Chapter 20: Magnetic field and forces
What will we learn in this chapter?
What is magnetism?
Magnetic fields and forces
Motion of charges in a magnetic field
Mass spectrometers
Magnetic forces on conductors
Magnetic forces on a loop
Magnetic field of a straight conductor
Magnetic forces between parallel
conductors
Magnetic field calculations
iron filings line up tangent
to the field lines
Why is magnetism important?
Technological relevance:
Most common electric motor technology relies
on magnetism
Generators require magnetism to produce
electricity
All disk drives are based on magnetic materials
Two types of magnets:
Permanent magnets: materials, which intrinsically have magnetic
moments.
Electromagnets: when charges move, a magnetic field is induced.
Monopoles?
Typical behavior of magnets:
Unlike poles attract
Same poles repel
What happens when we cut a magnet into two pieces?
S
N
S
N
+
SN
There are no magnetic monopoles. Note that this would have
huge implications in theoretical physics.
Earth: a huge magnet
The reason why a
compass points to the
north pole is because the north pole
is actually a magnetic south pole.
Earth’s axis is slightly offset
from the magnetic axis.
Earth behaves like a large
bar magnet. The field is
caused by currents in the
planet’s core. It periodically
flips (106 years…).
Birds and bacteria use this magnetic
field to orient themselves.
Brief history of magnetism
Permanent magnets (500 BC):
Fragments of magnetized iron are found near the city of Magnesia
(today Manisa, TK). When suspended from a string, the magnetized
iron would point north.
Electromagnetism (1819):
Hans Christian Ørsted (DK) observes that a compass needle is
deflected when placed next to a current-carrying wire.
I=0
I>0
I<0
Electric vs magnetic fields
Electric field:
A charge distribution at rest creates an electric field E at all points
in the surrounding space.
The electric field exerts a force F = qE
on any other charge q present in the field.
Magnetic field:
A permanent magnet, a moving charge, or a current
create a magnetic field B at all points in the surrounding space.
The magnetic field B exerts a force F on any
other moving charge or current present in the field.
Note:
Like the electric field, the magnetic field is a vector field.
Magnetic field lines
As for electric fields, we can draw magnetic field lines.
!
The magnetic field vector B
is tangent to the field lines.
The more dense the lines
the stronger the field.
At each point, the field lines
point in the same direction
as a compass would.
Because the field is unique,
field lines never cross.
Convention: Magnetic field
lines point away from N
poles and toward S poles.
Some example cases…
C-shaped magnet
The field is almost
uniform between the poles.
Loop and coil
similar field lines to a bar magnet...
Straight wire with current
Visualizing field lines…
Iron shavings can be used to visualize field lines.
field lines go from N to S
field lines go from N to S
Magnetic force
Magnitude of the magnetic force: When a charged particle moves with
! , the magnitude F of the force exerted
velocity !v in a magnetic field B
on it is
F = |q|v⊥ B = |q|vB sin(φ)
Here |q| is the magnitude of the charge and φ is the angle measured
!.
from the direction of !v to the direction of B
right index
right thumb
right middle finger
F =0
F = |q|v⊥ B = |q|vB sin(φ)
! the force is maximal. Note the difference to electric fields!
If !v ⊥ B
Magnetic force contd.
One can equivalently also write F = |q|vB⊥.
Thus the direction of the force is not specified completely.
Solution: introduce a right-hand rule…
Right-hand rule for magnetic forces (positive charge):
! with their tails together.
1. ! Draw the vectors !v and B
! until it points in the
2. ! Imagine turning !v in the plane containing !v and B
! . Turn it trough the smaller of the two possible angles.
! direction of B
3.! The force then acts along a line perpendicular to the plane
! . Using your right hand, curl your fingers around
! containing !v and B
! this line in the same direction (clockwise/counterclockwise) that
! you turned !v . Your thumb now points in the direction on of the
! force F! on a positive charge.
If the charge is negative, the force points opposite the direction given
by the right-hand rule.
Right-hand rule in pictures
Right-hand rule for positive charges:
Note: Mathematically, the force is actually given by the “cross
!.
product” (or “curl”) between the field and the velocity, F! = q !v × B
Negative charges
(left hand)
Different charges will move in opposite directions if they move in the
same direction in a magnetic field.
Units
The units of B must be the same as the units of F/qv, i.e., N/(Cm/s).
Because 1A = 1C/s it follows [B] = 1 N/(Am).
The unit for a magnetic field is called Tesla:
!
!
1 Tesla = 1 T = 1 N/(Am)
The cgs unit (often found) of B is gauss (1G = 10-4T).
Typical field strenghts:
Earth’s magnetic field is about 0.5G.
Permanent magnet or microwave oven: 0.1T.
MRI machines have about 1.5T fields (erase credit cards).
Strongest steady field in laboratories are around 45 - 60T.
