nihuss
19-04-2010, 17:17
Magnetic Fields
• Familiar permanent magnets produce a dipole
field that looks from afar just like the electric
dipole field from two point charges, but
– It is a magnetic field, not an electric field!
– The magnetic dipole can never be split into two
monopoles!
V/m
Tesla
r
B
r
Magnetic Dipole
Electric Dipole
E
Splitting a Magnetic Dipole
S
N
S
N
S
N
S N
S
N
S N
S
N
and etcetera, all the way down to individual atoms.
In fact, the building blocks of atoms, electrons, protons, and neutrons, are
all magnetic dipoles! (The electron is the strongest, by far.)
Magnetic Forces
• Like poles repel each other with a long-
range force that decreases with distance.
• Unlike poles attract each other.
– This sounds like the rule for electric charges,
but experimentally, nobody has ever found an
isolated N pole without a corresponding S
pole and vice versa. Magnetic monopoles
have never been found, but modern theories
predict they exist.
Magnetic Dipoles
• Two magnetic dipoles exert forces and
torques on each other and have minimum
energy when oriented like this:
S
N
Earth is a Magnetic Dipole
Compass needle
is another
permanent
dipole.
Announcements
Announcements
HW 6 deadline extended to 5/17
Makeup midterm will be held after class, from
3:30-4:45 on Friday 5/30, in ISB 455.
Rules for midterm 2: No eqn storing calculators,
cell phones or PDAs except turned off in a
closed bag. You will be given a sheet with
equations. NO OTHER NOTES allowed. A-M
here, N-Z in Classroom Unit 1. You MUST bring
your picture IDs, as you did last time.
Magnetic Fields are Produced by Moving
Charges
•
A static charge will produce only an
electric field.
•
A moving charge (or spinning charge)
will produce both electric and magnetic
fields.
• The textbook gives the following formula
for the magnetic field of a point charge
moving with constant velocity:
r
r
µ
ˆ
q
v
×
r
0
=
B
7
µ
=
4
×
10
T
m/A
0
2
4
r
This formula is only valid for speeds slow
compared to the speed of light, but it does
show several features of the exact result.
The B field from a moving charge falls off like
1/r
, just like the Coulomb Electric field of a
2
point charge. The B field is always
perpendicular to the velocity.
Static charged (and neutral) particles can also
have permanent magnetic dipoles. The proton,
neutron and electron all do: the electron’s is
about 1000 times larger. The dipole fields fall
off like 1/r
, just like electric dipole fields.
3
The source of all known magnetic fields is
currents or permanent dipoles of e,p etc.
The Vector Cross Product
• For vectors in a plane:
A
= (A
, A
),
B
= (B
, B
),
x
y
x
y
A
X
B
= (A
B
-A
B
) = +/- AB sin .
x
y
y
x
• It vanishes if
A
||
B.
Any two non-|| vectors in
3D determine a plane and there is a unique axis
perpendicular to this plane.
•
A
X
B
points in this perpendicular direction and
has magnitude AB |sin |
• Sign determined by right hand rule, or
equivalently by (
A
X
B
)
= (A
B
-A
B
) and
z
x
y
y
x
two
cyclic
permutations, (x,y,z) (z,x,y)
(y,z,x) , of this equation
r
r
µ
×
ˆ
q
v
r
r
µ
qv
sin
0
B
=
0
B
=
2
4
2
4
r
r
Direction follows from
right-hand rule.
Magnetism is full of
right hand rules. You
must learn them!
Sources of the
B
Field
1. Permanent magnets:
• Historically the first magnetic field
sources.
• But definitely the most
complicated, so we defer this to
later.
2. Currents:
• Magnetic field lines make closed
rings around a wire.
•
B
field lines never have a
beginning or end!
• Special case: infinite wire:
Magnetic field of an infinite wire
µ
I
0
=
B
2
r
B
B
I
End View
Field of a Permanent Magnetic
Dipole
Note how magnetic field lines
never end
.
This reflects the fact that there is no
magnetic monopole (no magnetic charge).
Magnetic Monopole Charge
• Dirac argued that IF there were magnetic
monopoles, this would explain the
quantization of electric charge.
