Week 6: Moving Charges and Magnetic Force 199
directlyobserved in aCloud Chamber^59 placed in a magnetic field. Observations of many tracks
(plus doing various current-based experiments) leads one to conclude that the force acting on a
charged particle with chargeqtravelling at velocity~vin a uniform magnetic fieldB~is:
F~=q(~v×B~) (408)Ooo! That peskycross productrears its ugly^60 head! Sorry about that, but if youdon’tfeel
completely comfortable with a cross product yet, it is time to start really working on it. See the
associated mathematical physics documentation linked to this course and start reviewing the good
old right hand rule and the two or three ways available to compute them.
This law is (as you can see)quitedifferent from the electrostatic rule, and the force depends on
both themagnitudeanddirectionof thevelocityof the charge in the magnetic field, and doesn’t
point in the direction of the magnetic field at all! In fact, it points in thedirectionperpendicularto
the plane determined by the magnetic field and the velocity vectors. Cross products are “twisty”
beasts, always pointing off at right angles compared to any of the directions one might expect.
This twistiness, however, doesn’t represent insoluble complexity, and you shouldn’t throw your
hands up in disgust or tremble in fear. As we will see, the motion produced by the magnetic force
acting on a point charge is often quitesimpleand easy to understand and compute. To see this, we
will begin at the beginning and solve for the motion in the simplest case,motion when the velocity
is perpendicular to the (uniform) magnetic field.
Example 6.2.1: A Charged Particle Moving in a Uniform Magnetic Field
rFvBoin+qFigure 63: A charge particle with velocity perpendicular to a uniform magnetic field moves in a
circle.
In figure(63 above, we see a charged particle +qmoving with initial velocity~v 0 perpendicular to
a uniform magnetic fieldB~ 0. The little crosses in this figure should be thought of as the “feather”
ends of vector arrows and stand for a vector that pointsintothe page – a circle with a dot will stand
for the “tip” of the arrow and a vector pointingoutof the page should we ever need it.
(^59) Wikipedia: http://www.wikipedia.org/wiki/Cloud Chamber. Cloud chambers are actually quite easy to build,
and I have had the building of an operational cloud chamber used for the extra credit/honors project my students
often undertake. They are very cool – literally, as they are often cooled with e.g. dry ice or liquid nitrogen – and
theydirectly reveal to the eyethe tracks of otherwise invisible charged microscopic/elementary particles from the
environment, from radioactive sources, from cosmic rays.
Just something to bear in mind if you are using this text in oneof my classes with this third-of-a-letter-grade option!
(^60) To introductory level students, at least. Actually, the cross product is amazinglybeautiful, an essential part of a
geometric algebrathat generalizes the idea of complex variables to higher “grade” (number of complex dimensions).
But to a student, “ugly” in this context is code formore complicatedthan the ordinary arithmetical multiplicative
product or the scalar inner product between two vectors, andyetessential to learnin order to do well in the course!