Physics 222 Lab Report Electron Charge To Mass Ratio

e

m

Electron Charge-to-Mass Ratio Emily Gatlin Partner: Whitney Heaston Date Performed: 25 February 2009 T.A. John Caruth 4 March 2009

OBJECTIVE This experiment uses the study of the motion of an electron that moves perpendicular to a magnetic field and measures the charge-to-mass ratio of electron.

INTRODUCTION A beam of electrons shot through a known potential difference allows the electrons to possess a known

1 2 kinetic energy. This factor yields the know velocity[1] KE  mv . At this velocity, the electrons 2

move a constant speed due to the presence of a uniform magnetic field that exerts a force on them and the

its mass. The apparatus uses Helmholtz coils to create a uniform magnetic field at right angles to the initial direction of the electron beam. The magnitude of the magnetic field is the current in the coils. This magnetic field exerts a centripetal force on the moving electrons and causes them to move in a circular path. The centripetal force depends on the magnetic field strength. Thus, with these variables, it is possible to find the

e (electron: mass) ratio from the accelerating potential, the current in the m

Helmholtz







A positive charge q moves with a velocity v in a magnetic field with induction B experiences a force F



 



given by the Lorenz force equation[2] F  q  v  B . The expression for the magnitude of F is[3]

  F  qv B sin  . From this equation, the direction of the force is:    1. F direction  plane that v and B are positioned    2. F direction  v and B

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Physics 222

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|4 March 2009|

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 F direction is direction right-hand screw will advance if rotated in same direction  3. that v is rotated through the smallest angle to   make v align with B 4. For negative charges, the force is opposite direction .

 F  B

 v

Since this force on a moving charge in a magnetic field is always perpendicular to the velocity, this force is a centripetal force that always moves the charge in a circle with a radius r .

v2 The centripetal force is [4] Fc  m where v is the magnitude of the velocity of the charge (its speed) r and m equals the mass of the charge. Since the Lorentz-force supplies the centripetal force, the forces from equations [2] and [4] are equal F  Fc . Thus, using equations [3] and[1], then the expression

F  Fc becomes

v2 [5] q v B sin   m r



[6]

q v  m Br sin 

Since it is electrons within the magnetic field, electrons possess a negative charge symbolized e . Thus, the force will be opposite than the one given by the right-hand rule for positive charges. Since the apparatus  aligns the velocity of the moving electrons perpendicular to the magnetic field where   90 . Equation

[6] becomes [7]

e v  . m Br

Therefore, the electron charge to mass ratio requires velocity, radius, and magnetic induction to be determined. The apparatus display the circular path of electrons using helium gas at low pressure to illuminate the electrons generated by a hot filament and accelerated though a potential difference. This electrical potential equals the charge of an electron times the potential difference through which is accelerates[8]

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|Emily A. Gatlin|

|4 March 2009|

|e/m |

1 KE  mv 2  eV . Electrons move at constant speed inside the magnetic field experiencing only the 2 directional changes along the arc of a circle that completes at the filament. Using equation[8], the electron charge-to-mass ratio becomes

e v2 [9]  m 2V The speed of electrons is [10] v 

e 2V m

Using Equation [10] into Equation [7] after squaring both sides yields  e 2 1 e [11]    2V  m  2 2 m

B r

e e 2V in Equation[11]  [12]  m m B2r 2 e Equation [12] shows the relationship between and the accelerating voltage and the radius of the m Now, solving for

circular path the electron take in the magnetic field. Both V and r are easily measured. The magnetic induction B for a set of the Helmholtz coils having N turns of wire with radii a , carrying current I is [13] B 

N 0  I 3

 5 4 2 a

[14] B  I .

The Helmholtz constant,  is given by [15]  

N 0  3

 4 2 a 5

Equation [15] uses equations [13] and [14] to calculate the electron charge-to-mass ratio 3 2 5

 

2V a e 4 [16]  m N  2 I 2r 2 0

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Physics 222

[17]

e 2V  . m I 2 r 2

Spring 2009

|Emily A. Gatlin|

|4 March 2009|

Equations [16] and [17] are used to measure the

|e/m |

e ratio. The parameters with equations [16]and m

equation [17] listed below show the measured values needed in order to calculate the electron charge -tomass ratio.