• Modern theories of high energy physics
indeed predict monopoles, but they are
extremely heavy, probably at least 10
17
times the mass of the proton.
Announcements
Announcements
HW 6 deadline extended to 5/17
Makeup midterm will be held after class, from
3:30-4:45 on Friday 5/30, in ISB 455.
Rules for midterm 2: No eqn storing calculators,
cell phones or PDAs except turned off in a
closed bag. You will be given a sheet with
equations. NO OTHER NOTES allowed. A-M
here, N-Z in Classroom Unit 1. You MUST bring
your picture IDs, as you did last time.
Biot Savart Law
r
r
r
µ
ˆ
l
×
I
d
r
ˆ =
d
l
×
r
sin
d
l
0
=
B
4
2
Direction follows from
r
right-hand rule.
• This formula is
not
approximate.
•
r
is the distance from the current element
to the location where the field is being
r
evaluated.
d
l
ˆ
r
r
d
B
r
d
l
r
I
Field of a Finite Straight Wire
( )
r
ˆ
×
ˆ
=
ˆ
d
l
r
dy
sin
z
z
ˆ
k
r
µ
I
sin
dy
0
ˆ
into
d
B
=
z
page
2
2
4
x
+
y
a
r
µ
I
x
dy
( )
0
ˆ
B
=
z
3
2
4
2
2
+
x
y
a
r
µ
I
1
0
=
ˆ
B
z
()
2
2
+
x
1
x
a
From an Integral Table
dy
y
( )
=
3
2
2
2
2
2
2
x
x
+
y
x
+
y
Field of 2 Antiparallel Infinite Wires
B
B
1
B
c
2
b
a
B field applet:
http://www.falstad.com/vector3dm/
Circular Current Loop
From symmetry, all the field components
perpendicular to the x axis (e.g.
dB
) will add up to
y
zero, giving
B
=
B
=0.
y
z
dl
All the
dB
are equal, since the distance
r
from
to
x
P
is the same everywhere on the ring, so the integral
is very easy to do.
r
ˆ
d
l
×
r
=
d
l
sin
=
d
l
2
µ
I
d
0
= l
dB
4
2
2
+
a
x
a
=
=
dB
dB
cos
dB
x
2
2
a
+
x
(
)
B
= symmetry
dB
=
0
by
y
y
2
µ
Ia
2
a
µ
( )
I
a
0
=
B
( )
0
B
=
dB
=
d
l
on axis only!
x
3
2
x
x
2
2
3
2
4
+
2
a
x
2
2
a
+
x
0
Circular Current Loop
• This loop is an example of a
magnetic
dipole
, with dipole moment:
µ
2
=
=
NIA
NI
a
N
= number of turns
µ
µ
( )
0
=
B
on axis only!
x
2
3
2
2
2
+
a
x
µ
µ
• Field at the center of the loop:
0
=
B
x
2
3
a
Field far from Current Loop
• Field on axis very far from the loop:
µ
µ
0
B
x
2
3
x
• More generally
B
= (µ
/ 2 r
) µ , where r is the distance
3
0
from the center of the loop and µ is the magnetic dipole
moment vector of the loop. It is perpendicular to the plane
of the loop, sign determined by right hand rule.
• Do not confuse the dipole moment vector with the
permittivity µ
!!!!!
0
Magnetic (Dipole) Moment
(This formula works even if the loop is not circular.)
Field of a Current Loop
A Current Loop is a “Magnet”
Current Loop vs Bar Magnet
“north pole”
“north pole”
Magnetic Dipole Field
•For
any
shape current loop (not just
circular), the field
on the axis
of the
magnetic dipole is to an excellent
approximation given by
r
r
µ
B
r
dipole
r
µ
µ
µ
0
=IA
=
(
if
the
B
r
dipole
loop is flat)
2
3
r
r
•
as long as the distance
from the dipole is
large compared with the size of the dipole
Analogy With Electric Dipole
(Note that the complete expression for the
field off axis is not much more complex, but
we won’t use it.)
Note the similarity to the electric dipole on-
axis field:
r
r
r
p
E
r
dipole
1
p
E
=
r
dipole
2
3
r
0
Magnetic Dipole Fields
• Since magnetic monopoles have not been
found, far from a current loop of finite
extent the magnetic field always falls off
1
/r
3
with distance at least as fast as
.
