Transcript Slide 1

Lecture 7
o Aim of the lecture
 Introduction to
 Magnets
 Magnetic field, B
 Interaction between current and B field
 Force
 Right Hand Rule
o Main learning outcomes
 familiarity with
 Magnets
 Magnetic Field
 Lorentz Force
Magnets:
o Everyone (should be) familiar with magnets
 Every permanent magnet has
 Two ‘poles’
 Called North and South poles
 Two North poles or two South poles repel each other
 Opposite poles attract each other
This origin of this force can be represented with field lines,
Just as for electric charges.
The field is a magnetic field
(as opposed to an electric field)
These filed lines can be visualised using ‘iron filings’, as in the picture
o Magnets like this are called
 Permanent Magnets
 Come in many shape and sizes
The earth is a big dipole
magnet
And the sun
o In fact without the earth’s magnetic field
o Life on earth would not exist
o The field protects us from most
 cosmic rays (high energy charged particles from space)
 why – see later
Some animals, such as this loggerhead turtle, can
even directly detect the earth’s field and use it to navigate
o All these magnets are characterised by being
 Dipoles
 Always have a North and a South
 Very similar to an electric dipole:
Electric Dipole
Magnetic Dipole
o Far away ( >3 times pole separation) from dipole
Electric and Magnetic Field lines have same pattern
o Close to dipole
 Field lines differ because
 Magnetic lines are in a magnetic medium
o If the electric dipole charges had a dielectric between them
The field lines could look identical for both
Important Points
 There are no observed magnetic monopoles
 Only magnetic dipoles
 Magnetic dipoles always have a magnetic medium between them
 Magnetic fields are analogous to Electric Fields
 For a dipole they differ in shape close to magnet only because of
the magnetic medium that makes up the magnet
Interesting fact: There is no theoretical reason why monopoles should not
exist. Particle Physicists look for them. It is thought they may just be very
(very!) heavy and therefore difficult to make.
Classical atomic hydrogen model
Electron in orbit around proton
Electon is charged
Electron ‘moving’
Therefore a current loop
Therefore a magnetic dipole moment
If all the ‘small’ dipoles in an object were to point the same way
It would make a macroscopic permanent magnet (a dipole)
This is how we have permanent dipole magnets
But no magnetic monopoles
All the magnetic fields we observe are cause by moving charges
Note: this is NOT a good model of the reality of the
origin of atomic magnetic dipole moments
Quantum mechanics needs to be used
It is the spin angular momentum which matters most
Spin is an entirely quantum mechanical effect
It is ‘sort of’ the ‘internal’ spin of the electron
More is beyond the scope of this course
James Clerk Maxwell
If we had magnetic monopoles, then a
monopole moving in a loop
would produce a dipole ELECTRIC field.
Electricity and Magnetism are both aspects
of the same force,
Electromagnetism – and it is described by
Maxwells Equations
Permanent Magnets can be made only from a few materials
Iron (Fe)
Nickel (Ni)
Cobalt (Co)
gadolinium and dysprosium (at low temperature)
Today’s permanent magnets are made of alloys. Alloy materials include
• Aluminium-Nickel-Cobalt (Alnico)
• Neodymium-Iron-Boron
(Neodymium magnets or "super magnets",
a member of the rare earth category)
• Samarium-Cobalt (a member of the rare earth category)
• Strontium-Iron (Ferrite or Ceramic)
The reason that these materials exhibit this property is
They have unpaired electron spins in the atom
Which spontaneously align in the bulk material
The spin is associated with a dipole magnetic moment (see later)
So all the ‘small’ dipoles align and add up to a macroscopic one
This is real data!
It shows measurements
of the spin of individual
atoms in a material
Atomic separation is just
a few angstoms (10-10m)
When spins align the
‘mini’ dipoles add up to
give a permanent magnet
Magnetic Field
Magnetic Field is given the symbol B
The unit of magnetic field strength is the Tesla, T
In fact there is a complication. There are two types of magnetic field,
The B-field and the H-field.
We will not consider this in the course, the H-field is different only
inside magnetic media
In free space the B-field and the H-field only differ by a constant.
This is also usually true inside a magnetic medium as well.
Note that B is a vector quantity
B
The Tesla is a big unit
An MMR machine for medical use
has a field ~2T
The earth’s field is ~ 30mT – 60mT
at the surface
The worlds strongest continuously
operable magnet is 45T
Stronger is possible, but the
magnets explode when operated,
so the strong field is only available
for a short time.
Picture shows a 1000T magnet,
Which provides 1ms of operation.
Some other examples for reference
Factor
(tesla)
SI prefix
Value
Item
10−18
attotesla
5 aT
SQUID magnetometers on Gravity Probe B gyros measure fields at this level
10−15
femtotesla
2 fT
SQUID magnetometers on Gravity Probe B gyros measure fields at this level in about one second
10−12
picotesla
0.1 - 1.0 pT
human brain magnetic field
10−9
nanotesla
nT
magnetic field strength in the heliosphere
10−6
microtesla
24 µT
strength of magnetic tape near tape head
10−5
31 µT
strength of Earth's magnetic field at 0° latitude (on the equator)
1/1000
5 mT
the strength of a typical refrigerator magnet -
1 T to 2.4 T
coil gap of a typical loudspeaker magnet[3].
