Week 6: Moving Charges and Magnetic Force
(Est 2/13-2/18)
- A charge moving through space is observed to deflect according tothe rule:
F~=q(~v×B~) (400)which we use todefinethe magnetic fieldB~much as we defined the electric field in terms of
the force observed and described by Coulomb’s Law.
For the moment we will ignore just howvBgot there, as we live in a locally uniform magnetic
field due to the Earth all the time and can discover magnetic materialsin nature so natural
sources of magnetism are ubiquitous.- This translates into:
dF~=I(d~ℓ×B~) (401)
for a small (differential) segment of wire carrying a currentIin a magnetic fieldvB. Magnetic
fields exert forces on current carrying wires. - Motion of a point charge in the plane perpendicular to a uniform magnetic field is therefore
circular:
|F~|=qvB=
mv^2
r(402)
(Newton’s second law plus definition of centripetal acceleration). It has an angular velocity
given by:
ωcyclotron=qB
m(403)
independent of itsspeed. This is called thecyclotron frequency.- You should be able to derive/explain:
- A cyclotron.
- A velocity selector (region of crossed fields).
- Thomson’s apparatus for measuringme.
- A mass spectrometer
- The Hall effect (region of crossed fields in a conductor).
- The magnetic dipole moment of a plane current loop is:
~m=N IAˆn (404)whereN is the number of turns,Iis the current,Ais the area, andˆnis the right-handed
normal to the plane of the loop.197