in this video we're asked to measure the charge to mass ratio for electrons by running them
through cyclotron motion so i actually took the specifications for this device from a popular
commercial product i think my numbers are pretty close here and we have an accelerating potential
of 300 volts for the electrons we're going to run those electrons into a magnetic
field with a strength of 5.84 gauss that's perpendicular to the velocity of the electrons
and the radius of curvature is observed to be 10 centimeters what i'm asked to find here is
a formula for e over m in general so the charge to mass ratio and then we're going to plug in
all the numbers from our experiment and get a numerical value for the charge to mass ratio
for the electron so we begin by looking at the accelerating potential that 300 volt potential
and i want to have a symbolic name for that it's just another capital v
and we can use energy conservation to figure out how fast the electrons are popping out of that
and just as a magnitude here the drop in potential energy for the electrons as they pass through this
potential difference is given by the magnitude of their charge we'll call that qe times the size of
the potential difference that's v and that drop in potential energy goes into the kinetic energy
of the electrons and i'm taking care to make my little v for the speed a lot smaller than the big
v for the potential difference and then i analyze the cyclotron motion part of the experiment
and in cyclotron motion we have the lorentz force on this moving charge providing the centripetal
force and so i can start by just writing down sort of generically newton's second law for the
cyclotron motion and it might be nice to visualize my electron over here and the force on it pointing
to the center of curvature and if we use the right hand rule on this for the lorentz force
we point our fingers in the direction of v sweep out to the direction of b our thumb points away
from the center of curvature here but it's a negative charge so our force points back
toward the center of curvature the magnitude of the lorentz force is given by qvb so i have
the magnitude of my charge and there's v for the velocity b for the magnitude of the magnetic field
and then the mass we're dealing with is the mass of the electron and then the centripetal
acceleration is given by v squared over r where r is that radius of curvature for the path
of the electrons so one factor of v is going to cancel and so now it's a matter of putting these
two things together and finding a solution for the charge to mass ratio so i'm going to go ahead and
solve for v in the second equation and then plug that into the first equation and when i solve for
v in that second equation i just multiply both sides by r and divide by m now i plug that into
my first equation and i'll go ahead and square everything in those parentheses and i can go ahead
and cancel one factor of q e on the left and right and i can cancel one factor of m e on the top and
bottom on the right side there and i notice that i get a q e over me in other words the charge to
mass ratio so part of the whole point of this is just to illustrate how the charge and mass are
tangled up and why in the early experiments on the mass of the electron there was no way to resolve
this you just end up with a ratio of the charge to mass instead of being able to directly measure
the mass of the electron that's why milliken's oil drop experiment was necessary so i'm going to
isolate the qe over me on the right hand side i'm multiplying by 2 to get rid of the one-half and
then i need to divide by the b squared r squared and i end up with this simple general solution
2 times v that's the accelerating potential over b squared times r squared so now we're ready to
plug some numbers into this just a quick reminder my magnetic field was given in gauss and i have
to change that to tesla and that's a factor of 10 to the fourth so i get .000584 tesla
so teslas are the bigger unit by a factor of 10 to the fourth and the charge to mass ratio
for the electron to three significant digits comes out to 1.76 times 10 to the 11th coulombs
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