How do MRI machines work?
Superconductors generate a
huge magnetic field of 1.5T.
By pulsing the field, the magnetic
moment of the hydrogen atoms
in the tissue flip back and forth.
Different tissues have different
amounts of hydrogen. Their
flipping is detected and the
figure is formed.
How is the signal of the atoms
detected?
Use use tiny superconducting pick-up coils which
can sense the faintest fields.
Many applications use beams of charged particles.
Common particle beam sources produce a range of
particle velocities.
How can we select a given particle speed?
Send the particles trough a capacitor (E field
points left) and a magnetic field (B field in plane).
When the fields are tuned properly F!E = F!B
−qe = qvB
It follows: v = E/B
Only particles with a speed of
E/B will fly trough the capacitor.
Thomson’s experiment: in 1894 Sir J. J. Thomson used this method to
determine the ratio e/me. He noted that it was independent of the
used material, i.e., he discovered subatomic particles. Later Millikan
computed e and thus the electron mass was determined.
Motion of charges in magnetic fields
When a charged particle moves in a magnetic field, B, v, and F are
always perpendicular to each other.
The magnitude of the magnetic force, qvB, does not change. Only the
direction with the aforementioned constraint.
It follows that the charged particles follow a
circular path on a plane perpendicular to the
field B.
The radial acceleration is v2/R. Using
Newton’s 2. law:
v2
F = |q|vB = m
R
The radius R of the circle is:
mv
R=
|q|B
Note: if q < 0 the particle moves clockwise.
particle source
Velocity selector
Motion in a magnetic field contd.
Angular velocity of a particle:
ω=
|q|B
m
ω = 2πf
Note:
The frequency is called the
cyclotron frequency.
It is independent of the radius R.
This is used in particle accelerators
(CERN, microwave oven, radar)
where particles are given a boost
twice each revolution.
Helical motion
If a particles moves in a uniform
magnetic field and the initial velocity
is not perpendicular to the field,
there is a parallel velocity component.
Because there is no force parallel to
the field, this velocity component is
constant.
The particle moves in a helix of
radius R:
mv⊥
R=
|q|B
This is the reason for aurora borealis:
Particles from the sun are trapped at
the poles and collide with air molecules
emitting light.
Recall that a velocity selector can select particles with
an exact velocity v = E/B .
Ions are sent trough slits S1 and S2 to make a narrow
beam.
The ions are sent into a region where only a given
magnetic field (no E field) perpendicular to the figure.
mv
It follows:
R=
|q|B !
If we know how many electrons an ion has lost,
we can determine m.
A counter then counts the number of hits
and a database search determines the
relative frequencies of the particles in the
unknown material.
This is how isotopes were discovered.
velocity selector
Mass spectrometers
Mass spectrometers contd.
Selected applications:
Isotope count (14C dating)
Trace gas analysis
Pharmacokinetics
Protein characterization
Space exploration
typical data set
Often shown in TV shows (CSI). Note that the process never is as
fast as it seems on TV…
Magnetic force on a conductor w. current
What makes an electric motor work?
How can we generate energy from e.g., water, wind, gas, fission?
The forces that make motors turn are forces a magnetic field exerts
on a current-carrying conductor.
Calculation of the force:
Assume positive charges in a wire of length l
and cross section A in a in-plane field B.
The force on a single charge is
F = |q|vB sin(φ)
Since the drift velocity vd, field B and force F are
perpendicular F = |q|vd B .
The total force can be computed by studying the
total charge moving in the conductor…
Magnetic force calculation contd.
The time needed for a charge Q to move from one end of the
conductor to the other is
l
∆t =
vd
The total charge flowing during this time trough
the wire is
Q = I∆t
F
=
Qv
B
Use now
:
d
F = Qvd B = (I∆t)(l/∆t)B = IlB
Magnetic force on a current-carrying conductor: The magnetic force on a
segment of length l carrying a current I in a field B is
F = IlB⊥ = IlB sin(φ)
The force is always perpendicular to both the conductor and the field,
with the direction determined by the right-hand rule.
Reversing the field or current
When the current in the wire or the
field B are reversed the force changes
direction (use the RH rule…).
What happens when the charges are
negative? Nothing! The sign of the
velocity and the charge cancel.
Note: moving the wire in a field produces a current (generator).
Force and torque on a current loop
For simplicity, assume the loop is rectangular.
Due to symmetry considerations, the force on the loop is zero, but
there is a net torque.
On the long side of the
loop there are forces
F and –F acting on the
wire with
|F | = IaB
Along the short sides the
force makes an angle 90◦ − φ
with the field, hence
F ! = IbB sin(90◦ − φ) = IbB cos(φ)
(along the y axis).
Force & torque on a current loop contd.
Since the magnitudes of the forces on opposing sides of the loops
are the same, but with different signs, they cancel.