1.25 T
strength of a modern neodymium-iron-boron (Nd2Fe14B) rare earth magnet.
1.5 T to 3 T
strength of medical magnetic resonance imaging systems
45 T
strongest continuous magnetic field yet produced in a laboratory
2.8 kT
strongest (pulsed) magnetic field ever obtained (with explosives)
1
tesla
100
1000
kilotesla
AN ‘ORDINARY’ MAGNET
More on units
Because the SI unit, the Tesla, is so large,
it is common for magnetic fields to be
quoted in a different (non-SI) unit,
the Gauss, G
104 G = 1 T
Nicola Tesla
Carl Feidrich Gauss
Magnetic Field and Current
The origin of permanent magnetic field is
The dipole moments of atoms
These are related to ‘moving’ electric charges in the atom
A current (moving charge) has a magnetic field associated with it
B
Electron flow
I
Conventional current flow
Fingers point in the direction of
the B field.
The Right-hand Thumb Rule applies to
Conventional current (ie flow of positive charge)
Remember this is conventional current, not electron flow.
m0 is called the
permeability of free space
it is a constant of nature
B = m0 I
2pr
Imagine sitting on one of the
moving charges.
Then you would see no magnetic field
as from your frame of reference the
charge would be stationary.
You would see an electric field.
So electric and magnetic fields are
VERY closely related.
A magnetic field is actually just a relativistic transformation of an electric field
- this is well beyond the scope of this course – but interesting!
Solenoids
By coiling a wire into
a solenoid
A uniform B field can
be created inside
This is an example of how particular magnetic field geometries
can be created. A uniform field in this case.
n – number of
turns of wire
I – current in wire
m0 permeability
of free space
k is the relative permeability, it is a property of
the material, and is usually given the symbol mr
The ATLAS detector
at the LHC has a large
solenoid to enable it
to measure charged
particles
The ATLAS solenoid
creates a 1.5T uniform B field in
a very large volume.
B
The energy stored is huge.
Magnetic fields store energy,
just like electric fields – see later
B
These are coils of a ‘torroidal’ magnet in ATLAS - a different field configuration
used to measure the muon particles.
Forces on charged particles in a magnetic field
A charged particle moving in a B field experiences a force:
The direction of the force is perpendicular to the motion
The direction of the force is perpendicular to the B field
The right hand rule can be used to relate the directions
This force is called
The Lorentz Force
The force is given by:
charge
velocity
B-field
Prof. Lorentz
The vector equation above uses a cross product,
(NOT a dot product) to combine the two vectors.
Electric and Magnetic forces
combined
• Because the force is always
perpendicular to the velocity
• speed never changes
• Particle will circle in uniform B
A x (blue here) indicates that
the B field is pointing INTO page
A . would mean it was pointing
out of the page
This is the basis for keeping
charged particles circling in an
accelerator like the LHC.
Force on a Current
This is the basis of an
electric motor
• A current is simply a ‘line’ of moving charge, so
• A wire will have a force on it
General case is:
= ILB for simple case
Where
• I is current,
• L length of wire in the field
• B is the (uniform) B-field
why
Force on a charge = qvxB
A current = quantity of charge crossing a point per unit time
= Qv where Q is the linear charge density
= qv / s where q is total charge, s is linear length
so Is = qv
Where we have defined I to be a vector current having a
direction as well as a magnitude.
Hence we can write F = qvxB as IsxB
The force on a short length, dL can then be written as
F = I(L)dLxB where
I(L) is the magnitude of the current and
dL points in the direction of the current flow
Arbitrary shaped Wires
For a current loop in a uniform B-field
• The net force will be zero
• Independent of its shape
• Because the integral round a closed loop must be 0
• But the torque is not zero (otherwise motors would not work!)
Basic DC motor
Actual motor has
• several coils
• Several magnets
• ‘Capacitors’ to stop interference
Slip rings and ‘brushes’
organised to ensure that
force is always in same
(rotational) direction on coil
Maxwell's Equations
We will explain these
by the end of the course
The ‘easiest’ of Maxwell’s equations is
divB = 0
also written as
.B=0
If F = Ui + Vj + Wk
where I,j,k are orthogonal unit vectors, then
o What this says is that there is no source of magnetic field in the theory,
o it is the same as saying that there are no magnetic monopoles
o At any point in space the number of B-field lines ‘in’ = number ‘out’
Like conservation of charge, quantity of current into node = quantity out
If there is no electric charge present, then can also say
.E=0
but this is not the general case.
Maxwell’s equations in free space (no electric charges present and
in vacuum)
o What this equation is saying is that the source of the electric field is
the charge.
o It relates the charge density r to E
o The divergence of the electric field is equal to the local charge density
o Integrating the equation shows that
the charge contained inside closed surface is proportional to the
total electric field penetrating the surface.
We will return
to the coffee
cup later.
To finish the
second half!