Thomson Tube: Charge-to-Mass (e/m) Ratio of an Electron

Matthew Boyd, Matthew Slahetka

Department of Physics, Towson University

Due Date: October 19, 2015 at 9:30am

Abstract

In this experiment, we will mirror the Nobel Prize winning experiment done by J.J. Thomson to

measure the charge to mass ratio of an electron. Thomson first discovered the electron when the

charge to mass ratio remained the same throughout each cathode ray while it usually varied from

anode to anode. This meant there was a basic building block to the atom that he personally

discovered. We try to find the same results by finding the standard e/m ratio of 1.769E11 using

two different methods. Method one uses the process of null deflection where the electric and

magnetic fields are crossed so that the forces cancel on the electron. The plate voltage is varied

and the cathode ray doesn’t bend. This method resulted an average e/m ratio of 1.789E11, which

is a 1.702% error. Method two uses the deflection of the magnetic field alone to curve the ray

into the scaled fluorescent screen. The anode voltage is kept constant while the current through

the Helmholtz coil bends the ray, and where it intersects the edge of the screen is measured.

When the current is flipped, the ray bends the other direction and different trajectory is measured

then averaged. The resulting values present an average e/m ratio of 1.740E11, a percent error of

only 1.097%. Each method proved to be accurate and resulted in outstanding data.

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I. Intro

In 1897 the British physicist Joseph John Thomson was believed to discover the electron

in a series of experiments designed to study the nature of electric discharge in a high-vacuum

cathode-ray tube. While many other scientists were investigating this same subject, he was the

first to figure it all out. He interpreted the deflection of the rays by electrically charged plates and

magnets as evidence of “bodies smaller than atoms” that he eventually calculated as having a

massive charge-to-mass ratio. He noticed that for each cathode, he used the ratio remained the

same, were as in his anode rays experiments, the charge-to-mass ratio would vary from anode to

anode. He concluded that the rays were composed of very light, negatively charged particles

which were a universal building block of atoms. Thompson was awarded the 1906 Nobel Prize in

Physics for the discovery of the electron and for his work on the conduction of electricity of

gases. Later in 1897 after he announced his discovery, he found that he could deflect the rays by

an electric field if he placed the discharge tube in a very low-pressure space. By comparing the

beam deflection of the cathode rays in magnetic and electric fields, he obtained a more accurate

determination of the ratio. This design is what we mimic for our experiment to determine our

own electron charge-to-mass ratio.

II. Theory

For this experiment, we determine the charge-to-mass (e/m) ratio of an electron by using

two different methods. The first is the null deflection method and the second is the method using

the deflection of the magnetic field alone. When the electron travels from the cathode, it first

goes through two collimating anodes that focus the beam into a ribbon so it skims the edges of

the titled fluorescent screen so the trajectory can be easily observed.

Figure 1. Thomson’s tube with dotted lines showing the different paths the electrons can take.

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The beam then travels through two different fields that we adjust and control to manipulate the

beam how we want. The first of the two fields is an electric field that is present when the two

upper and lower metal plates, that have a separation distance of 8mm, are given a charge shown

by the two solid blue bars in Figure 1. The lower plate is grounded while the upper plate is

positively charged. This energy field in the y-direction causes the cathode ray particles to bend

upwards. The second type of field is a perpendicular magnetic field (B) created by a pair of

Helmholtz coils. A uniform magnetic field that is oriented perpendicular to the beam results in a

circular beam trajectory. When these two field clash, a Lorentz Force is created. The magnetic

field (B) has a force in the (–z)-direction that is perpendicular to the electron velocity in the x-

direction and the electric field (e) in the y-direction. So we have,

F⃑Lorentz=(−e ) v⃗ x B⃗ (1)

Since v and B are perpendicular to each other,

F⃑Lorentz=−evB (2)

With the velocity along the +x-direction and the magnetic field along the (-z)-direction, the

Lorentz force will be along the (-y)-direction. This force is opposite to the force –eE due to the

electric field E. Combining this with equation 2 gives,

e vxB=eE (3)

vx=EB

=V p

dB (4)

We obtain our (e/m) equation from the angle (θ) caused by the electric field. For a small angle θ,

tan(θ) ~ θ. So we have,

tan (θ )=v yvx

=θ (5)

From this, Vx can be solved using the kinematic equation and substituting in the known a y.

v y=0+ay t (6)

v y=eEmt (7)

t is equal the length of tube between the plates divided by Vx. Hence,

v y=eElm vx

(8)

θ= eElmv x

2=( em

) Vld vx

2 (9)

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Substitute equation 4 into equation 9 now gives us,

θ=( em

) B2ldV

(10)

( em )= V θB2ld

(11)

This finally gives us what we need for the experiment because we measure everything on the

right hand side of equation 11.

Figure 2. This shows how to measure the ray on the screen with values that are in our equations.

It is also worth noting Figure 2 that shows how to measure and give variables that are

derived in our equations. For the first experiment (A.13) when the fields nullify each other, the

angle (θ ¿ is now nullified or irrelevant. Substituting equation 4 into equation 11 gives,

e /m=2V p

d B2(ld ) (12)

Noting l∗d= y / x2 (13) and,

y=d /2 (14)

We finally have,

e /m=V p

B2 x2 (15)

This is how we obtain the e/m value for experiment A.13.

