Download Ch. 30 - Sources of magnetic fields

Sources of Magnetic Fields
Chapter 30
Biot-Savart Law
Lines of Magnetic Field
Ampere’s Law
Solenoids and Toroids
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).
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.
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.
dl
I
r
0 Idl  rˆ
dB 
2
4 r
dB
Add up all the bits!
q
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.
r  r rˆ
dl
I
is the vector from dl to
the observation point
r
0 Idl  rˆ
dB 
2
4 r
dB
Add up all the bits!
q

Biot-Savart Law
dl
I
q
r
0 Idl  rˆ
dB 
2
4 r
dB
The constant 0 = 4 x 10-7 T m/A
is called the permeability of free space.
It turns out that 0 and eo are related in a simple
way: (e00)-1/2 = 3x10 8 m/s = c, the speed of light.
Why? Light is a wave of electric and magnetic fields.
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.
P
R
I
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.
x
dx
x
dl
r
P
0
R
I
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.
Example: Magnetic field from a long wire
0 I dl x rˆ
dB 
2
4 r
x
q
dx
x
0
rˆ
Direction of dB: into page.
r
 R
I
+
Example: Magnetic field from a long wire
0 I dl x rˆ
dB 
2
4 r
x
q
dx
x
0
rˆ
r
 R
I
Direction of dB: into the page.
This is true for every bit; so we
+ don’t need to break into
components, and B also points
into the page.
Example: Magnetic field from a long wire
0 I dl x rˆ
dB 
2
4 r
x
q
dx
x
0
rˆ
r
 R
I
Direction of dB: into the page.
This is true for every bit; so we
+ don’t need to break into
components, and B also points
into the page.
Moreover, lines of B go around a
long wire.
Example: Magnetic field from a long wire
Moreover, lines of B go around a
long wire. Perspective:
x
q
dx
x
0
rˆ
r
i
R
I
+
B
P
Another right-hand rule
Example: Magnetic field from a long wire
0 I dl x rˆ
dB 
2
4 r
x
q
dx
x
0
rˆ
Direction of dB (or B): into page
r
 R
I
+
0 I dx sin q
dB 
2
4 r
0 I
 B   dB 
4

x 
x
sin qdx
2
r
Example: Magnetic field from a long wire
sin q dx
 x r 2
R
2
2
r  x  R , sin q 
r
Rdx
0 I x 

B

3
+
x
2
2
4
(x  R ) 2
0 I
 B   dB 
4
x
q
dx
x
0
rˆ
r
R
I
x
0 I

1
2
2
4R (x  R ) 2
x 
x 
x
0 I

2R
Force between two current-carrying wires
B2
d
I1
B1
I2
Current 1 produces a magnetic
field B1 =0I/ (2d) at the position
of wire 2.
This produces a force on current 2:
Force between two current-carrying wires
B2
d
I1
B1
I2
Current 1 produces a magnetic
field B1 =0I/ (2d) at the position
of wire 2.
This produces a force on current 2:
F2 = I2L x B1
Force between two current-carrying wires
B2
d
I1
F2
B1
I2
Current 1 produces a magnetic
field B1 =0I/ (2d) at the position
of wire 2.
This produces a force on current 2:
F2 = I2L x B1
Force between two current-carrying wires
B2
d
I1
F2
B1
I2
Current 1 produces a magnetic
field B1 =0I/ (2d) at the position
of wire 2.
This produces a force on current 2:
F2 = I2L x B1
This gives the force on a length L of wire 2 to be:
0 I1 I2 L
F2  I2 LB1 
2 d
Direction: towards 1, if the currents are in the same direction.
Force between two current-carrying wires
Current I1 produces a magnetic
field B1 =0I/ (2d) at the position
I1 of the current I2.
B2
d
F2
B1
I2
This produces a force on current I2:
F2 = I2L x B1
Thus, the force on a length L of the conductor 2 is given by:
0 I1 I2 L
F2  I2 LB1 
2 d
[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?
Magnetic field from a circular current loop
0 Idl  rˆ
dB 
2
4 r
0 Idl
Only z component
dB 
2
is nonzero.
4 r
0 Idl cos a
dBz  dBcosa 
2
4
r
r
R  z ,cosa  R
2
along the axis only!
B
dBperp
r
dBz
z
a
dl
R
2
R2  z2
I
Magnetic field from a circular current loop
B
 dB
z


0
IR
dl
3
4  (R 2  z 2 ) 2
along the axis only!
0
IR
B
dl 

3
4  (R 2  z 2 ) 2
0
IR
0
IR 2
B
2R 
3
3
2
2
2
2
2
4  (R  z )
2 (R  z ) 2
At the center of the loop B 
At distance z on axis
from the loop, z>>R

0 I
2R
0 IR 2
B
2z 3
B
dBperp
r
dBz
z
a
dl
R
I
Magnetic field in terms of dipole moment
Far away on the axis,
B
0 IR
2z
2
3
B
The magnetic dipole moment of the
loop is defined as = IA =IR2.
The direction is given by the right
 hand rule: with fingers closed in
the direction of the current flow,
the thumb points along .
z

