Chapter 19: Magnetic Forces and
Fields
•Magnetic Fields
•Magnetic Force on a Point Charge
•Motion of a Charged Particle in a Magnetic Field
•Crossed E and B fields
•Magnetic Forces on Current Carrying Wires
•Torque on a Current Loop
•Magnetic Field Due to a Current
•Ampère’s Law
•Magnetic Materials
1
Magnetism (Just another Aspect of Electricity!)
“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”
LUCRETIUS THE ROMAN EPICUREAN SCIENTIST - Roman philosopher
at beginning of the first century B.C.
Magnetism has been known since antiquity:
Iron oxide minerals like lodestone (Magnetite Fe3O4) can act as bar
magnets.
What we will learn: Motion of electric charge creates a magnetic field
If charges or charged particles are moving, there will be magnetism
2
Magnetic Fields
All magnets have
at least one north
pole and one south
pole.
Magnetic Dipole
Field lines emerge from north
poles and enter through south
poles.
As for charges: there is a magnetic field , field
lines (the pictorial representation of a magnetic
field) and equipotential lines/planes. is always
tangential to magnetic field lines.
3
Magnets exert forces on one another.
Opposite magnetic poles attract and like magnetic
poles repel.
4
Magnetic field lines are closed loops. There is no (known!)
source of magnetic field lines. (No magnetic monopoles)
If a magnet is broken in half you just end up with two
magnets.
5
Nearly homogenous
magnetic field between
two large parallel
magnets
(horseshoemagnet)
6
Geomagnetism
Near the surface of
the Earth, the
magnetic field is
that of a dipole.
Note the
orientation of
the magnetic
poles!
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)
The magnetic field of the earth change with time
7
Away from the Earth, the magnetic field is distorted by the
solar wind. The solar wind is a stream of charged
particles (i.e., a plasma) which are ejected from the upper
atmosphere of the sun. It consists mostly of high-energy
electrons and protons (about 1 keV)
Evidence for magnetic pole reversals has been found on the
ocean floor. The iron bearing minerals in the rock contain a
record of the Earth’s magnetic field.
8
Magnetic Force on a Point Charge
The magnetic force on a point charge is:
FB = q(v × B )
The unit of magnetic field (B) is the tesla (1T = 1 N/Am).
A further unit for the magnetic field is gauss: 1G = 10-4 T
(Magnetic field of earth around 0.5G)
9
The magnitude of FB is:
FB = qB(v sin θ )
where vsinθ is the component of the velocity perpendicular
to the direction of the magnetic field. θ represents the angle
between v and B.
v
Draw the vectors tailto-tail to determine θ.
θ
B
10
The direction of FB is found from the right-hand rule.
For a general cross product:
C = A×B
The right-hand rule is: using your right hand, point your
fingers in the direction of A and curl them in the direction
of B. Your thumb points in the direction of C.
Note : C = A × B ≠ B × A
11
Example (text problem 19.15): An electron moves with
speed 2.0×105 m/s in a 1.2 Tesla uniform magnetic field. At
one instant, the electron is moving due west and
experiences an upward magnetic force of 3.2×10-14 N.
What is the direction of the magnetic field?
y
FB = qBv sin θ
FB
sin θ =
= 0.8323
qBv
∴θ = 56°
θ
v (west)
θ
F (up)
x
The angle can be either north
of west OR north of east.
12
§19.3 Charged Particle Moving
Perpendicular to a Uniform B-field
⊗ ⊗ ⊗
⊗ ⊗ ⊗
⊗ ⊗ ⊗
⊗
⊗
⊗
⊗
⊗
⊗
⊗
⊗
⊗
⊗ ⊗ ⊗
⊗ ⊗ ⊗
⊗
⊗
⊗
⊗
⊗
⊗
A positively charged particle
has a velocity v (orange
arrow) as shown. The
magnetic field is into the page.
The magnetic force, at this instant, is shown in blue. In this
region of space this positive charge will move CCW in a
circular path.
