PH202 – Winter 2006 – Lecture Notes - Bjoern Seipel
PH202 – Winter 2006 – Lecture Notes - Bjoern Seipel
Magnetism (Just another Aspect of Electricity!)
r
As for charges: there is a magnetic field B , field lines (the pictorial
r
representation of a magnetic field) and equipotential lines/planes. B is
always tangential to magnetic field lines.
“To pass on I will begin to discuss by what law of nature it comes about that
iron can be attracted by that stone which the Greeks called magnet from the
name of its home, because it is found within the national boundaries of the
Magnetes.
This astonishes men….
In matters of this sort many principles have to be established
Before you can give reason for the thing itself, you must approach by
exceedingly long and roundabout ways”
Difference to E-field: Field lines never start
or end. Field lines exit at the north pole and
enter at the south pole and continue in the
body of the magnet.
LUCRETIUS THE ROMAN EPICUREAN SCIENTIST - Roman philosopher at
beginning of the first century B.C.
Magnetic field can be monitored with a
compass needle (as we did with a positive
test charge in case of electricity)
Magnetism has been known since antiquity:
Iron oxide minerals like lodestone (Magnetite Fe3O4) can act as bar
magnets.
North pole of magnetic needle will point to magnetic south pole
What we will learn: Motion of electric charge creates a magnetic field
If charges or charged particles are moving, there will be magnetism
Geomagnetism
Earth produce, like many other places, an own magnetic field. North
pole of compass points towards geographical north direction.
Magnetic south pole of the Earth
is quite close to geographic North
pole
So magnetic north of compass
needle points to geographic
north of earth
(Everything is only a convention)
r
The magnetic field of the earth B
change with time (reversed a couple of times in past)
Magnets have two poles (North and South), they come always as a
pair. There are no magnetic monopoles (and that is fundamentally
different to electricity –> point charges).
As for charges: Like poles repel, unlike poles attract each other
If a magnet is broken into pieces, each piece
will have two poles no matter how small it is.
There is nearly homogenous magnetic field between two large parallel
magnets (horseshoemagnet)
1
2
PH202 – Winter 2006 – Lecture Notes - Bjoern Seipel
PH202 – Winter 2006 – Lecture Notes - Bjoern Seipel
The magnetic force on moving charges
The direction of the force
depends on the sign of the
charge. The force exerted on a
negatively charged particle is
opposite in direction to the
fore exerted on a positively
charged particle
Magnetic field does not affect charges at rest
F = ⏐q⏐ v B sin θ
Vector cross product, so F ⊥ v and
F⊥
r Br r
F =q v x B
or
B = F/(|q| v sinθ)
[T, Tesla] 1T = 1N/(Am)
A further unit for the magnetic
field is gauss: 1G = 10-4 T (Magnetic field of earth around 0.5G)
Magnetic force right hand rule for a positive point charge (to
determine the direction of the force on a moving charge or current)
r
r
The magnetic force points (vector product) perpendicular to B and v .
1. Fingers point in the
r
direction of v
2. Curle fingersrin the
direction of B
3. Thumb pointsr in the
direction of F
3
4
PH202 – Winter 2006 – Lecture Notes - Bjoern Seipel
PH202 – Winter 2006 – Lecture Notes - Bjoern Seipel
Motion of a particle in a magnetic field
(No magnetic force exert on particle when moving parallel or anti
parallel with respect to the magnetic field lines)
A positively charged particle
experience a constant
downward force
parabolic path
Circular motion
F = |q| v B sin 90°
Newton´s 2nd law
F=ma
A positively charged particle
entering a magnetic field
experiences a horizontal force
a right angles to its direction
of motion. In this case the
speed of the particle remains
constant.
acp = v2/r
centripetal acceleration
m v2/r = ⏐q⏐ v B
A constant magnetic field
cannot do work on a charged
particle because the force is
perpendicular to velocity.
W = F s cosθ and cosθ = 0
Radius of circular path
r = mv/(|q| B)
The faster the velocity or the mass the larger the circle (mass
spectrometer).
