Jason Fong
702847140
Lab Section 3
Exp Date: 2-20-00
Physics 4BL: Experiment 5
This experiment investigated the motion of charged particles in a magnetic field.
A solenoid supplied the magnetic field, and a cathode ray tube supplied the charged
particles (electrons) and the means to view their deflection within the magnetic field.
The cathode ray tube was place inside the solenoid and the orientation of the tube and the
current of the solenoid were varied. The data gathered from the deflections of the
electrons in a magnetic field will lead to a calculation of the charge to mass ratio of an
election. It will also lead to an estimate of the earth’s magnetic field.
Part A
A compass was placed in the center of the solenoid while the current to the
solenoid was off. The current was slowly increased until the compass made a 45 degree
angle with the axis of the coil. When the compass was moved within the solenoid, the
deflection of the compass needle remained at 45 degrees. This showed that the magnetic
field was constant while inside the solenoid.
The axis of the solenoid was then aligned to the earth’s magnetic field. The
accelerating, focusing, and grid voltages on the CRT were adjusted until a well-defined
spot was seen on the CRT display. The voltage reading of the CRT was 443 V. The
CRT was then placed at an 90 degree in the solenoid and the current of the solenoid was
set to give the maximum visible deflection on the CRT display. With the current
constant, the angle of the CRT was varied from 90 to 270 degrees, and the resulting
deflection on the CRT display was noted.
Since the current and voltage were not changed during the rotation, the deflection
D can be given in terms of some constant and sinθ, where θ is the angle of the rotation of
the CRT.
1
 e 
 sin 
D  0.832  0 nIL 
8
mV
A 

D  K sin 
2
2
1
 e 
  const .
K  0.832  0 nIL 
8
mV
A 

2
2
Since K is a constant, deflection D is linearly related to sinθ. A graph of deflection
versus sinθ was plotted, and a line fitted the graph. This shows that deflection and sinθ
are linearly related. Also, the correlation coefficient can be calculated to show how well
the linear relation fits. If the correlation coefficient is near 1 or –1, then the deflection
and sinθ are correlated. The correlation coefficient was found to be 0.999, so the
deflection and sinθ are correlated, and they should have a linear relationship.
Part B
The CRT was then reoriented to 90 degrees so that it was perpendicular to the
magnetic field. The current was varied and the deflection on the CRT display was noted.
A graph of deflection D versus current I was made. The predicted slope of the graph can
be derived as follows:
1
 e 
 sin 
D  0.832  0 nIL 
8
mV
A 

D
 slope  C
I
2
2
1
 e 
 sin   const .
C  0.832  0 nL2 
 8mVA 
2
Since the angle is unchanged, sinθ will be constant, so C will be constant. Thus, the
slope of the graph should be constant. A graph with a constant slope will be a straight
line, which is consistent with the straight line fit of the deflection versus current graph.
Using Excel’s regression calculation feature, the slope of the graph was found to be
0.0136 ± 0.0001 m/A.
The length of the CRT was taken as 14.4 ± 0.9 cm. The 14.4 cm was obtained by
taking the length from the front of the curved part of the CRT to the front of the vertical
deflecting plate, then subtracting the horizontal length of the curved part of the CRT, and
then adding the length of the vertical deflecting plate. The length of the vertical
deflecting plate was added since the magnetic field is acting on the electrons right after
they leave the accelerating device. The horizontal length of the curved section of the
front of the CRT was subtracted since the electrons strike at different locations on the
curved surface, so the horizontal length at that section will be different. The length of the
curved section was taken as the uncertainty since the horizontal distance there varies
depending on the deflection of the electron.
The diagram in figure 5-2 differs from the actual setup in a couple of ways. The
first way is that the magnetic field is acting on the electron at all times since the CRT is
completely within the solenoid. The diagram only shows the magnetic field acting on the
electron after it leaves Region A, the part that accelerates the electron. Also, the diagram
shows the electrons striking a flat surface at the end. In the actual CRT, the electrons
strike a curved surface so that the length L varies depending on the amount of deflection.
Since some of the equations used for this experiment were derived based on figure 5-2,
the values calculated from those equations will not be entirely accurate.
The formula for the charge to mass ratio of an election can be calculated as
follows:
1
 e 
 sin 
D  0.832  0 nIL 
8
mV
A

2
2
1
 1  e2
D
   sin 
 0.832  0 nL2 
I
 8VA   m 
1
2
1
2
1
e
8VA 12 1 D
  
2
0.832 0 nL
sin  I
m
e
8VA
 D

 
2
2
2
m 0.832 0 nL sin   I 

2

Since D/I is the slope of the graph, the charge to mass ratio of an electron e/m can be
calculated using the slope of the graph.
e
8443V 
0.01362

2
m 
 57 
2
 0.832 4  10 7 H m 
 sin 2 90



.
144
m



 .585 m 


e
 1.47  1011 C kg
m


In order to find the uncertainty, we rearrange the formula to extract the constants and
introduce the variable S for the slope of graph D/I
e
8VA
 D