13
A positively charged particle
experience a constant downward
force (parabolic path)
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.
A constant magnetic field cannot
do work on a charged particle
because the force is
perpendicular to velocity.
14
Applying Newton’s 2nd Law
to the charge:
∑F = F
B
= mar
v2
qvB = m
r
Radius of circular path
r = mv/(|q| B)
15
Mass Spectrometer
B
A charged particle is
shot into a region of
known magnetic field.
⊗ ⊗ ⊗
⊗ ⊗ ⊗
⊗ ⊗ ⊗
⊗
⊗
⊗
⊗
⊗
⊗
⊗
⊗
⊗
⊗ ⊗ ⊗
⊗ ⊗ ⊗
⊗
⊗
⊗
⊗
⊗
⊗
Detector
v2
Here, qvB = m
r
or qBr = mv
V
Particles of different mass will
travel different distances before
striking the detector. (v, B, and
q can be controlled.)
16
Other devices that use magnetic fields to bend particle
paths are cyclotrons and synchrotrons.
Cyclotrons are used in the production of
radioactive nuclei. For medical uses see the
website of the Nuclear Energy Institute.
Synchrotrons are being tested for use in
treating tumors.
17
Motion of a Charged Particle in a
Uniform B-field
If a charged particle has a component
of its velocity perpendicular to B, then
its path will be a circle. If it also a
component of v parallel to B, then it
will move forward as well. This
resulting path is a helix.
18
Crossed E and B Fields
If a charged particle enters a region of space with both
electric and magnetic fields present, the force on the
particle will be
F = Fe + FB
= qE + q(v × B ).
19
Consider a region of space with crossed electric and
magnetic fields.
Charge
q>0 with
velocity v
⊗ ⊗ ⊗
⊗ ⊗ ⊗
⊗ ⊗ ⊗
⊗
⊗
⊗
⊗
⊗
⊗
⊗ B (into page)
⊗
⊗
⊗ ⊗ ⊗
⊗ ⊗ ⊗
⊗
⊗
⊗
⊗
⊗
⊗
E
20
The value of the charge’s speed can be adjusted so that
Fnet = Fe + FB = 0.
The net force equal zero will occur when v=E/B.
This region of space (with crossed E and B
fields) is called a “velocity selector”. It can be
used as part of a mass spectrometer.
21
Magnetic Force on a Current
Carrying Wire
The force on a current carrying
wire in an external magnetic field
is
F = I (L × B )
L is a vector that points in the
direction of the current flow. Its
magnitude is the length of the wire.
22
The magnitude of
F = I (L × B )
is
F = ILB sin θ
and its direction is given by the right-hand rule.
23
Example (text problem 19.43): A 20.0 cm by 30.0 cm loop
of wire carries 1.0 A of current clockwise.
(a) Find the magnetic force on each side of the loop if the
magnetic field is 2.5 T to the left.
I= 1.0 A
Left: F out of page
Top: no force
B
Right: F into page
Bottom: no force
24
Example continued:
The magnitudes of the nonzero forces are:
F = ILB sin θ
= (1.0 A )(0.20 m )(2.5 T )sin 90°
= 0.50 N
(b) What is the net force on the loop?
Fnet = 0
25
Torque on a Current Loop
Consider a current carrying loop in a magnetic field. The
net force on this loop is zero, but the net torque is not.
Axis
Force
into
page
B
L/2
L/2
Force
out of
page
26
The net torque on the current loop is:
τ = NIAB sin θ
N = number of turns of wire in the loop.
I = the current carried by the loop.
A = area of the loop.
B = the magnetic field strength.
θ = the angle between A and B.
NIA “magnetic moment”
27
The direction of A is defined with a right-hand rule. Curl
the fingers of your right hand in the direction of the current
flow around a loop and your thumb will point in the direction
of A.
Because there is a torque on the current loop, it must have
both a north and south pole. A current loop is a magnetic
dipole. (Your thumb, using the above RHR, points from
south to north.)