The greater the charge the smaller the circle
Charge in E-field and B-Field
Helical motion
No force or deflection occur on
the particle when electric and
magnetic force point in opposite
direction and have same
magnitude
Particle has a velocity with a
certain angle to magnetic
field.
Fel=Fmag
qE = qvB
v= E/B
(velocity selector, to measure precisely the speed of charged particles)
5
The velocity can be resolved
into parallel and
perpendicular components
with respect to magnetic
field.
Parallel component gives a
constant velocity and
perpendicular component
circular motion.
6
PH202 – Winter 2006 – Lecture Notes - Bjoern Seipel
PH202 – Winter 2006 – Lecture Notes - Bjoern Seipel
Superposition gives a helical motion.
NIA “magnetic moment”
Magnetic Force on a current carrying wire
Electric Currents/ Magnetic Fields and Amperes Law
F = |q| v B sin 90°
The connection between current
and magnetic field was discovered
by Hans Christian Oersted. He
discovered that a current creates a
magnetic field (His compass
deflected when it came close to
the a current carrying wire).
q = I Δt
time for charge travelling through wire
Δt = L/v
F = (I L/v) v B sinθ
Magnetic force on a current carrying wire
F = I L B sinθ [N]
Direction of magnetic field
produced by a current through
a long straight wire:
The wire will bent!!
Loops
Magnetic Field Right Hand Rule
To find the direction of the magnetic
field due to a current carrying wire,
point the thumb of your right hand a in
the direction of the current (+ -).
Then your fingers are curling around
the wire in the direction of the magnetic field.
⏐B⏐ ∼ ⏐I⏐
the larger the electric current,
the larger the magnetic
induction (field strength)
If there is a rotation axis at center of
loop, it is obviously that the forces
exert a torque (τ=F w/2) for each
vertical segment. Sum of torque
produced by each segment:
⏐B⏐ ∼
1
r gets smaller further
away from wire
τ = I h B (w/2) + I h B (w/2)
τ = I B hw = I B A
⏐B⏐=
Torque exerted on
rectangular (and any other planar)= loop: τ = I B hw = I B A sinθ
Torque exerted on a general loop of area A and N turns
τ = I B hw = N I A B sinθ
μo
I
2π r
μ0= 4 π 10-7 Tm/A (permeability of free space, a fundamental
constant of magnetism)
7
8
PH202 – Winter 2006 – Lecture Notes - Bjoern Seipel
PH202 – Winter 2006 – Lecture Notes - Bjoern Seipel
So Ampère’s Law becomes:
Bparallel ΔL1 + Bparallel ΔL2 + Bparallel ΔL3 + Bparallel ΔL4 + … = μ0 I
Ampère’s Law
Bparallel Σ Li = μ0 I = Bparallel 2 π r
(some similarity to Gauss’s
law, but over a loop not a
surface, a fundamental
law relating a current to
the magnetic field it
creates)
since I always perpendicular to B,
B = Bparallel , i.e. μ0 I = B 2 π r
μ0 I
e.g. rearranged for a straight wire, we get B = 2π ⋅ r
The tangential component
of the magnetic field
vector around any closed
path is equal to the
product of the permeability
(of free space, μ0) and the
enclosed net current that
pierces the loop.
if the close path (e.g. circular loop with radius R) is inside a
wire of any shape we get after rearrangements
μ0 I R
B = 2π ⋅ R 2 ⋅ r
Σ Bparallel ΔL ≈ μ0 I
If we add an (infinitely) large number of (infinitesimal) small scalar
products B Δs along any close path outside a wire of any shape we get
μ0 I (times μr to be accurate) a constant times the net current that
flows trough area enclosed by path
i.e. the magnetic field depends on the current IR within the
enclosed path – but there is a magnetic field inside a current
conducting wire, on the other hand, Gauss’s law states that
there can’t be any electric field inside a conductor
μ0= 4 π 10-7 Tm/A (permeability of free space, a fundamental
constant of magnetism)
very important: it does not matter what the path is, any close path
will do
From that we can derive expressions for magnetic field B, around
wires of any shape, but if we have a wire, we take a circle as a closed
path, locate the circle perpendicular to the axis of the wire, and B will
be tangent at any point to that circle
9
10
PH202 – Winter 2006 – Lecture Notes - Bjoern Seipel
PH202 – Winter 2006 – Lecture Notes - Bjoern Seipel
Force between two current carrying parallel
wires
Definition of an Ampere:
1 Ampere is that unvarying current which, if present in each
of two parallel conductors of infinite length and one meter
apart in empty space, causes each conductor to experience a
force of exactly 2 10-7 N m (- it is one of the basic SI units)
If there are currents in two
parallel wires, each of
them will set up a
magnetic field, each of the
magnetic fields will affect
the moving electrons in the
Ampère’s Law for a loop / an ideal solenoid
Loop:
B = μ0 I N/(2R) in the center of circular loop
other wire (moving charge
experiences a force in a
magnetic field).
