

(𝑁𝜇˳𝐼𝑟)2

The radius of the beam is read by lining up the beam with its reflection and noting down where each side of the beam is. The mean of each diameter measurement is then taken to provide the radius. This part of the experiment should be done extremely carefully as it is very easy to misread measurments. This is known as a parallax error. The precision of this measurment is limited by how well the experimentor can focus on, the edges while accuratly reading the measurments at a distance. This will not be precise.

Image 4: Helmholtz Coils Field Lines

Equation 1

Equation 2 Equation 3 Equation 4

Equation 5 Equation 6 Equation 7

Equation 8

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3. Equipment Description and Procedure

1. Install hood around Helmholtz coils. Insert e/m tube into the controls. 2. Ensure toggle switch is in up position to “e/m MEASURE position” and that all

adjustment knobs are in the off position. 3. Connect all power supplies and equipment into the apparatus controls as shown in

image 9 below. 4. Set voltage of heater up to (but not exceeding) 6.3V. The Helmholtz coils should be set

to 6-9 VDC and the electrodes to between 150 and 300 VDC. Ensure that current on the ammeter does not go past 2 Amps to prevent damage to equipment. Adjust the ammeter clockwise.

5. Wait for cathode to heat up. The appearance of the blue electron beam will indicate that it is properly heated up.

6. Adjust tube so that the beam is parallel to the Helmholtz coils by rotating it. The beam should spiral out a little to prevent emerging beams from being affected. Adjust focus knob if the beam is not spiraling out a bit. Ensure that the angle is not too large so that it minimizes its effects on the calculations and data.

7. Record the current applied to the coils along with its voltage. 8. At set accelerating voltages applied to the beam, measure and record the radius of the

beam. To record the diameter, align the beams reflection with the beam itself, and look at the far right and left measurements of each side of the beam. Take the average as the radius.

9. Repeat for accelerating voltages between 150 and 250 volts. 10. Calculate e/m from data. Find the mean e/m value along with standard deviation. This

should be compared to the accepted value. Conduct further error analysis by estimating the error in beam radius, accelerating voltage, and current in coils.

11. Compare statistical errors with estimated errors.

Image 8: PASCO Model SE-

9638 e/m Apparatus Controls Image 7:

Helmholtz Coils Image 6: Outside-e/m Hood,

inside-e/m Tube and Mirror

Image5: Top: PASCO Model SF-9585

High Voltage Power Supply, Bottom: PASCO Model SF-9584 Low

Voltage Power Supply

Image 9: e/m wiring & setup

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4. Data

Throughout the experiment, most of the variables were kept constant or could not be altered. All of the values within the second chart below are only showing the changing values and e/m once applied to all of the unchanging variables in chart 1 and equation 8 on page 3.

Constant Value Symbol

Coil Radius 0.15m a

Coil Current 1.95A I

Coil Constant (5/4)3/2 (5/4)3/2

Permeability 4𝜋−7N/𝐴2 μ˳

The calculations done from the data below is in meters but the chart is showing the data in centimeters. The left and right side of each beam was done by first lining one side up to an easily read value then by observing as closly as possible the other value while attempting to keep the other side visually on the known measurement. Even though the measurements were done only up to two significant figures, results are put into three significant figures so that the radius measurements would not get rounded.

