Magnetism
Magnetic Fields
Permanent magnets (lodestones) have been known since the ancient Greeks. They exert magnetic force on each other, and are surrounded by a magnetic field B.
A bar magnet is a magnetic dipole
Note: no isolated magnetic charge, or "monopole" has been seen - some theories predict they should exist.
Michael Faraday showed that magnetic fields are also produced by an electric current: basis for electric motors, electric generators, etc.
Electric and magnetic fields are "unified" in Maxwell's theory of electromagnetism.
Definition of B
Recall the definition of E:
or F = qE
for the electrostatic force on a test charge q
In the same way, we define
FB = qv x B
Where FB is the magnetic force on a charged particle, charge q, velocity v, and B is the magnetic field vector.
The experimental facts:
Magnetic force FB is proportional to charge q, speed v and field strength B
FB is perpendicular to the velocity v and field B as given by the Right Hand rule.
The direction of the force is opposite for positive and negative charges.
Unit of magnetic field:
1 N/(C.m.s-1) = 1 "Tesla" = 1 T
(older unit was the "gauss": 1 T = 104 gauss)
The world record field ~ 100 T. The Earth's magnetic field ~ 10-4 T (1 gauss).
Magnetic lines of force are drawn in the same way as electric lines of force.
Example: A uniform B field 1.2 mT points vertically upwards. A 5.3 MeV proton moves horizontally from south to north. What is the magnetic force on the proton?
[proton mass mp = 1.6 x 10-27 kg, 1 eV = 1.6 x 10-19 J]
Firstly, find velocity of the proton:
K = (1/2) mv2 = 5.3 MeV = (5.3 x 106) x (1.6 x 10-19) J
Can find the velocity:
The magnetic force magnitude:
FB = qvBsinq = (1.6 x 10-19) x (3.2 x 107) x (1.2 x 10-3) x 1 = 6.1 x 10-15 N
Direction is given by the right hand rule: to the East
Circulating charge
If a charged particle travels at right angles to a magnetic field, it will move in a circle:
F = qvB inwards
For circular motion :
radius can be calculated
The radius is proportional to the velocity v, and inversely proportional to the field strength |B|.
The frequency f = velocity/circumference:
Frequency and period are independent of velocity
If the particle has a component parallel to B, that remains unchanged. The particle then moves in a spiral, or a helix.
e.g. the ionosphere or "tokamak"
Applications
1) Velocity selectors
Fire a beam of charged particles q through a region with "crossed" electric and magnetic fields:
Electric force qE points downwards opposing the magnetic force qvB pointing upwards.
At balance, qvB = qE and 
Otherwise, the particles deviate and collide with the plates.
Hall effect (also see example)
2) Mass spectrometer
Ions, charge q, are fired through a velocity selector into a region of constant magnetic field, where they move in circular path.
We can measure the mass m as proportional to the radius r
Magnetic force on a current-carrying wire
The moving electrons in a current (in a wire) experience a force in a magnetic field.
Find the force F on a length L of conductor.
The time for charge carriers in this segment to pass point A is t = L/v
Charge transferred q = it = (iL)/v
Force F = qvBsinq = 
Force on a current-carrying conductor, where L is in the direction of the current.
Example:
a) A straight copper rod carries a current 50 A from west to east in a region between the poles of a large electromagnet. In this region there is a horizontal magnetic field toward the north east with magnitude 1.2T. Find the magnitude and direction of the force on a 1 m section of the rod.
b) How should the rod be oriented to maximize the magnitude of the force.
F = iLB sinq = (50)(1)(1.2)sin 45o = 42.4 N
Direction by the right hand rule: out of the page.
b) The force will be maximum, when sinq is maximum, when q = 90o, rod points south east.
In the examples when the wire is not straight or the field is not uniform, we must divide the wire into infinitesimal segments and write:
dF =idLx B
Applications
The direct current motor
A current carrying coil in a magnetic field experiences forces causing it to rotate. This is the principle of the electric motor.
Equal and opposite forces produce a "torque" or turning moment. The current in the coil must be reversed each half cycle to maintain steady rotation.
Sources of the magnetic field
Moving charges experience force in magnetic field. Where does the magnetic field come from? Magnetic fields are produced by moving charges and currents.
Magnetic field of a moving charge:
Experimentally, a moving charge produces a magnetic field forming circles around its direction of motion. At point P:
Similarly to electrostatic constant, there is a constant factor
,
where m0 is the permeability of vacuum. The magnitude of B is proportional to the charge q and speed v, and drops off like 1/r2.
r is the distance vector from the "source point" to the "field point" P
is
a unit vector in the same direction
B is perpendicular to both v and 
, in a direction given by the right hand rule.
Magnetic field of a current element
In practice, moving charge mostly in a wire, part of a current.
Current element of length dl a source:
Charge dq travels distance dl in time dt:
dqv = dq (dl/dt) = idl (taking i = dq/dt)
The magnetic field produced by this current element:
This is the Biot-Savart law.
An infinitesimal current element produces an infinitesimal field. The total field of a long wire must be found by integration.
Example: Field due to a long straight wire
Apply Biot-Savart law:
By symmetry, same result applies for the other half of the wire, so multiply
the result by 2.
for long straight straight wire.
Direction given by another right hand rule: fingers curl as B field, thumb in direction of the current.
Force between two parallel conductors
Two parallel wires: each wire will produce a force on the other:
Example: Two long parallel wires, each with current i, are placed as shown. What is the resultant field B at P, midway between them?
Use superposition B = B1 + B2
By symmetry, resultant field is to the right.
Magnitudes: B1 = B2 = 
Wire 1 produces a field at wire 2:
Wire 2 experiences a force F on length L given by:
F = i2L x B1
Magnitude: 
Force/unit length: 
Parallel currents attract, antiparallel currents repel
The force on wire 2 is due to the field of wire 1. There will be an equal and opposite force on wire 1 in accordance with Newton's third law.
This force is the basis for the definition of the ampere:
"The ampere is that constant current which, if maintained in two straight parallel conductors of infinite length, of negligible circular cross section, and placed 1 m apart in a vacuum, would produce on each of these conductors a force equal to 2 x 10-7 Newtons per metre of length."
Ampere's Law
This is the integral form of the Biot-Savart law:
Left hand side denotes the "line integral" of B around a closed loop ("Amperian loop"), ds is an element of distance along the path and i is the current through the loop, in a direction given by the right hand rule.
We interpret the integral as the circulation of B around the loop: it is driven by the current i through the loop.
Example: Find the "circulation" of B around the Amperian loops shown.
A long straight wire
Check that Ampere's Law gives the same answer as the Biot-Savart law:
The system is symmetric under rotation about the axis, and axial translations; therefore the field must be also.
Conclusion: the field lines must form circles. Choose our amperian loop as a circle about the axis, radius r:
B is parallel to ds,
B.ds = B(r)ds
as loop integral of ds is the distance round the circle.
same as Biot-Savart Law!
Field inside a solenoid
A solenoid is a long tightly wound coil, which produces an almost uniform magnetic field.
Choose the rectangle abcd as an Amperian loop: only the length ab contributes
Total current iloop = nLi
n = number of windings per unit length.
From Ampere's law
field inside an ideal solenoid
The direction of the field is given by the right hand rule: fingers in the direction of the current, thumb gives direction of B.
Field due to a current loop
Near the loop the field is like that of the straight wire.
Current loop acts as a magnetic dipole, with direction given by the right hand rule.