• Contrast this with the electric case
– If the total charge of a finite distribution of
electric charges is not zero, then the electric
1
/r
field falls off like
at large distance.
2
Magnetic Dipoles
–If the total charge is zero, but there is an
electric dipole moment,
–then the electric field falls off like 1
/r
3 at
large distance.
•The magnetic dipole concept is extremely
important in practice, because it gives us a
simple form of the magnetic field when far from
the current.
Helmholtz Coils: 2 current loops
Used in this week’s lab to
a
produce the magnetic field. 2
I
a
a
x
µ
µ
µ
0
B
(
x
)
=
+
() ()
2
3
2
3
2
+
2
2
2
2
+
a
+
a
a
x
a
x
2
2
The flatness in
2
2
this region is
what makes this
1.5
Plot of the field
configuration
strength along
B
useful in practice
fx
()
1
x
the
axis
(including some
=
B
( =
0
)
B
(
0
)
0
of the newer MRI
0.5
machines).
0
0
a
1 0.5 0 0.5 1
a
x
1
1
This “open
geometry” MRI
machine has a
configuration
very much like
Two
coils
a Helmholtz
coil.
Ampère’s Law
• This is mathematically equivalent to the
Biot-Savart Law (no new physics).
– I’m not going to ask you to become proficient
in using this law to calculate magnetic fields,
but you need to be aware of the basic content
in the law. It is the analog of Gauss’ Law for
Electric Fields. But note also an analogy to
Kirchoff’s loop rule
Ampere’s Law
The line integral of the magnetic
field around any closed loop in
space is proportional to the
current passing through the
loop.
r
r
µ
=
B
d
s
I
0
through
Arbitrary loop
in space
Ampère’s Law Simple Example
Infinite straight wire.
r
r
µ
B
d
s
=
I
0
through
B is the same magnitude
everywhere on the circle and
tangent to the circle, so the
integral is trivial:
µ
=
B
2
d
I
0
µ
I
0
B
=
2
d
Identical to the result obtained
from the Biot-Savart law.
Ampère’s Law Example
r
r
B
d
s
What is around the closed curve below?
4 wires, going in/out of the page.
Be careful to use the
2.0A
right-hand rule to get
X
7.0A
the signs on the
currents.
1.0A
X
X
7.0A
r
r
( )
7
6
B
d
s
=
4
10
×
2
7
+
1
=
5
.
0
×
10
Tm
Field of a Solenoid
Intense, almost uniform field inside
Weak field outside
MRI scanner machine, built with a
superconducting solenoid magnet.
Field of an Ideal Solenoid
• Very long and narrow. To a good
approximation we can say that the field
inside is uniform, while outside it is zero.
r
r
µ
µ
=
=
B
d
s
I
NI
0
through
0
µ
BL
=
NI
0
µ
NI
0
µ
B
=
=
nI
0
L
We will use this formula in many
problems, both because it is simple
(uniform field) and because solenoids
are widely used in practice.
:s_thumbup::s_thumbup:
• Familiar permanent magnets produce a dipole
field that looks from afar just like the electric
dipole field from two point charges, but
– It is a magnetic field, not an electric field!
– The magnetic dipole can never be split into two
monopoles!
V/m
Tesla
r
B
r
Magnetic Dipole
Electric Dipole
E
Splitting a Magnetic Dipole
S
N
S
N
S
N
S N
S
N
S N
S
N
and etcetera, all the way down to individual atoms.
In fact, the building blocks of atoms, electrons, protons, and neutrons, are
all magnetic dipoles! (The electron is the strongest, by far.)
Magnetic Forces
• Like poles repel each other with a long-
range force that decreases with distance.
• Unlike poles attract each other.
– This sounds like the rule for electric charges,
but experimentally, nobody has ever found an
isolated N pole without a corresponding S
pole and vice versa. Magnetic monopoles
have never been found, but modern theories
predict they exist.
Magnetic Dipoles
• Two magnetic dipoles exert forces and
torques on each other and have minimum
energy when oriented like this:
S
N
Earth is a Magnetic Dipole
Compass needle
is another
permanent
dipole.