Voltage Left Right Diameter Radius e/m

150V 3.0cm 3.9cm 6.9cm 3.45cm 1.32x1011 𝐽

𝑚

160V 3.0cm 4.0cm 7.0cm 3.50cm 1.37x1011 𝐽

𝑚

170V 3.0cm 4.3cm 7.3cm 3.65cm 1.34x1011 𝐽

𝑚

180V 3.0cm 4.9cm 7.9cm 3.95cm 1.21x1011 𝐽

𝑚

190V 3.0cm 4.5cm 7.5cm 3.75cm 1.42x1011 𝐽

𝑚

200V 3.0cm 4.8cm 7.8cm 3.90cm 1.38x1011 𝐽

𝑚

210V 3.5cm 4.8cm 8.3cm 4.15cm 1.28x1011 𝐽

𝑚

220V 4.0cm 4.2cm 8.2cm 4.10cm 1.38x1011 𝐽

𝑚

230V 4.0cm 4.4cm 8.4cm 4.20cm 1.37x1011 𝐽

𝑚

240V 4.0cm 4.8cm 8.8cm 4.40cm 1.30x1011 𝐽

𝑚

250V 4.0cm 4.7cm 8.7cm 4.35cm 1.39x1011 𝐽

𝑚

mean e/m value 1.48x1011 𝐽

𝑚

The error was calculated in two ways. The first way simply took the statistical standard deviation of the above e/m values. This gave the e/m an error of ±0.05921011 𝐽

𝑚. To get to this

error, the following formula was done.

𝜎𝑒/𝑚 = √∑((𝑒/𝑚)𝑖 − 𝑒/𝑚̅̅ ̅̅ ̅̅ )2

𝑛 − 1

This gave the reported value of 1.48x1011±0.0592𝑥1011 𝐽

𝑚.

The second error calculation done shows the estimated errors with coil current, beam radius, and voltage. The partial derivatives with respect to each of the three error sources is done then added together. The mean error is then found. The beam radius error is more of a

Equation 9

Chart 1: Constants

Chart 2: Variables & e/m calculations

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guess, since there was a good amount of parallax error which cannot be accurately measured. In general, the radius error was guessed to be between 0.1cm and 0.4cm. The equation used to find this error is given below.

𝛿𝑒/𝑚 = √(𝛿𝑒/𝑚

𝛿𝐼)

2

𝜎𝐼2 + (

𝛿𝑒/𝑚

𝛿𝑟)

2

𝜎𝑟2 + (

𝛿𝑒/𝑚

𝛿𝑉)

2

𝜎𝑉2

By calculating the error of each iteration in this fashion and then taking the mean an error of plus or minus 2.13x1011 was observed. This would give a reported value of 1.48x1011±2.13x1011 𝐽

𝑚. Obviously, this indicates a very imprecise observation of e/m. The

accepted value of e/m is 1.76 x1011 𝐽

𝑚. The calculations showing this error is provided below in

chart 3. The percent error is calculated by taking the absolute value of the accepted e/m value and subtracting it from the observed value. This is then divided from the accepted value then multiplied by 100. The percent error is 23.5%. With the first method of calculating error, no iteration is within one sigma of the actual value. With the second method, every point is well within one sigma of the actual value.

%𝐸𝑟𝑟𝑜𝑟 =|𝑚𝑒𝑎𝑛 𝑣𝑎𝑙𝑢𝑒−𝑎𝑐𝑐𝑒𝑝𝑡𝑒𝑑 𝑣𝑎𝑙𝑢𝑒|

𝑎𝑐𝑐𝑒𝑝𝑡𝑒𝑑 𝑣𝑎𝑙𝑢𝑒∗ 100

The error of the coil current and accelerating voltage was the same for each iteration. Current error is ±0.01A and voltage error is ±1V.

𝜎𝑟 𝛿𝑟 𝛿𝐼 𝛿𝑉 𝛿𝑒/𝑚

0.01 5.90x1021 7.38𝑥1018 7.80𝑥1017 7.68𝑥1010

0.03 5.54x1022 7.93𝑥1018 7.36𝑥1017 2.35𝑥1011

0.02 2.16x1022 7.57𝑥1018 6.22𝑥1017 1.42𝑥1011

0.03 3.39x1022 6.19𝑥1018 4.54𝑥1017 1.84𝑥1011

0.04 9.18x1022 8.49𝑥1018 5.59𝑥1017 3.03𝑥1011

0.04 8.04x1022 8.04𝑥1018 4.78𝑥1017 2.84𝑥1011

0.03 3.43x1022 6.91𝑥1018 3.72𝑥1017 1.84𝑥1011

0.03 4.05x1022 7.96𝑥1018 3.91𝑥1017 2.01𝑥1011

0.03 3.83x1022 7.90𝑥1018 3.55𝑥1017 1.96𝑥1011

0.03 3.16x1022 7.14𝑥1018 2.95𝑥1017 1.78𝑥1011

0.03 6.52x1022 8.11𝑥1018 3.09𝑥1017 2.55𝑥1011

Equation 10

Equation 11

Chart 3: Error Calculations

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Below is the graph showing the measured radius of the beam compared to the applied accelerating voltage. From this, the radius is clearly not accurately measured. The e/m relationship cannot be determined from the graph.