28
Magnetic Field due to a Current
Moving charges (a current) create
magnetic fields.
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).
29
The magnetic field at a distance r from a long, straight wire
carrying current I is
μ0 I
B=
2πr
where μ0 = 4π×10-7 Tm/A is the permeability of free space.
The direction of the B-field lines is given by a right-hand
rule. Point the thumb of your right hand in the direction of
the current flow while wrapping your hand around the wire;
your fingers will curl in the direction of the magnetic field
lines.
30
A wire carries
current I out of
the page.
The B-field lines of
this wire are CCW.
Note: The field (B) is tangent to the field lines.
31
Example (text problem 19.62): Two parallel wires in a
horizontal plane carry currents I1 and I2 to the right. The
wires each have a length L and are separated by a
distance d.
1
I
d
2
I
(a) What are the magnitude and direction of the B-field of
wire 1 at the location of wire 2?
μ 0 I1
B1 =
2πd
Into the page
32
Example continued:
(b) What are the magnitude and direction of the magnetic
force on wire 2 due to wire 1?
F12 = I 2 LB1 sin θ
μ 0 I1 I 2 L
= I 2 LB1 =
2πd
F12 toward top of
page (toward wire 1)
(c) What are the magnitude and direction of the B-field of
wire 2 at the location of wire 1?
μ0 I 2
B2 =
2πd
Out of the page
33
Example continued:
(d) What are the magnitude and direction of the magnetic
force on wire 1 due to wire 2?
F21 = I1 LB2 sin θ
μ 0 I1 I 2 L
= I1 LB2 =
2πd
F21 toward bottom of
page (toward wire 2)
(e) Do parallel currents attract or repel? They attract.
(f) Do antiparallel currents attract or repel? They repel.
34
The magnetic field of a current loop:
The strength of the B-field at
the center of the (single) wire
loop is:
B=
μ0 I
2R
35
The magnetic field of a solenoid:
A solenoid is a coil of wire that is wrapped in a cylindrical
shape.
The field inside a solenoid is nearly uniform (if you stay
away from the ends) and has a strength:
B = μ 0 nI
Where n=N/L is the number of turns of wire (N) per unit
length (L) and I is the current in the wire.
36
37
Ampère’s Law
Ampère’s Law relates the magnetic field on a path to the
net current cutting through the path.
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
38
Example (text problem 19.65): A number of wires carry
current into or out of the page as indicated.
(a) What is the net
current though the
interior of loop 1?
39
Example continued:
Assume currents into the page are negative and current
out of the page are positive.
Loop 1 encloses currents -3I, +14I, and -6I. The net
current is +5I or 5I out of the page.
(b) What is the net current though the interior of loop 2?
Loop 2 encloses currents -16I and +14I. The net
current is -2I or 2I into the page.
40
Define circulation:
circulation = ∑ B|| Δl
41
Consider a wire carrying current into the page. Draw a
closed path around the wire.
Here the B-field is tangent
to the path everywhere
(hence the choice of a
circular path). The
circulation is
∑ B Δl = B(2πr ).
||
42
Ampere’s Law is
∑ B Δl = μ I
||
0
where I is the net current that cuts
through the circular path.
If the wire from the previous page carries a current I then
the magnetic field at distance r from the wire is
μ0 I
.
B=
2πr
43
§19.10 Magnetic Materials
Ferromagnetic materials have domains, regions in which
its atomic dipoles are aligned, giving the region a strong
dipole moment.
44
When the domains are
oriented randomly there
will be no net
magnetization of the
object.
When the domains are
aligned, the material will
have a net magnetization.
45
Summary
•Magnetic forces are felt only by moving charges
•Right-Hand Rules
•Magnetic Force on a Current Carrying Wire
•Torque on a Current Loop
•Magnetic Field of a Current Carrying Wire (straight wire,
wire loop, solenoid)
•Ampère’s Law
46