Physics I
Class 24
e/m Ratio for the Electron
Rev. 14-Apr-04 GB
24-1
Joseph John (“J.J.”) Thomson
J.J. Thomson
1856-1940
J.J. Thomson was appointed in 1884 as
the third Cavendish Professor (head of
the Cavendish Laboratory) at Cambridge,
after James Clerk Maxwell and Lord
Rayleigh. In 1899, his experiments with
cathode ray tubes led him to postulate the
existence of a new particle with a ratio of
charge to mass (e/m) far larger than the
same ratio for a positive hydrogen ion.
The word “electron” was coined in 1891 by G. Johnstone Stoney.
Today we will measure e/m for the electron.
24-2
Calculating Change in K.E.
from Electric Potential (Review)
final
e
 K   U  0 or  K   U
U  q V  (e) V or  U  (e) V
 K  ( e)( V )  (1.6  10 19 ) (100) 
 1.6  10 17 J
If the electron starts at rest (or very close to it),
then
V = 100
initial
V = 50
e
1
2
m v 2  e V
V=0
24-3
Magnetic Force on a
Moving Charge (Review)

 
F  q vB
charge of the particle (C; + or –)
v : velocity of the particle (m/s)
B : magnetic field (T)
q:

 Force is at a right angle to velocity.
 Force is at a right angle to magnetic field.
Important: If q is negative, that reverses the direction of force.
24-4
The Radius of the Circle (Review)
F
r
v
Although the directions of the vectors are
changing, the magnitudes stay the same.
v2
F  ma  m
r
F  qvB
v2
qvB  m
r
v2
mv
rm

qvB qB
24-5
Apparatus for Measuring e/m
We will set
 Potential in tube (V).
 Current in coils (I).
We will observe
 Radius of circular
electron path (r).
We will calculate
 e/m.
24-6
The Cathode Ray Tube
Electrons
Electrons are randomly kicked
out of the metallic cathode by
thermal energy.
Once free of the metal, the electrons are accelerated through
a potential difference of V from cathode to anode.
24-7
Analysis of Electron Acceleration
cathode
anode
 K   U
 U  (e) V
 K  (e)[ V ] 
Electron Stream
pot = -V
pot = 0
e [0  ( V )]  e V
Electrons have very low kinetic
energy when they leave the
cathode – essentially zero.
m v2  e V
2eV
v
m
1
2
or
24-8
Helmholtz Coils
Magnetic
Field
Helmholtz Coils are designed to have a
nearly uniform (constant) magnetic
field in the center. The field direction is
along the axis of the coils. The
magnitude is proportional to I, the
current. The formula to calculate this is
a Physics II topic, but for our coils,
B  7.8  10  4 I
where B is in tesla and I is in amperes.
24-9
Derivation of e/m Formula
r
mv
eB
e
v

m rB
v
2eV
m
2eV
e
2V
e
e 2V
m
m



m
rB
rB
m rB
2V
e

m
rB
or
2V
e
 2 2
m r B
We set or observe all of the variables
on the right side of the equation.
24-10
Experimental Procedure
1.
Turn up V to get an electron
beam. Record V.
2. Turn up I (to make B) until the
electrons make a complete circle.
Record I.
3. Observe r – use the mirrored
scale in the rear.
3a. Or (easier): Adjust I until the
electrons just hit the far side of
the tube. This is a known radius
(5.57 cm). Record I.
4. Repeat three times with different
values of V.
24-11
Viewing the Electron Path
24-12
A Similar Application -Mass Spectrometer
Definition from chemistry:
“Mass spectrometers use the difference in mass-to-charge ratio (m/e) of
ionized atoms or molecules to separate them from each other. Mass
spectrometry is therefore useful for quantitation of atoms or molecules and
also for determining chemical and structural information about molecules.
Molecules have distinctive fragmentation patterns that provide structural
information to identify structural components.”
24-13
Basic Operation of a Magnetic
Sector Mass Spectrometer
Stage 1: Ionization
The atom is ionized by knocking one or more electrons off to give a positive
ion. This is true even for things which you would normally expect to form
negative ions (chlorine, for example) or never form ions at all (argon, for
example). Mass spectrometers always work with positive ions.
Stage 2: Acceleration
The ions are accelerated so that they all have the same kinetic energy.
Stage 3: Deflection
The ions are then deflected by a magnetic field according to their masses. The
lighter they are, the more they are deflected.
The amount of deflection also depends on the number of positive charges on
the ion – in other words, on how many electrons were knocked off in the first
stage. The more the ion is charged, the more it gets deflected.
Stage 4: Detection
The beam of ions passing through the machine is detected electrically.
24-14
Mass Spectrometer Diagram
Adjusting the accelerating
potential (V) and/or the
magnetic field (B) “tunes”
the spectrometer to detect
only ions with a specific
ratio of charge to mass.
24-15
Class #24
Take-Away Concepts
1.
2.
The ratio of e/m is determined by accelerating a stream
of electrons through an electric potential difference and
observing their path in a magnetic field.
The formula for e/m (know how to derive it):
2V
e
 2 2
m r B
3.
Mass spectrometers work by the same principle.
24-16
Activity #24
Measuring e/m for the Electron
Objective of the Activity:
1.
2.
Work through the analysis steps to get the e/m formula.
Take your own measurements and compare your value of e/m
for an electron with the known value. (1.76 x 10+11 C/kg)
24-17
Optional Material for Class #24
How do we know the mass of an electron?
This is a great question!
We just measured e/m for an electron using table-top instruments that
would have been available (in some form) 100 years ago.
The first accurate measurement of the mass of the electron was made
(indirectly) by Robert Andrews Millikan in 1910. How did he do it?
24-18
Robert Andrews Millikan
R.A. Millikan
1868-1953
Oil Drop
Experiment
(1910)
At the University of Chicago in 1910, Millikan developed his famous oil drop
experiment to determine the charge on an electron. An atomizer sprayed small oil
droplets into the top chamber. Some of them fell through a small hole into a second
chamber. X-rays caused the air in the second chamber to become ionized. The
drops in the second chamber picked up a small (variable) number of electrons.
They were acted upon by both the force of gravity and an opposing electrical force
created by an adjustable electrical potential difference between the top and bottom.
24-19
Analysis of Oil Drop Experiment
QE
1.
2.
Q
Oil Drop
3.
Mg
With E = 0, the terminal velocity of the drop falling through
air allows calculation of its mass (M) via a known
mathematical relationship.
Once M is known, E is adjusted so that the drop is in
equilibrium (drop stopped). Then Q is determined:
4.
Mg
E
Steps 1 and 2 are repeated many times. Each drop has a
different M and Q, but when enough drops are examined, the
smallest difference between different Q values is found to be
e, the charge on one electron (ignoring the – sign).
The mass of the electron is
1
e
m    e  (1.76  10 11 ) 1 (1.6  10 19 )  9.1  10  31 kg
m
24-20