For the second part of this experiment, the method using the deflection of the magnetic

field alone requires a more careful measurement and is dependent on different variables. Hence

we derive from the Lorentz force (equation 2),

F=eVB (16)

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In uniform magnetic field (B) perpendicular to the electron trajectory, the electron beam moves

in a circle whose radius is determined from Newton’s second law, mass times centripetal

acceleration.

mV 2

R=eVB (17)

e /m= VBR (18)

Where R is derived from Figure 2 and ∆ is equal to the length of the side of the diamond figure,

R= ∆2+L2

√2(∆−L) (19)

The accelerating potential (Va) of the electron gun determines the speed (V) of the electrons

leaving the gun. Conservation of energy gives

eV a=12mV 2 (20)

Solving for V and substituting this into equation 18 gives us the necessary equation,

e /m=2V a

(RB)2 (21)

We solve for Va, R, and B in experiment A.14 and this determines the e/m ratio a second time.

Now we have two methods and equations to determine our e/m ratio.

To correct for some inequalities in the equipment during the second method, we use the

same Va but flip the current so that we measure the beam reflecting above the middle point and

below. For this, we have a simple equation,

L=L+¿+

L−¿

2 ¿¿ (22)

This gives us an average L that will make our determined e/m that much more precise.

III. Experimental Technique

The central focus of the entire setup is connecting the Tel 2501 Universal Stand correctly.

The Thomson Tube is connected into this stand where in the back there are 5 labeled circuit

connections. Also on either side of the tube there are two Helmholtz coils that have an input and

output plug for each. A Daedalon Corporation 5kV Power Supply powers the anode voltage (Va)

while also powering the filament voltage (Vf). A 350V HiVolt Bias Tel 2811 powers the plate

voltage (Vp), therefore controlling the electric field. A GP-4303D DC Power Supply, which

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powers the Helmholtz coils is connected in series to a Keithley 2000 Multimeter so that the

current flowing through the coils can be read accurately to the ten-thousandths decimal place.

For Method 1 (Experiment A.13), we want to obtain null deflection. To do this, we cross

the electric and magnetic fields so that the forces cancel on the electron in the y-direction. When

the forces are cancelled, the beam is almost perfectly straight and touches the far corner, point G

in Figure 2. We first apply a deflecting electric field (E) by setting the plate voltage (Vp) to a

specific value; 100, 150, 200, 250. Then we adjust the anode voltage (Va), usually between 1.5-

4.5kV, to force the trajectory of the electron so that it intersects at the fiducial pin located at

x=47mm. Finally, we apply the balancing magnetic field (B) by adjusting the Helmholtz coil

current (IH) until null deflection is achieved. We then record the Vp and IH , multiply the current

by B=4.15E-3 (T/A) to obtain the B needed to substitute into Equation 15. The e/m ratio is then

determined. We repeat this process while varying Vp to get an average e/m ratio.

For Method 2 (Experiment A.14), we determine the e/m ratio using the magnetic field

alone. This process requires careful measurements of the radius of curvature R of the electron

beam path. To make an accurate determination, we correct for the errors due to non-idealities in

the experiment by alternating the path of the current through the coils. This means that we read R

while the beam is curved upwards, then again after the current is flipped to measure the R when

it is pointing downwards. This time, the anode voltage (Va) is kept at a constant while we adjust

the current in the coils so that it bends the ray to marked points on the scaled fluorescent screen

that are easy to determine R, usually 1.0cm increments. From this we obtain the Va, average R,

and B that are substituted into Equation 21 to determine the e/m ratio. We repeated this process

with a different Va value that is again kept constant while the beam is manipulated to determine

and average R and resulting B. These attempts are all averaged to obtain the e/m ratio.

IV. Results and Analysis

From Method 1, we adjusted the Vp to set increments on the 350V HiVolt, but 50V,

300V, and 350V were either to dim or bright to obtain accurate readings. In Figure 3, the four

incremented Vp voltages all give accurate determinations of the e/m ratio. These values averaged

equals 1.789E11. Each individual value, though, is slightly higher than the accepted e/m ratio.

This could be the result of a possible residual current flowing through and creating a residual

field, skewing our initial deflection measurement. Regardless, the average percent error is only

1.702%.

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For Method 2, we experimented with several different values of the magnetic field with a

fixed value of Va. For the first three trials, the Va was set to 2.5kV, while the Va was set to

3.5kV for the last three trials. Both sets of Va had the beam touching the points where L was

easily determined; 2, 3, and 4cm. Flipping the current and averaging L makes up for some of the

inequalities in the experiment makes our data significantly more accurate. The six trials averaged

to 1.740E11 and gives a percent error of 1.097%. This method proved to be more accurate and

tended to have values lower than the actual e/m ratio.

Conclusions

As a result, both methods proved to be accurate as no test runs were much over a 3%

error. The equipment all worked correctly and was easy enough to set-up and understand. Both

method’s final e/m ratio was around the same percent error, but method one was above the actual

ratio and method two was below. Method one’s error might be because of the residual field

skewing the initial deflection measurement. Method two was made significantly more accurate

because of averaging the length L, so this e/m ratio of 1.740E11 is appealing.

Acknowledgements

I acknowledge the guidance and supervision of Professor Kolagani and Jeffery Klupt.

Also, regards to Towson University for the opportunity to operate in their lab and use the testing

equipment.

References

[1] Melissinos, A.C. “Thomson Tube”, Experiments in Modern Physics. San Diego, CA:

Acedemic Press, pp. 110-113. (1966).

[2] “Joseph John Thomson,” Chemical Heritage Foundation. Retrieved from

http://www.chemheritage.org/discover/online-resources/chemistry-in-history/themes/atomic-

and-nuclear-structure/thomson.aspx

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