Objectives: by the end of this you will be able to
Understand some of the history of magnetism
FInd the force on a charged particle moving in a magnetic field
Find the force on a wire and hence on a loop of current
Find the magnetic field in some simple cases
Understand why there are no magnetic poles
Solve some simple examples
PHYS1008 Magnetic Forces
Magnetism is hard because for the first time it is really 3-D. Hence visualizing fields and forces is difficult. we will divide it up into
Magnetic Forces
Magnetic Fields
Induction
Historically:
First observed as "lodestones": lumps of magnetic iron-ore (magnetite,
or Fe₃O4). Can be suspended from a string and will point
North. Used by Vikings. "Letter on the Magnet" Petrus Peregrinus written
in 1269.
Basic Observations:
Some materials act as if they contain magnetic charges, or poles.
Note (confusingly): the north pole on a compass magnet points north, so it must be attracted to a south pole so the magnetic pole in the north of Canada is a south magnetic pole.....
With electrostatics, our process was:
Force between charges⇒
Field of charge⇒
Potential
With magnetism the relation is much more complicated, so we will go:
Field ⇒
Force on charge ⇒
Origin of field
(and we aren't even going
to get to a potential!).
Units: SI unit is Tesla, but still find fields quoted in Gauss: 104 G = 1 T.
For comparison:
Smallest detectable field ~ 10-12 G
Fields produced by currents in brain ~ 10-9 G
Earths field ~ .5 G
Permanent magnets ~ 10 3 G
Largest fields ever ~ 5x105 G ~ 50 T
What does a magnetic field look like?
We can "see" magnetic fields in various ways: e.g. magnetic fields are very important in sun: this shows a "coronal loop" (from TRACE project), which is (roughly) a dipole field produced by sunspots
Magnetic force on
a single charge
Simplest Electrostatic field is a uniform one, produced by two charged plates
Can make a uniform field with permanent magnets
or
so we will start with a uniform mag. field
Force on a single Particle
We will start with a single particle:
Field Out of Screen (Points of Arrows!). Experiments show that
No force if charge q = 0
No force if vel. of charge v = 0
Force ⊥ field B
Force ⊥ velocity v
Only way to satisfy this is
F = qvBsin(θ)
where θ is the angle between the field and the velocity
Note
Force on electron is in opposite direction.
No force if particle moves along magnetic field v || B ⇒ sin(θ) = 0
For an proton:
Force ⊥ Vel.
causes circular motion ⇒
or in equations:
mv² = qvB so mv = rqB
r
or
r = mv
qB
r is the cyclotron radius.
e.g. a proton with v = 107ms-1,B = 10²G: what is the radius?
If we have a positive and negative charge entering a
region of constant mag. field, what happens?
How do we define the direction?
Right hand
Point fingers along velocity
Curl the fingers towards the magnetic field
Thumb points along force
Suppose motion is not ⊥ field
B: vparallel will be unaffected so
linear motion + circular motion ⇒ helical motion
Note:
F ⊥ to Velocity v
F ⊥ Mag. Field B,
hence mag fields do no work. (why?)
This means you cannot change the K.E. of a particle with a mag. field (why?)
Non-Uniform Fields
What happens if the field is not uniform?
Non-Uniform Mag. Field gives rise to Magnetic Mirror
At this point, Velocity is into screen:
Force is directed to centre and away from converging field lines,
so particle is reflected
Mag Bottle Consists of 2 Mag. Mirrors Confines charge particles
--> fusion???
Van Allen belts
Electrons (and Protons) Bound in Belts Round Earth Reflected from North to South
Pole by and back by magnetic mirror Effect
Hence "belts" of particles form near earth of particles. It is this process that leads to Aurora
Aurora from Space (NASA picture from shuttle)
Jupiter Aurora (NASA picture from Galileo)
Force on Wire
This is for one particle: what happens to a wire?
How does the force on a charge turn into a force on a wire?
Each charged particle will feel force due to mag field, but charges are confined to wire, so
force is applied to wire.
Force on wire, length L,
F = B I L
= Field x current x length, provided everything is at right angles).
Seen from "above": forces on side add to give torque:
force
F = B I a
so torque
τ = -B I (ab) sin(θ)
= -BIA sin(θ)
(A = area of loop)
This quantity (IA) is important: it is the "magnetic moment" of the loop.
e.g.
suppose we have an electron in orbit in an atom with radius of orbit r: what
is the mag. mom.?
Area of loop
A = πr²
Current =charge = q = qv
time 2πr/v 2πr
so mag. mom.
M = IA = qv π r²=qrv
2πr 2
but angular momentum L = mvr,
so
M = qL
2m
(will return to this later on when we talk about atoms)
Applications of this:
Galvanometer: uses torque on loop to counterbalanced by spring to measure current
Motors:
A D.C. electrical motor consists of many loops of wire in a mag. field, Current
flows in both sides of coil, producing torque.
with a device (commutator) to flip direction of flow.
Max. torque in (b), in (c) approaches "dead" position with no torque
and the in (d) direction of current flips
Hall Effect
Path of electrons in wire is altered by Mag. Field B out of screen
Both + and - charge carriers drift to top until
Electrostatic Force FE
balances Magnetic Force FB or
qE = q v B
Negative charges would give a negative Hall voltage. What is measured is Hall
voltage: sign measures sign of charge carrier, and confirms metals have electrons,
but semi-conductors have +ve carriers (holes)
Now we really need to worry about where magnetic fields come from.
The magnetic field created by the moving charges in a long, straight current-carrying
wire curls around the wire, and falls off as I/r. The direction of field is given by right-hand rule:
If the right thumb points in the direction of conventional current
(red arrow: moving positive charges), the fingers curl around the wire
in the direction of the magnetic field.
B = 2 k'I = μ₀ I
r 2π r
We introduce this quantity
μ₀ = 4πx10-7 Tm A-1
for the same reason as we introduced ε₀: it makes the more important equations simpler.
Suppose we have two parallel wires carrying currents. Can think of one as producing the field which the other then feels.
Suppose we have two parallel wires carrying currents. Can think of one as producing the field which the other then feels.
What direction will the force be in? (Hint: you have to assume that one wire, say I₁, produces the field, and the second one I₂ feels it)
B = μ₀ I
2π r
F = B I' L
to give
F = μ₀ I I' L
2π r
Note that it doesn't matter which wire we think of as producing the field and which feeling it. What happens if we reverse one of the currents? This gives us the real definition of the ampere: two wires carrying a current of 1 A , separated by 1m have a force of ....
In E.S., the detailed distribution of charge didn't matter too much at large
distance
Field due to current loop: want the field on the axis at a large distance from the loop. Do this by adding up the contribution due to all the parts of the loop
We have already introduced Mag. Mom. of loop
At large distances the field is the same.
as that due to two equal and opposite charges
You can't tell whether this arises from a dipole (i.e. 2 charges)
or from a current loop.
Even if mag. charges don't exist,
we can still get fields that look as though they do exist!
One more very important aspect of Magnetism is Induction