Announcements
Announcements
HW 6 deadline extended to 5/17
Makeup midterm will be held after class, from
3:30-4:45 on Friday 5/30, in ISB 455.
Rules for midterm 2: No eqn storing calculators,
cell phones or PDAs except turned off in a
closed bag. You will be given a sheet with
equations. NO OTHER NOTES allowed. A-M
here, N-Z in Classroom Unit 1. You MUST bring
your picture IDs, as you did last time.
Magnetic Fields are Produced by Moving
Charges
•
A static charge will produce only an
electric field.
•
A moving charge (or spinning charge)
will produce both electric and magnetic
fields.
• The textbook gives the following formula
for the magnetic field of a point charge
moving with constant velocity:
r
r
µ
ˆ
q
v
×
r
0
=
B
7
µ
=
4
×
10
T
m/A
0
2
4
r
This formula is only valid for speeds slow
compared to the speed of light, but it does
show several features of the exact result.
The B field from a moving charge falls off like
1/r
, just like the Coulomb Electric field of a
2
point charge. The B field is always
perpendicular to the velocity.
Static charged (and neutral) particles can also
have permanent magnetic dipoles. The proton,
neutron and electron all do: the electron’s is
about 1000 times larger. The dipole fields fall
off like 1/r
, just like electric dipole fields.
3
The source of all known magnetic fields is
currents or permanent dipoles of e,p etc.
The Vector Cross Product
• For vectors in a plane:
A
= (A
, A
),
B
= (B
, B
),
x
y
x
y
A
X
B
= (A
B
-A
B
) = +/- AB sin .
x
y
y
x
• It vanishes if
A
||
B.
Any two non-|| vectors in
3D determine a plane and there is a unique axis
perpendicular to this plane.
•
A
X
B
points in this perpendicular direction and
has magnitude AB |sin |
• Sign determined by right hand rule, or
equivalently by (
A
X
B
)
= (A
B
-A
B
) and
z
x
y
y
x
two
cyclic
permutations, (x,y,z) (z,x,y)
(y,z,x) , of this equation
r
r
µ
×
ˆ
q
v
r
r
µ
qv
sin
0
B
=
0
B
=
2
4
2
4
r
r
Direction follows from
right-hand rule.
Magnetism is full of
right hand rules. You
must learn them!
Sources of the
B
Field
1. Permanent magnets:
• Historically the first magnetic field
sources.
• But definitely the most
complicated, so we defer this to
later.
2. Currents:
• Magnetic field lines make closed
rings around a wire.
•
B
field lines never have a
beginning or end!
• Special case: infinite wire:
Magnetic field of an infinite wire
µ
I
0
=
B
2
r
B
B
I
End View
Field of a Permanent Magnetic
Dipole
Note how magnetic field lines
never end
.
This reflects the fact that there is no
magnetic monopole (no magnetic charge).
Magnetic Monopole Charge
• Dirac argued that IF there were magnetic
monopoles, this would explain the
quantization of electric charge.
• Modern theories of high energy physics
indeed predict monopoles, but they are
extremely heavy, probably at least 10
17
times the mass of the proton.
Announcements
Announcements
HW 6 deadline extended to 5/17
Makeup midterm will be held after class, from
3:30-4:45 on Friday 5/30, in ISB 455.
Rules for midterm 2: No eqn storing calculators,
cell phones or PDAs except turned off in a
closed bag. You will be given a sheet with
equations. NO OTHER NOTES allowed. A-M
here, N-Z in Classroom Unit 1. You MUST bring
your picture IDs, as you did last time.
Biot Savart Law
r
r
r
µ
ˆ
l
×
I
d
r
ˆ =
d
l
×
r
sin
d
l
0
=
B
4
2
Direction follows from
r
right-hand rule.
• This formula is
not
approximate.
