Recommended problems for Lecture 18: FGT 29.8, 29.10, 29.11, 29.20
Recommended reading for Lecture 19: FGT pp 779-790
There are many analogies between electrostatics and magnetostatics (time independent magnetic fields). Just as there are two types of electric charge, there are two types of magnetic ``charge''. They are called the North pole ``N'' and the south pole ``S''. Magnetic field lines come out of north poles and go into south poles. Two like poles repel and unlike poles attract. However there are no elementary particles which carry an isolated magnetic charge. That is, there is no particle that has magnetic charge ``N''. Instead elementary particles often have a property called spin which is quantized (for example the electron has spin half). Associated with spin is a quantum of magnetic dipole moment, not a quantum of magnetic charge. No free magnetic charges or ``magnetic monopoles'' have been observed, though there has been a lot of theoretical(Dirac) and experimental research looking for them. Magnetic dipoles occur as a property of elementary particles, for example a free electron which has spin 1/2 has magnetic dipole moment which is the Bohr magneton. The magnetic field lines due to a magnetic dipole (e.g.a bar magnet or an electron spin) look the same as the electric field lines due to an electric dipole.
However there is another way to produce magnetic fields. It was observed by Oersted and Ampere in 1820 that a DC current produces a magnetic field. The magnitude of the magnetic field increases as the current increases. A wire wound into a dense coil (e.g. a solenoid) produces a very large and uniform magnetic field. A wire loop with a DC current also produces a magnetic field which looks like a magnetic dipole. A straight infinite wire carrying a DC current produces a magnetic field which circulates around the wire. Since a current produces a magnetic field, electrons in orbit around the nucleus can also produce a magnetic field and hence have a magnetic moment. Magnetism in materials like iron is due to a combination of the spin and orbital contributions to the magnetic moment.
The MKS unit of the magnetic field is the Tesla (T). Sometimes magnetic fields are given in units of Gauss or Oersted which are both names for the CGS unit of magnetic field. 1 Gauss = 1 Oersted = 10-4T. Typical values of magnetic fields: Near the earth's surface . A bar magnet produces 10-100mT. A research magnet produces 1-30 T
We shall analyse the way in which
currents produce magnetic fields in a couple of days.
First we look at the effects of magnetic fields
on charges and wires carrying currents, including current loops.
A charge moving in a magnetic field
A charge q moving in a magnetic field
experiences a force,
(i) The force on the charged particle is always perpendicular to both the velocity vector and to the magnetic field vector.
(ii) If the particle moves in the direction of the magnetic field, it experiences no magnetic force.
(iii) If the particle moves perpendicular to the magnetic field it experiences the maximum force.
(iv) Since the force is perpendicular to the magnetic field lines and to the velocity vector, the particle ``spirals'' around the magnetic field lines. The larger the magnetic field the larger the magnetic force and the tighter the spiral.
(v) No work is done by the magnetic field on the
charged particle. The kinetic energy of the particle
is therefore a constant, if no other forces act on the
Constant field perpendicular to
A useful notation for drawing vectors in three dimensions:
- A cross indicates a vector into the page
- A dot indicates a vector coming out of the page
Consider the simplest case in which
a particle of charge q has velocity
which is perpendicular to the direction
of the magnetic field, then,
|FB = q v B||(3)|
(i) Radius of the orbit - R=mv/qB.
(ii) Period of the orbit - .
(iii) Frequency (Cylotron frequency) of the orbit .
Notice that the period and frequency of the orbit do not depend on the radius. They only depend on the charge to mass ratio q/m and the applied magnetic field B. In addition, if we know the speed of a charged particle and the magnetic field, we can find the charge to mass ratio. This is used in mass spectrometers. Often the frequency of this orbit is called the cyclotron frequency.
The circular motion described about is the basis of understanding more complex motion. For example
(i) If a charged particle has a component of its velocity parallel to the magnetic field lines, then it spirals around the magnetic field lines in a helical motion.
(ii) If there is a non-uniform magnetic field,
then the radius of the spiral is larger in
regions of weak field and smaller in regions of
Electric and magnetic fields
If a charge moves through a region of space which has both
electric and magnetic fields, then the particle experiences
both the electric force (), as well as the
magnetic force .
A velocity selector
A charged particle in crossed electric and
magnetic fields can still have constant
velocity motion. This occurs if the
electric and magnetic forces balance
perfectly. This special case can
be used as a velocity selector. That is,
if we want to select particles with
speed v from a set of particles
with different speeds, we can do so
by arranging crossed electric and magnetic
fields in the correct manner. Here is
how to do it: Choose ,
all perpendicular to
each other. The direction of the
magnetic force is then either parallel
or antiparallel with the electric force.
If we choose the electric force
to be antiparallel to the magnetic
force, we can arrange for them to
cancel each other. This is achieved if,
|qE = qvB||(6)|
|v = E/B||(7)|
Thompson measurement of q/m of the electron
He used a velocity selector so that, v=E/B.
In addition, he accelerated electrons in
a cathode ray over a potential V, so that,