To the right is an image of the electron beam, its reflection and the ruler used to measure it. The right side of the beam should be noticed is much sharper than the left and by the time it goes full circle, it has noticeably faded away a little bit. The radius of the beam can appear to change depending upon angle of viewing, distance away from the beam, and movement of the head and eyes.

y = 97.457x - 184.51

150

160

170

180

190

200

210

220

230

240

250

3.5 3.7 3.9 4.1 4.3 4.5

Acc

ele

rati

ng

Vo

ltag

e

Beam Radius in cm

Voltage to Radius

Graph 1

Image 10: The Electron Beam

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5. Analysis

There were a few significant sources for error within this experiment such as parallax, velocity calculation, and equipment limitations. By far the greatest source of error was in measuring the radius of the electron beam. The beam was lined up with its reflection and the diameter was taken with a ruler printed on the mirror by looking at it at a distance. There was a significant parallax error that degraded the results. Despite this error, the e/m measurements still correlated closely with the accepted value. The next greatest source of error stemmed from measuring the velocity of the electrons. The electrons would collide with Helium atoms within the beam and as they are colliding, they will lose kinetic energy. This loss of kinetic energy results in a drop in velocity of the electron. As the velocity drops, the effect of the b-field on the beam also decreases. This was not directly observed but observed but was a factor. The next factor of error was the limitations of the equipment. As per the manufacturer specifications, to get the best results of the e/m experiment, the electron gun should have an accelerating voltage of over 250 volts but this was the highest value used. Additionally, there was a slight angle created by the b-field on the beam which caused it to spiral out a little bit. This angle was not measured and therefore cannot be calculated to observe its effects on the e/m ratio. Despite the errors, the experiment was a success. The reported e/m value for this is 1.48x1011±2.13x1011 𝐽

𝑚 with a percent error of 23.5%. The accepted value of e/m is 1.76x1011 𝐽

𝑚.

If the charge is assumed correct on the observed value and divided form it, a mass of 1.08x10−30Kg is seen. If the mass is assumed correct on the observed value and divided from it then the charge is 1.35x10−19C. The actual mass of the electron is 9.11x10−31Kg and the charge is 1.6010−19C. Clearly the results are very close to actual values.

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6. Conclusion The purpose of this experiment was to replicate J.J. Thompson experiments and measure the ratio of electron charge to electron mass in SI units. The observed e/m value was 1.48x1011±2.13x1011 𝐽

𝑚 and when compared to the accepted value of 1.76x1011 𝐽

𝑚 gave a

percent error of 23.5%. A strong correlation of the observed results was seen with the actual value and despite significant errors, verified the e/m constant. To accomplish this, a beam of electrons, accelerated by a high voltage, within a uniform b-field bent the beam into a circular path. As the voltage varied, the radius of the beam also varied. The linear relationship of voltage to radius was used to calculate the electrons charge to mass ratio which proved to be constant.

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8. References e/m apparatus. In (2008). pasco012-03471d. S.F. State: PASCO Scientific. Retrieved from

http://www.coursehero.com/file/1512907/pasco012-03471d/

Lab handout: PHYS 34300 Modern Physics; Robert Kramer, spring 2014 34300 lab.

Kramer, R. (2014). Modern physics notes. Informally published manuscript, Department of

Chemistry and Physics, Purdue University Calumet, Hammond, IN, Retrieved from

https://mycourses.purduecal.edu/webapps/portal/frameset.jsp?url=/webapps/blackboard/exec

ute/launcher?type=Course&id=_97498_1

Lab handout: PHYS 25100 Heat, Electricity, and Optics Laboratory Helmholtz Coils; Jackura,

Rengstorf, fall 2013 25100 lab.

Mettam, Dane. 2014. Photograph. n.p. Web. 1 Apr 2014.