3
012-03471E
e/m Apparatus
!
Current adjust knob
for Helmholtz coils
Focus knob
Power Supply
(Heater
6.3 VDC or
VAC)
Power Supply
(Accelerating
Voltage
150-300 VDC)
Voltmeter
(0-300
VDC)
DC
Ammeter
(0-2 A)
Power Supply
(Helmholtz
Coils 6-9 VDC,
ripple < 1%)
Toggle Switch:
Up for e/m experiment,
Down when using deflection plates.
Figure 4 Connections for e/m Experiment
-
+
+
-
+
-
Upper
Lower
+
+
+
-
-
-
-
+
on the mirrored scale. Measure the radius of the beam as you
see it on both sides of the scale, then average the results.
Record your result below.
Electron Beam Radius = r =
Analysis of e/m Measurement
The magnetic force (F
m
) acting on a charged particle of
charge q moving with velocity v in a magnetic field (B) is
given by the equation F
m
= qv X B, (where F, v, and B are
vectors and X is a vector cross product). Since the electron
beam in this experiment is perpendicular to the magnetic
field, the equation can be written in scalar form as:
F
m
= evB (1)
where e is the charge of the electron.
Since the electrons are moving in a circle, they must be
experiencing a centripetal force of magnitude
F
c
= mv
2
/r (2)
where m is the mass of the electron, v is its velocity, and r is
the radius of the circular motion. Since the only force acting
on the electrons is that caused by the magnetic field, F
m
= F
c
,
so equations 1 and 2 can be combined to give evB =
mv
2
/r or
e/m = v/Br (3)
Therefore, in order to determine e/m, it is only necessary to
know the velocity of the electrons, the magnetic field
produced by the Helmholtz coils, and the radius of the
electron beam.
The electrons are accelerated through the accelerating
potential V, gaining kinetic energy equal to their charge
times the accelerating potential. Therefore eV = 1/2 mv
2
.
The velocity of the electrons is therefore:
v = (2eV/m)
1/2
(4)
The magnetic field produced near the axis of a pair of
B =
[N
µ
0
] I
(5/4)
3/2
a
(5)
Helmholtz coils is given by the equation:
A derivation for this formula can be found in most introduc-
tory texts on electricity and magnetism.
Equations 4 and 5 can be plugged into equation 3 to get a
final formula for e/m:
e/m = v/Br = 2V (5/4)
3
a
2
(N
µ
0
Ir)
2
