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Charge to Mass Ratio e/m for an Electron
Studio Physics I
The apparatus available in the room will allow us to shoot a beam of electrons at velocity v into a uniform
magnetic field B. When we do this, we observe that the electrons move in a circle.
Deriving an Equation for e/m Ratio
The first thing that we will do is to derive the mathematical expression that we will use to calculate a
value of charge to mass ratio e/m from measurements of velocity v, magnetic field B, and the radius r of
the circle that the beam makes. We will use a pair of wire coils – called Helmholtz coils – to produce the
magnetic field in this experiment. Helmholtz coils are coils designed to make a very uniform magnetic
field in the center. The distance between the coils is the same as their radius.
1. Is the magnetic field produced by the coils perpendicular to the plane of the coils as shown is figure A
, parallel to the plane of the coils as in figure B or in some other orientation? You can use a compass and
the small wire coils to help figure this out or verify your prediction with your instructor or TA.
Figure A
Figure B
B
B
2. A particle moving in a circle has a centripetal acceleration. What is the expression for centripetal
acceleration? What is the expression for the centripetal force on a particle of mass m moving with
velocity v in a circle of radius r?
3. What is the expression for the force on a particle of charge –e moving at velocity v in a magnetic field
B? (This force is called the Lorentz Force).
4. Is the velocity of the electrons parallel to the plane to the coils, perpendicular to the plane of the coils
or in some other orientation?
5. Is the velocity of the emitted electrons perpendicular to the magnetic field? If so, does this remain true
even after the particles start to move in a circle? If not, what is the angle between the velocity of the
electrons and the magnetic field produced by the coils?
6. What is the expression for the force on a particle of charge q moving at velocity v perpendicular to a
magnetic field B?
7. Find the equation of e/m in terms of v, r, and B by setting the centripetal force equal to the force on the
particle due to the magnetic field and solving the resultant equation for e/m.
8. What is the equation for the kinetic energy of a mass m moving at a speed v?
1999 P. Laws, K. Cummings, Wiley and Sons Publishers; Rev. 2002 Bedrosian
9. The electrons in the tube are accelerated to a speed v by starting with a very small speed (essentially
zero) and then moving through a potential difference of V before being shot into the magnetic field. What
is the equation for the potential energy lost by a charge –e moving from a potential of 0 to a potential of V
volts?
10. Use conservation of energy to replace velocity in the expression that you found in step 7 with an
expression using the potential difference, V. Show that e/m is given by the expression
e
2V
 2 2
m r B
(Hint: Consider e/m to be a variable – call it x – replace e/m with x everywhere, and solve for x.)
Magnetic Field Produced by a Pair of Current Carrying Coils
We now need to determine the magnitude of the magnetic field that the beam of electrons moves through.
An approximately uniform magnetic field is produced by the two large current-carrying coils called
Helmholtz coils. (We’ll be studying the ability of currents to produce magnetic fields next semester.) For
now, you should just accept on faith that the magnetic field between the Helmholtz coils can be calculated
from well-accepted equations if the current in the coils is known.
Helmholtz coils to produce a magnetic field.
For the pair of Helmholtz coils we use:
B= 7.8x10-4 teslas per amp of current in coils
I
So B = 7.8 x 10-4 (I)
= the current through the coils in amps
11. Use the equation above for the magnetic field produced by the coils and the expression shown in # 10
to write a mathematical expression for the charge to mass ratio e/m for an electron in our experiment in
terms of the accelerating potential V, the current in the coils I and the radius that the beam of particles
makes.
Measuring e/m for electrons
Go to the experimental apparatus set-up. In the presence of an instructor, turn on an accelerating
potential between 200 and 300 volts. Vary the current in the coils until the beam bends in a circle which
just hits the far side of the tube. In this case, the electrons in the beam are moving in a circle of radius
r=0.055 meters. Record the exact values of the current in the coils and the accelerating potential.
12. Calculate the e/m ratio for an electron based on your measured values of current and potential.
13. Calculate the e/m ratio for an electron based on the values of charge and mass for an electron which
are cited in the cover of your textbook.
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14. What is the percent difference in your measurement? Show your calculation.
1999 P. Laws, K. Cummings, Wiley and Sons Publishers; Rev. 2002 Bedrosian