The forces along the y axis act along the same line, i.e., no net
torque.
φ = 90◦
φ = 0◦
One can compute
the torque to find
τ = (IaB)(b sin φ)
When φ = 0◦ we have
a stable equilibrium
because the torque is maximal torque
zero torque
zero.
When φ = 180◦ we have an unstable equilibrium position.
Since the area of the loop is A = ab, we can write
τ = IAB sin φ
Force & torque on a current loop contd.
Changing the shape of the loop:
Any arbitrarily-shaped planar loop can be
approximated by a set of rectangular loops.
It follows that τ = IAB sin φ still holds with
A the area of the loop.
Multiple loops:
The effect of each loop adds up. Thus for
N loops
τ → Nτ
Force & torque on a current loop summary
Torque on a current-carrying loop: When a conducting loop of area A
carries a current I in a uniform magnetic field B, the torque exerted on
the loop is
τ = IAB sin φ
where φ is the angle between the normal to the loop and the field B.
For multiple loops (solenoid with N loops) τ = N IAB sin φ .
The torque tends to rotate the loop in the direction of decreasing angle
(stable equilibrium). The product IA is called the magnetic moment of
the loop:
µ = IA
For a solenoid with N windings µ = N IA .
Note:
A solenoid rotates in a field and
become parallel to the field (compass).
Direct-current (dc) motor
Now that we know how to generate torque on a wire, we can build
a motor…
Center: rotor of soft steel
which rotates.
Copper wires are
embedded in the rotor.
Brushes (graphite) pass
current onto the rotor
via the commutator
which switches the
currents on and off.
In the field coils F and F’
a steady magnetic field is
set up.
Direct-current motor contd.
The brushes transmit
current onto contacts
1 and 1’ to coil 1, which
is oriented to have
maximal torque.
When the brushes are between
contacts, inertia keeps the
motor running.
After a rotation, the brushes touch 1’ and 1, but with opposed
polarity, hence the torque is still counterclockwise.
Magnetic field of a long straight conductor
So far:
We have only studied forces caused by magnetic fields.
We did not worry how these fields are produced.
Now:
Study some simple cases,
such as a long straight
wire carrying a current I.
The field depends on
the distance r from
the wire and the current I.
Along a field line B has the
same magnitude.
RH rule: the thumb
points in the direction of the field.
Application: coaxial cables
Goal:
Shield sensitive electromagnetic signals from noise.
Avoid signal leakage.
Solution:
Place a wire inside a hollow conducting tube. If currents in both
conductors have the same magnitude and are opposed, fields
cancel.
Do not introduce kinks! Experiment at home with the TV…
Magnetic field of a straight conductor contd.
Magnetic field of a long straight wire: The magnetic field B produced by a
long straight wire carrying a current I at a distance r has a magnitude
B=
µ0 I
2π r
µ0 is the permeability of vacuum. Its numerical value depends on the
used units. For SI units
µ0 = 4π × 10−7 Tm/A = 4π × 10−7 N/A2
Note:
Do not confuse the permeability µ0 with the magnetic moment µ !
Remember: Magnetic field lines encircle the current that acts as
their source. In contrast, electric field lines originate at charges
which act as their source.
Force between parallel conductors
Motivation:
The problem appears in many applications.
Significant for the definition of the Ampere.
Setup:
Two wires with currents I and I’
in the same direction at a
distance r.
The magnetic field of the lower wire
exerts a force on the upper wire and
vice versa.
Note: If the wires had currents in the opposite directions, they
would repel each other.
Force between parallel conductors contd.
Determination of the force per length:
The magnitude of the B vector at points
on the upper conductor is
µ0 I
2πr
Use now that the force on a wire of
length l is given by F = IlB to obtain
!
"
µ
I
µ0 II ! l
0
!
!
F = I Bl = I
l=
2πr
2πr
B=
The force per unit length is thus
F
µ0 II !
=
l
2πr
Using the right-hand rule it follows that the wires attract (repel)
each other when current flows in the same (opposite) direction.
Example: superconducting wires
What is the force of two superconducting cables 4.5mm apart which
each carry 15000A?
Solution:
F
µ0 II !
=
= 1.0 × 104 N/m
l
2πr
Note:
For 1m of wire, this would be equivalent to one ton (!).
This is why the structural design of high-intensity magnets is
crucial.
FYI: the definition of the ampere
The forces that two straight parallel conductors exert on each other
form the basis for the official SI definition of the ampere:
Definition of 1 ampere: One ampere is that unvarying current which, if
present in each of two parallel conductors of infinite length and 1
meter apart in vacuum causes a force of exactly 2.10-7 N per meter
length on each conductor.
Note:
This SI definition agrees with the definition of the constant µ0 .
There is a proposal to redefine the ampere as a number of
electrons per second.