R
I
Magnetic field in terms of dipole moment
In terms of , the magnetic field on axis (far from
the loop) is therefore
0 
B
3
2 z
This also works for a loop with N turns. Far from
the loop the same expression is true with the
dipole moment given by =NIA = INR2
Dipole Equations
Electric Dipole
t=p x E
U = -p·E
Eax = (2e0 )-1 p/z3
Ebis = (4e0 )-1 p/x3
Magnetic Dipole
t= x B
U = -·B
Bax = ( 0/2) /z3
Bbis = (0/4) /x3
Ampere’s Law
Electric fields
Coulomb’s law gives E
directly (as some integral).
Magnetic fields
Biot-Savart law gives B
directly (as some integral).
Gauss’s law is always true. It Ampere’s law is always true.
is seldom useful. But when it It is seldom useful. But when
is, it is an easy way to get E. it is, it is an easy way to get B.
Gauss’s law is a surface
integral over some
Gaussian surface.
Ampere’s law is a line integral
around some
Amperian loop.
Ampere’s 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
oIenc:
I2
I3
 B  dl   I
0 enc
Ampere’s law is to the Biot-Savart law exactly
as Gauss’s law is to Coulomb’s law.
Ampere’s 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
oIenc:
The sign of Ienc comes
from another RH rule.
I2
I3
 B  dl   I
0 enc
Ampere’s law is to the Biot-Savart law exactly
as Gauss’s law is to Coulomb’s law.
Ampere’s Law - a line integral
 B  dl 
blue - into figure
red - out of figure
a
 B  dl 
B  dl 
c
 B  dl 
d
I1
b
b

a
I2
c
I3
d
Ampere’s Law - a line integral
 B  dl   I
blue - into figure
red - out of figure
0 1
a
 B  dl 
B  dl 
c
 B  dl 
d
I1
b
b

a
I2
c
I3
d
Ampere’s Law - a line integral
 B  dl   I
blue - into figure
red - out of figure
0 1
a
 B  dl   I
a
I1
0 2
b
b

B  dl 
c
 B  dl 
d
I2
c
I3
d
Ampere’s Law - a line integral
 B  dl   I
blue - into figure
red - out of figure
0 1
a
 B  dl   I
a
I1
0 2
b
b
 B  dl   I )
0
c
 B  dl 
d
3
I2
c
I3
d
Ampere’s Law - a line integral
 B  dl   I
blue - into figure
red - out of figure
0 1
a
 B  dl   I
a
I1
0 2
b
b
 B  dl   I )
0
I2
3
c
 B  dl   I  I )
0
d
1
3
c
I3
d
Ampere’s Law on a Wire
What is magnetic field
at point P ?
i
P
Ampere’s 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
dl
TAKE ADVANTAGE OF SYMMETRY!!!!
B
Ampere’s 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
Then
i
P
B
dl
 B  dl =  Bdl  B(2 r)   I
0
TAKE ADVANTAGE OF SYMMETRY!!!!
Ampere’s 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:
i
P
B
dl
 B  dl =  Bdl  B(2 r)   I
0
0 I
B
2 r
A Solenoid
.. is a closely wound coil having n turns per unit length.
current flows
into plane
current flows
out of plane
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?
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?
A Solenoid
Consider longer and longer solenoids.
Fields get weaker and weaker outside.
Apply Ampere’s 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
Apply Ampere’s 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
BL  0 NI

B  n0 I
where
n  N /L
Gauss’s Law for Magnetism
For electric charges
Gauss’s Law is:

q
E  dA  e
0
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
 or end.
field lines to begin
Thus, Gauss’s Law for
magnetic charges must be:
 B  dA  0
Laws of Electromagnetism
We have now 2.5 of Maxwell’s 4 fundamental
laws of electromagnetism. They are:
Gauss’s law for electric charges
Gauss’s law for magnetic charges
Ampere’s law (it is still incomplete as it only
applies to steady currents in its present form.
Therefore, the 0.5 of a law.)
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
Diamagnetism
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:

Bapplied
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:
Bapplied
Binternal
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:
Bapplied
Binternal
Bint   M Bapp
Magnetic Materials


Bint   M Bapp
M is the relative permeability
(the magnetic equivalent of E )
Usually M is very close to 1.
- if M > 1, material is “paramagnetic” - e.g. O2
- if M < 1, material is “diamagnetic” - e.g. Cu
Because M is close to 1, we define the
magnetic susceptibility cM= M - 1
Magnetic Materials
Bint   M Bapp  (1 c M )Bapp
Hence:
For paramagnetic materials cM is positive
- so Bint > Bapp
For diamagnetic materials cM is negative
- so Bint < Bapp
Typically, cM ~ +10-5 for paramagnetics,
cM ~ -10-6 for diamagnetics.
(For both M is very close to 1)
Magnetic Materials
Ferromagnetic Materials:
These are the stuff permanent magnets are
made of.
These materials can have huge susceptibilities:
cM as big as +104
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!
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!