Sources of Magnetic Fields

1
Sources of Magnetic Fields

• Chapter 30
• Biot-Savart Law
• Lines of Magnetic Field
• Amperes Law
• Solenoids and Toroids

2
Sources of Magnetic Fields

• Magnetic fields exert forces on moving charges.
• Something reciprocal happens moving charges give
  rise to magnetic fields (which can then exert a
  force on other moving charges).
• We will look at the easiest case the magnetic
  field created by currents in wires.
• The magnetism of permanent magnets also comes
  from moving charges (the electrons in the atoms).

3
Magnetic Interaction
Rather than discussing moving charges in general,
restrict attention to currents in wires. Then

• A current generates a magnetic field.
• A magnetic field exerts a force on a current.
• Two conductors, carrying currents, will exert
  forces on each other.

4
Biot-Savart Law

• The mathematical description of the magnetic
  field B due to a current-carrying wire is called
  the Biot-Savart law. It gives B at a selected
  position.
• A current I is moving all through the wire. We
  need to add up the bits of magnetic field dB
  arising from each infinitesimal length dl.

Add up all the bits!
5
Biot-Savart Law

• The mathematical description of the magnetic
  field B due to a current-carrying wire is called
  the Biot-Savart law. It gives B at a selected
  position.
• A current I is moving all through the wire. We
  need to add up the bits of magnetic field dB
  arising from each infinitesimal length dl.

is the vector from dl to the observation point
Add up all the bits!
6
Biot-Savart Law
The constant m0 4p x 10-7 T m/A is called
the permeability of free space.

• It turns out that m0 and eo are related in a
  simple
• way (e0m0)-1/2 3x10 8 m/s c, the speed of
  light.
• Why? Light is a wave of electric and magnetic
  fields.

7
Example Magnetic field from a long wire
Consider a long straight wire carrying a current
I. We want to find the magnetic field B at a
point P, a distance R from the wire.
8
Example Magnetic field from a long wire
Consider a long straight wire carrying a current
I. We want to find the magnetic field B at a
point P, a distance R from the wire.
Break the wire into bits dl. To do that, choose
coordinates let the wire be along the x
axis, and consider the little bit dx at
a position x. The vector r r r is from this
bit to the point P.
r
P

9
Example Magnetic field from a long wire
x
Direction of dB into page.
q
dx
r
x
0
R
I
10
Example Magnetic field from a long wire
x
Direction of dB into the page. This is true for
every bit so we dont need to break into
components, and B also points into the page.
q
dx
r
x
0
R
I
11
Example Magnetic field from a long wire
x
Direction of dB into the page. This is true for
every bit so we dont need to break into
components, and B also points into the
page. Moreover, lines of B go around a long wire.
q
dx
r
x
0
R
I
12
Example Magnetic field from a long wire
Moreover, lines of B go around a long wire.
Perspective
x
q
dx
r
x
i
0
R
B
P
I
Another right-hand rule
13
Example Magnetic field from a long wire
x
Direction of dB (or B) into page
q
dx
r
x
0
R
I
14
Example Magnetic field from a long wire
x
q
dx
r
x
0
R
I
15
Force between two current-carrying wires
B2
Current 1 produces a magnetic field B1 m0I/ (2p
d) at the position of wire 2.
d
I1
I2
This produces a force on current 2
B1
16
Force between two current-carrying wires
B2
Current 1 produces a magnetic field B1 m0I/ (2p
d) at the position of wire 2.
d
I1
I2
This produces a force on current 2 F2 I2L
x B1
B1
17
Force between two current-carrying wires
B2
Current 1 produces a magnetic field B1 m0I/ (2p
d) at the position of wire 2.
d
I1
F2
I2
This produces a force on current 2 F2 I2L
x B1
B1
18
Force between two current-carrying wires
B2
Current 1 produces a magnetic field B1 m0I/ (2p
d) at the position of wire 2.
d
I1
F2
I2
This produces a force on current 2 F2 I2L
x B1
B1
This gives the force on a length L of wire 2 to
be
Direction towards 1, if the currents are in the
same direction.
19
Force between two current-carrying wires
B2
Current I1 produces a magnetic field B1 m0I/ (2p
d) at the position of the current I2.
d
I1
F2
This produces a force on current I2
F2 I2L x B1
I2
B1
Thus, the force on a length L of the conductor 2
is given by
Direction towards I1
The magnetic force between two parallel wires
carrying currents in the same direction is
attractive .
What is the force on wire 1? What happens if one
current is reversed?
20
Magnetic field from a circular current loop
along the axis only!
B
Only z component is nonzero.
dBz
dBperp
r
z
a
dl
R
I
21
Magnetic field from a circular current loop
along the axis only!
B
dBz
dBperp
r
z
a
dl
At the center of the loop
R
I
At distance z on axis from the loop, zgtgtR
22
Magnetic field in terms of dipole moment
Far away on the axis,
B
The magnetic dipole moment of the loop is defined
as m IA IpR2. The direction is given by the
right hand rule with fingers closed in the
direction of the current flow, the thumb points
along m.
z
m
R
I
23
Magnetic field in terms of dipole moment
In terms of m, the magnetic field on axis (far
from the loop) is therefore
This also works for a loop with N turns. Far from
the loop the same expression is true with the
dipole moment given by mNIA IpNR2
24
Dipole Equations

