in this video we're looking at the cyclotron motion of a solar wind
electron that enters the earth's magnetic field at right angles
so we're given a speed on this electron of 6 times 10 to the fifth meters per second and we're given
the strength of the earth's magnetic field at this point as 0.1 gauss now the conversion
to tesla is by a factor of 10 to the fourth gauss are the smaller units so this is .0001
teslas and in part a we're just looking for the qualitative direction of the magnetic force on
this electron so when i look in the diagram our point of view is such that the magnetic
field points directly into the page the velocity of the electron is 90 degrees to that because
it's in the plane of the page and we're trying to figure out the direction of the force here
so to get this done we have to use the lorentz force right hand rule and i'll post a link real
quick to where this was first introduced there are actually three different ways of using this
right hand rule that i'm aware of my favorite method is to put my fingers of my right hand
in the direction of the velocity and then sweep out to the direction of the magnetic field which
is into the page in this example so it looks like this now notice that my thumb is pointing this way
that's the direction the force would be on a positive charge so in the lorentz force right
hand rule it gives you the direction of force on a positive charge if the charge is negative you have
to reverse that direction now there are other ways of using the lorentz force right hand rule
and you can just pick whatever one is your favorite another one that you might use here
is to put your fingers in the direction of the magnetic field so you pretend your fingers on
your right hand are the field lines put your thumb in the direction of the velocity vector
the force would point out of your palm which again is down into the left and then you have to say
wait a second this charge is negative and i need to reverse the direction of that and we arrive at
the same direction for the force vector so there it is qualitatively and what this force is going
to do is it's going to cause the velocity vector to rotate in the clockwise direction
that velocity will stay in the plane of the page so the electron is always moving perpendicular to
the magnetic field the speed never changes and the velocity vector simply rotates and
the path of this thing is curved into a circle so there is some radius of curvature to this motion
and that's what we're looking for in part b so to get at the radius of curvature we have to
do a force analysis and i'm just starting by stating newton's second law f net equals m a
on this electron and what's the net force it feels all it feels is the lorentz force here
and its acceleration that's related to the speed and the radius of curvature that's v squared over
r now the magnitude of the magnetic force on this electron is given by qvb times the sine of theta
where theta is the angle between the velocity vector and the magnetic field lines well that's
90 degrees and the sine of 90 is just 1. all right so we're left with q v b
equals mv squared over r we can cancel one factor of v and what we're trying to solve for in this
problem is the radius of curvature so i multiply both sides by r and then divide by qb and i find
that the radius of curvature is mv over qb now all we have to do is plug in our mass for the electron
the mass of an electron is 9.11 times 10 to the negative 31 kilograms the speed of the electron
we were told is 6 times 10 to the fifth meters per second the charge on the electron i'm just working
in magnitude here is 1.6 times 10 to the negative 19 coulombs and the strength of the magnetic field
was .0001 tesla and when i plug in the numbers i get 0.342 meters just keeping three sig figs which
might be nicer to state as 34.2 centimeters if you enjoyed this video or at least found it useful
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