V  the accelerating potential a  the radius of the Helmholtz coils N  the Number of turns on each Helmholtz coil  130 0  the permeability constant of free space = 4 107 N  A I  the current through the Helmholtz coils r  the radius of the electron beam path The theoretical value for charge and mass of electron ar e

e  the charge of the electron  1.61019C m= the mass of the electron  9.11031 kg

PROCEDURE The apparatus in the experiment is a Pasco® Model SE-9638 e/m apparatus, a Pasco® Model SF-9584A low voltage power supply, Pasco® Model SF-9585A high voltage power supply, assorted leads with banana plugs, and permanent bar magnet. The e/m tube is centered between a pair of Helmholtz coils and filled with helium at 10 -2 torr with an electron gun that generates an electron beam that excites the helium atoms to higher energy levels leaving a visible trail as the atom radiate light from returning to their energy states. The electron gun consists of a heater that heats the cathode and emits electrons. Electrons are accelerated by potential between cathode and Electron gun

Helium Filled Vacuum Tube

anode. The Helmholtz coils are two identical circular coils of wire separated by a distance equal to the radii of the coils.

Grid

Anode

Cathode

The radius and

separation of the coils for the Pasco® e/m apparatus is 15 Heater

Deflecti on Plates

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|Emily A. Gatlin|

|4 March 2009|

|e/m |

cm and each coil has 130 turns of wire. The control panel has banana jacks with labels. The cloth hood covers the coils to see the trail of the electron beam path. The mirrored scale is on the back of the rear Helmholtz coil. The apparatus is properly assembled with particular care to the connections. The voltage is set to approximately 150-volts and the Helmholtz current is approximately 1-Amp. The radius of the beam path is measured using the position of the left side of the path and the right side of the path with the left value as the negative value. The readings were recorded to the accelerating voltages, coil current, and left scale readings for this combination. The measurements and calculations were repeated for the rest of the combinations of accelerating voltage and coil current.

DATA/CALCULATIONS P e r m e a b i l i t y o f F r e e Sp a c e

N

0  1.25664106

R a di u s o f C o i l s

a  1.50101 m

Number of Turns Helmh oltz Cons tant

N 130

A2

T A e  1.601019 C

  7.79104

C h a r g e o f E l e ct r o n

m  9.101031 kg e C  1.758241011 m kg

M a s s o f E l e ct r o n Charge- to-Mass Ratio

Measurement

Accelerating

Coil

Left Scale

Right Scale

Number

Voltage

Current

Reading

Reading

1

150

1

-3.8

5.3

0.0455 2.38619E+11 35.71442%

2

150

1.25

-3.8

4.6

0.042 1.79229E+11 1.93661%

3

150

1.5

-3

3.8

0.034 1.89927E+11 8.02106%

4

150

1.75

-2.3

3

0.0265 2.29699E+11 30.64144%

5

150

2

-2.7

3.3

0.03 1.37222E+11 -21.95478%

7

200

1.25

-4

5.1

0.0455 2.03621E+11 15.80964%

8

200

1.5

-3.8

4.5

0.0415 1.69976E+11 -3.32619%

9

200

1.75

-3

3.8

0.034 1.86051E+11 5.81655%

10

200

2

-2.8

3

0.029 1.95799E+11 11.36059%

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Physics 222

Radius

Measured e/m

% Error

Spring 2009

|Emily A. Gatlin|

|4 March 2009|

|e/m |

12

250

1.5

-4.4

5

0.047 1.65652E+11 -5.78516%

13

250

1.75

-3.9

4

0.0395 1.72308E+11 -1.99974%

14

250

2

-3.1

3.5

0.033 1.89012E+11 7.50030%

15

300

1.5

-4.4

5.2

0.048 1.90587E+11 8.39613%

16

300

1.75

-4.1

4.5

0.043 1.74479E+11 -0.76479%

Average e/m Value

1.87299E+11 -0.76479%

The data reflected a little error on the absence and inability to record some of the readings. Additionally, the voltage readings represent approximations and the measurement of the radius relied heavily upon the human ability to read the radii measurements accurately. The electron beam did not appear as strong and this is probably due to mechanical error. Additionally, the current readings often reflected estimates and not the exact current measurements that would also account for the discrepancies in the data. However, despite the human, mechanical, and technical difficulties, this experiment still exhibited the core concepts behind the calculation the electron charge -to-mass ratio.

CONCLUSION This experiment showed the principle of the interactive forces of an electron moving in a magnetic field. A beam of electrons accelerated through a known potential difference gives a known kinetic energy used to find the velocity based on the mass. From this, the knowledge that once accelerated, the electrons maintain a constant speed in the presence of a uniform magnetic field exerting a force that causes them to move in a circular path. This relates the speed of the electron to the mass by using a magnetic field to induce a centripetal force. Using the centripetal force, the e/m ratio is found by measuring the

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