• Magnetic Dipole
• t m x B
• U - m B
• Bax ( m0/2p) m/z3
• Bbis (m0/4p) m/x3

• Electric Dipole
• t p x E
• U - p E
• Eax (2pe0 )-1 p/z3
• Ebis (4pe0 )-1 p/x3

25
Amperes Law
Magnetic fields Biot-Savart law gives B directly
(as some integral). Amperes law is always true.
It is seldom useful. But when it is, it is an
easy way to get B. Amperes law is a line
integral around some Amperian loop.
Electric fields Coulombs law gives E directly
(as some integral). Gausss law is always true.
It is seldom useful. But when it is, it is an
easy way to get E. Gausss law is a surface
integral over some Gaussian surface.
26
Amperes Law
Draw an Amperian loop around the sources of
current. The line integral of the tangential
component of B around this loop is equal to
moIenc
I2
I3
Amperes law is to the Biot-Savart law exactly as
Gausss law is to Coulombs law.
27
Amperes Law
Draw an Amperian loop around the sources of
current. The line integral of the tangential
component of B around this loop is equal to
moIenc
The sign of Ienc comes from another RH rule.
I2
I3
Amperes law is to the Biot-Savart law exactly as
Gausss law is to Coulombs law.
28
Amperes Law - a line integral
29
Amperes Law - a line integral
30
Amperes Law - a line integral
31
Amperes Law - a line integral
32
Amperes Law - a line integral
33
Amperes Law on a Wire
What is magnetic field at point P ?
i
P
34
Amperes Law on a Wire
What is magnetic field at point P? Draw Amperian
loop through P around current source and
integrate B dl around loop
TAKE ADVANTAGE OF SYMMETRY!!!!
35
Amperes Law on a Wire
What is magnetic field at point P? Draw Amperian
loop through P around current source and
integrate B dl around loop
i
P
B
Then
dl
TAKE ADVANTAGE OF SYMMETRY!!!!
36
Amperes Law for a Wire
What is the magnetic field at point P? Draw an
Amperian loop through P, around the current
source, and integrate B dl around the loop.
Then
37
A Solenoid
.. is a closely wound coil having n turns per
unit length.
current flows into plane
current flows out of plane
38
A Solenoid
.. is a closely wound coil having n turns per
unit length.
current flows into plane
current flows out of plane
What direction is the magnetic field?
39
A Solenoid
.. is a closely wound coil having n turns per
unit length.
current flows into plane
current flows out of plane
What direction is the magnetic field?
40
A Solenoid
Consider longer and longer solenoids.
Fields get weaker and weaker outside.
41
Apply Amperes Law to the loop shown. Is there a
net enclosed current? In what direction does the
field point? What is the magnetic field inside
the solenoid?
current flows into plane
current flows out of plane
42
Apply Amperes Law to the loop shown. Is there a
net enclosed current? In what direction does the
field point? What is the magnetic field inside
the solenoid?
current flows into plane
current flows out of plane
L
43
Solenoids and Toroids

• Solenoid B moIn n of turns/m
  of length of the solenoid
• This is valid inside, not too near the ends.
• A toroid is a solenoid bent in a circle.
• A similar calculation gives B m0IN/2pr,
• where in this case N is the total number of
  turns.

44
Gausss Law for Magnetism
For electric charges Gausss Law is
because there are single electric charges. On
the other hand, we have never detected a single
magnetic charge, only dipoles. Since there are no
magnetic monopoles there is no place for magnetic
field lines to begin or end.
Thus, Gausss Law for magnetic charges must be
45
Laws of Electromagnetism

• We have now 2.5 of Maxwells 4 fundamental
• laws of electromagnetism. They are
• Gausss law for electric charges
• Gausss law for magnetic charges
• Amperes law (it is still incomplete as it only
• applies to steady currents in its present form.
• Therefore, the 0.5 of a law.)

46
Magnetic Materials
The phenomenon of magnetism is due mainly to the
orbital motion of electrons inside materials, as
well as to the intrinsic magnetic moment of
electrons (spin).
There are three types of magnetic behavior in
bulk matter Ferromagnetism Paramagnetism Diama
gnetism
47
Magnetic Materials

• Because of the configuration of electron orbits
  in atoms, and due to the intrinsic magnetic
  properties of electrons and protons (called
  spin), materials can enhance or diminish
  applied magnetic fields

48
Magnetic Materials

• Because of the configuration of electron orbits
  in atoms, and due to the intrinsic magnetic
  properties of electrons and protons (called
  spin), materials can enhance or diminish
  applied magnetic fields

49
Magnetic Materials

• Because of the configuration of electron orbits
  in atoms, and due to the intrinsic magnetic
  properties of electrons and protons (called
  spin), materials can enhance or diminish
  applied magnetic fields

50
Magnetic Materials

• kM is the relative permeability
• (the magnetic equivalent of kE )
• Usually kM is very close to 1.
• - if kM gt 1, material is paramagnetic -
  e.g. O2
• - if kM lt 1, material is diamagnetic -
  e.g. Cu
• Because kM is close to 1, we define the
• magnetic susceptibility cM kM - 1

51
Magnetic Materials

• Hence
• For paramagnetic materials cM is positive
• - so Bint gt Bapp
• For diamagnetic materials cM is negative
• - so Bint lt Bapp
• Typically, cM 10-5 for paramagnetics,
• cM -10-6 for diamagnetics.
• (For both kM is very close to 1)

52
Magnetic Materials

• Ferromagnetic Materials
• These are the stuff permanent magnets are made
  of.
• These materials can have huge susceptibilities
• cM as big as 104

53
Magnetic Materials

• Ferromagnetic Materials
• These are the stuff permanent magnets are made
  of.
• These materials can have huge susceptibilities
• cM as big as 104
• But ferromagnets have memory - when you turn
  off the Bapp, the internal field, Bint ,
  remains!

54
Magnetic Materials

• Ferromagnetic Materials
• These are the stuff permanent magnets are made
  of.
• These materials can have huge susceptibilities
• cM as big as 104
• But ferromagnets have memory - when you turn
  off the Bapp, the internal field, Bint ,
  remains!
• permanent magnets!