Current loops and solenoids
As we have seen in the case of electric motors,
many applications require current loops and coils
made from many circular loops.
Magnetic field in the center of a loop:
µ0 I
B=
2R
If we have a coil of N loops
B → NB
Note:
The field lines (determined by the RH rule)
are not circular.
Note the similarity with the equation for
a straight wire.
Magnetic field of a solenoid
A solenoid is basically a multi-wind coil where the length L is much
larger than the radius R of the windings.
The field is strongest in the
center and almost uniform along
the x-axis.
The magnetic field depends
on the number of turns per
unit length n = N/L.
At the center we find
B = µ0 nI
Remember: This expression is only valid for the case where L is much
larger than R.
Special case: toroidal solenoids
When the windings are tied very
closely, the field is enclosed
almost entirely in the area
enclosed.
If there are N turns then the
field B at a distance r from the
center of the toroid is
µ0 N I
B=
2πr
If the radial thickness of the
core compared to the radius of
the toroid is small, then the
field is uniform and B = µ0 nI
with n = N/2πr.
Law of Biot and Savart
So far: Only results for the B-field with different geometries.
These results stem from the law of Biot & Savart.
Description:
! produced by
The law of Biot & Savart gives the magnetic field ∆B
a current I in a tiny segment of conductor of length ∆!. The
vector∆!" is the line segment in the direction of the current.
The segment is called a source point, the place we want the field at
at a distance r, the field point.
! due to a
Law of Biot & Savart: The magnitude of the magnetic field ∆B
segment of conductor with length ∆! carrying a current I is
∆B =
µ0 I∆" sin θ
4π
r2
Law of Biot and Savart contd.
zero field
cross-section view
maximal field
Note: the right-hand rule applies.
Law of Biot and Savart contd.
How can the magnetic field for a large object be computed?
Solution:
Represent the conductor of length l as a
superposition of tiny wire segments ∆!.
! ‘s.
! is then the vector sum over all ∆B
The total field B
Note: mathematically one would perform a line integral along a
given path. This problem can usually only be solved analytically for
simple cases.
Here we sketch a simple example (conducting ring) where we can
avoid cumbersome integrations.
Field of a conducting ring revisited
Start from Biot & Savarts law:
µ0 I∆" sin θ
4π
r2
We can represent the loop as a sum
of tiny ring segments ∆!i .
Because of the ring geometry
! ⊥R
!
θ = 90◦
∆!" ⊥ B
∆B =
It follows:
µ0 I
B=
(∆"1 + ∆"2 + ∆"3 + . . .)
4πR2
Use the fact that the sum over line segments is the circumference of
the loop, i.e., ∆!1 + ∆!2 + ∆!3 + . . . = 2πR
We obtain:
µ0 I
B=
2R
Ampere’s law
closed loop
Ampere’s law provides an alternative
formulation of the relationship between
a magnetic field and its source.
It is analogous to Gauss’ law.
Ampere’s law is useful when the problem
has some symmetry which can be exploited.
parallel component
Ampere’s law: When a path is made up of a series of segments ∆s and
when the path links conductors carrying a total current Iencl
!
B! ∆s = µ0 Iencl
Note: If there are more conductors in the path, then the field is the
vector sum of the individual contributions,
whereas the current is
!
the algebraic sum. In general:
! s = µ0 Iencl
Bd!
Example of Ampere’s law: long straight wire
The magnetic field lines produced by a long straight wire with a
current are circles.
In this case the parallel component of the magnetic field is always
tangential to the circle.
Ampere’s law:
Result:
!
B! ∆s = B(2πR) = µ0 Iencl
B=
µ0 I
2πR
Magnetic materials
In general, magnetic materials can be classified in different categories:
Paramagnets (iron oxide, uranium, platinum, tungsten, oxygen, … )
Diamagnets (bismuth, copper, carbon, water, silver, lead, …)
Ferromagnets (cobalt, iron, nickel, gadolinium, dysprosium, …)
Antiferromagnets (chromium, iron-manganese, nickel oxide, …)
Ferrimagnets (magnetite, yttrium-iron-garnet, …)
Spin glasses (iron-gold alloys, diluted LiHo, …)
…
Most common are the first 3:
paramagnets, diamagnets and ferromagnets (“stick to fridge”).
Magnetic materials contd.
Paramagnetism:
All equations shown are valid for paramagnets.
The strength of the field is quantified with the permeability Km
which typically is 1.00000002 to 1.0004 for typical materials.
To take this into account, replace µ0 → µ = Km µ0 .
Diamagnetism:
In this case Km is smaller than 1, i.e., a deflection due to a field
would happen in the opposite direction.
Ferromagnetism:
Km is typically 103 to 104 due to cooperative
B=0
effects between domains.
Some materials retain the magnetization
and thus become permanent magnets.
B>0