We had ⏐B⏐=
=lIB
Iμ o
2πr and F
Combining the two gets us:
r
rr
F
I1 I 2 μ 0
l = 2πr
Force per unit length [ N m ]
if I1 parallel I2 both wires will bent towards each other
if I1 anti-parallel to I2, both wired will bent away from each other
Solenoid:
Nothing fancy, just coils of wire in the geometry of a helix which
contain an electric current, so
there will be a magnetic field
and we want to calculate it’s
Magnetic field B
11
12
PH202 – Winter 2006 – Lecture Notes - Bjoern Seipel
PH202 – Winter 2006 – Lecture Notes - Bjoern Seipel
Ideal scenario: solenoid is infinitely long, i.e. magnetic field outside
is negligible
Magnetism in Matter
Closed path is composed of 4 segments, remember I can choose any
path for the calculation
Electron orbiting around
the nucleus
Magnetic field
(very small)
Spin up or spin
down
Spin up + spin
down = zero net
field
ab – outside solenoid, parallel to
the solenoid axis, infinitely far
away
bc – perpendicular to the
solenoid axis, outside – in,
infinitely long
Ferromagnetism
For certain atom like iron, nickel or cobalt the net spin is
nonzero. In those kind of materials the magnetic spin is strong
enough to self align the magnetic atom (areas with aligned
atoms are called magnetic domains). This ability is called
ferromagnetism the material ferromagnets (latin: ferrum =
iron).
cd - inside the solenoid, parallel
to the solenoid axis, length l
da – perpendicular to the solenoid, inside – out, infinitely long
Bparallel1 Δsab + Bparallel2 Δsbc + Bparallel3 Δscd + Bparallel4 Δsba ≈ μ0 I
Bparallel3 = small, if the solenoid becomes infinitely long, this
projection of B becomes zero
Bparallel2 = Bparallel4 = zero because there is no parallel B component
either if the solenoid becomes infinitely long
Paramagnetism
Net spin is non zero but the effects are very small. It needs a
strong external magnetic field to align atoms.
Diamagnetism
Is the effect of the production of the production by a material
of a magnetic field in the opposite direction to an external
magnetic field. All materials show at least a small diamagnetic
effect.
with l = length cd, N number of current currying coils
B = μ0 I N/l , or with n = number of coils loops per unit length = N l
B = μ0 n I
independent from cross sectional area
Use of solenoid: Electromagnet
13
14
PH202 – Winter 2006 – Lecture Notes - Bjoern Seipel
Power Plant – a nice piece of physics
German “Kraftwerk” = Force factory
we start with potential energy, which can be either
gravitational (water in a dam), chemical (coal, oil, gas),
on nuclear (U235 92 or Pu94)
we convert potential energy into kinetic rotational energy
we put a metal wire (with charges that are free to move and
which is connected to the ground on one side and to the
grit on the other side) into a magnetic field – and let it be
turned around and around by kinetic rotational energy we
gained from potential energy
moving wire in magnetic field results in separated
charges, i.e. an electric potential difference (voltage), an
electric field, electric energy, electricity
effectively, with our rotational mechanical energy we pump
electrons out of the ground and up to the grit
electrons are like charges, so they disperse themselves all
over the grit
electric energy - that is stored in electric field – with is due
to separated charges - is sold to us as electricity –
which we then have work done for us!
15