 
2
2
2
m 0.832  0 nL sin   I 

2

e
8VA

m 0.832  0 n 2 sin 2 
 1  D 
 4  
 L  I 
2
 S2 
e
 K  4 
m
L 
8VA
8443V 
K

2
2
2
0.8320n  sin  
57  
7 H 
2

 0.832  4  10
m 
 .585 m   sin 90




11
K  3.415  10


 S2 
e
 R  K  4 
m
L 
where R is the charge to mass ratio, S is the slope, and K is a constant.
Thus, the uncertainty in e/m is:
2

 R   R 
 2 KS    4 KS
R   S    L    4 S   
L 
5
 S   L 
 L
  L

2
2
2
2
 23.415  1011 0.0136
   43.415  1011 0.01362

R  
0.0001  
0.009
4


0.1445
0.144

 
2




2
R  4  1010  0.4  1011
The measured charge to mass ratio of an electron (e/m) is (1.5 ± 0.4) x 1011 C/kg
The charge to mass ratio of an electron can also be calculated using the true
values for the charge and mass of an electron.
e 1.60  1019 C

 1.76  1011 C kg
 31
m 9.11 10 kg
This true value for the charge to mass ratio is within the uncertainty range for the value
calculated from the measurements. Thus, the calculated value agrees with the true value.
Part C
The CRT was then removed from the solenoid and isolated as much as possible
from any ambient magnetic fields. The CRT was rotated in the horizontal plane in order
to find the position with the maximum deflection of the beam. This deflection was found
to be 11.5 mm. This is the deflection due to the horizontal component of the earth’s
magnetic field BH. BH can be found by using the measured deflection.
1
 e 
 BH sin 
D  L2 
 8mVA 
sin   1
1
2
 e 
 BH
D  L2 
 8mVA 
2
1

2
1  m
1
D
0.0115
8  443 2 1.5  10-11
BH  2 8VA  2   
2
L
.144
e

1
2
 1.28  10 4 tesla
The measured horizontal component of the earth’s magnetic field is 1.28 x 10-4 tesla.
Using a compass, the angle between the earth’s magnetic field BE and the horizontal was
found to be 19 degrees. This angle is the magnetic inclination Φ. From the
measurements of BH and Φ, the earth’s magnetic field can be found.
Φ
BH
BH = BE cos Φ
BE
BH  BE cos 
BE 
BH
1.28  10 4

 1.35  10 4
cos 
cos19
The earth’s magnetic field BE is 1.35 x 10-4 tesla, and the horizontal component of the
earth’s magnetic field BH is 1.28 x 10-4 tesla.
The field generated by the solenoid when the CRT is at 90 degrees is given by:
1

2
1  m
D
0.024
2
8  44312 1.5 10-11
B  2 8VA    
2
L
.144
e

1
2
 2.67 10 4 tesla
The measured field generated by the solenoid is 2.67 x 10-4. The field generated by the
solenoid is only twice as large as the horizontal component of the earth’s magnetic field,
so the effect of the earth’s magnetic field needs to be considered since the current into the
solenoid would not be consistent with the total magnetic field experienced by the CRT.
There would not be much of an effect on part A because that part only investigated the
correlation between the deflection and sinθ. The current to the solenoid compared to the
strength of the magnetic field did not matter as long as they stayed constant. In part B,
the earth’s magnetic field would not affect the value of the slope, since the added
magnetic field strength would increase the deflection and only shift the graph. But the
slope would remain the same. The computation of the charge to mass ratio also would
not be affected either since the current of the solenoid is taken into account in the slope of
the deflection versus current graph. And since the slope is unaffected, the charge to mass
ratio computation will also not be affected. Thus, as long as the solenoid’s magnetic field
is aligned with the horizontal component of the earth’s magnetic field, the experiment
will be unaffected.
Conclusion
This experiment gave further insight into the behavior of charged particles in a
magnetic field. The deflection of the electrons in the CRT and the sine of the angle of the
CRT within the solenoid were found to be correlated. The data gathered from the
deflections of the electrons in the magnetic field of the solenoid gave a value of (1.5 ±
0.4) x 1011 C/kg for the charge to mass ratio of an electron. This value is within
agreement of its uncertainty to the true value of 1.76 x 1011 C/kg. The measurement of
the deflection due to the earth’s magnetic field and the previous value for the charge to
mass ratio lead to a calculation of the earth’s magnetic field. The earth’s magnetic field
BE was found to be 1.35 x 10-4 tesla. The horizontal component of this value is about
half as strong as the solenoid’s field of 2.67 x 10-4 tesla, so the earth’s magnetic field
could have had an effect on the experiment. Since the solenoid was aligned with the
earth’s magnetic field, however, the experiment was unaffected and the results should be
accurate.