•
r
is the distance from the current element
to the location where the field is being
r
evaluated.
d
l
ˆ
r
r
d
B
r
d
l
r
I
Field of a Finite Straight Wire
( )
r
ˆ
×
ˆ
=
ˆ
d
l
r
dy
sin
z
z
ˆ
k
r
µ
I
sin
dy
0
ˆ
into
d
B
=
z
page
2
2
4
x
+
y
a
r
µ
I
x
dy
( )
0
ˆ
B
=
z
3
2
4
2
2
+
x
y
a
r
µ
I
1
0
=
ˆ
B
z
()
2
2
+
x
1
x
a
From an Integral Table
dy
y
( )
=
3
2
2
2
2
2
2
x
x
+
y
x
+
y
Field of 2 Antiparallel Infinite Wires
B
B
1
B
c
2
b
a
B field applet:
http://www.falstad.com/vector3dm/
Circular Current Loop
From symmetry, all the field components
perpendicular to the x axis (e.g.
dB
) will add up to
y
zero, giving
B
=
B
=0.
y
z
dl
All the
dB
are equal, since the distance
r
from
to
x
P
is the same everywhere on the ring, so the integral
is very easy to do.
r
ˆ
d
l
×
r
=
d
l
sin
=
d
l
2
µ
I
d
0
= l
dB
4
2
2
+
a
x
a
=
=
dB
dB
cos
dB
x
2
2
a
+
x
(
)
B
= symmetry
dB
=
0
by
y
y
2
µ
Ia
2
a
µ
( )
I
a
0
=
B
( )
0
B
=
dB
=
d
l
on axis only!
x
3
2
x
x
2
2
3
2
4
+
2
a
x
2
2
a
+
x
0
Circular Current Loop
• This loop is an example of a
magnetic
dipole
, with dipole moment:
µ
2
=
=
NIA
NI
a
N
= number of turns
µ
µ
( )
0
=
B
on axis only!
x
2
3
2
2
2
+
a
x
µ
µ
• Field at the center of the loop:
0
=
B
x
2
3
a
Field far from Current Loop
• Field on axis very far from the loop:
µ
µ
0
B
x
2
3
x
• More generally
B
= (µ
/ 2 r
) µ , where r is the distance
3
0
from the center of the loop and µ is the magnetic dipole
moment vector of the loop. It is perpendicular to the plane
of the loop, sign determined by right hand rule.
• Do not confuse the dipole moment vector with the
permittivity µ
!!!!!
0
Magnetic (Dipole) Moment
(This formula works even if the loop is not circular.)
Field of a Current Loop
A Current Loop is a “Magnet”
Current Loop vs Bar Magnet
“north pole”
“north pole”
Magnetic Dipole Field
•For
any
shape current loop (not just
circular), the field
on the axis
of the
magnetic dipole is to an excellent
approximation given by
r
r
µ
B
r
dipole
r
µ
µ
µ
0
=IA
=
(
if
the
B
r
dipole
loop is flat)
2
3
r
r
•
as long as the distance
from the dipole is
large compared with the size of the dipole
Analogy With Electric Dipole
(Note that the complete expression for the
field off axis is not much more complex, but
we won’t use it.)
Note the similarity to the electric dipole on-
axis field:
r
r
r
p
E
r
dipole
1
p
E
=
r
dipole
2
3
r
0
Magnetic Dipole Fields
• Since magnetic monopoles have not been
found, far from a current loop of finite
extent the magnetic field always falls off
1
/r
3
with distance at least as fast as
.
• Contrast this with the electric case
– If the total charge of a finite distribution of
electric charges is not zero, then the electric
1
/r
field falls off like
at large distance.
2
Magnetic Dipoles
–If the total charge is zero, but there is an
electric dipole moment,
–then the electric field falls off like 1
/r
3 at
large distance.
•The magnetic dipole concept is extremely
important in practice, because it gives us a
simple form of the magnetic field when far from
the current.
Helmholtz Coils: 2 current loops
Used in this week’s lab to
a
produce the magnetic field. 2
I
a
a
x
µ
µ
µ
0
B
(
x
)
=
+
() ()
2
3
2
3
2
+
2
2
2
2
+
a
+
a
a
x
a
x
2
2
The flatness in
2
2
this region is
what makes this
1.5
Plot of the field
configuration
strength along
B
useful in practice
fx
()
1
x
the
axis
(including some
=
B
( =
0
)
B
(
0
)
0
of the newer MRI
0.5
machines).
0
0
a
1 0.5 0 0.5 1
a
x
1
1
This “open
geometry” MRI
machine has a
configuration
very much like
Two
coils
a Helmholtz
coil.
Ampère’s Law
• This is mathematically equivalent to the
Biot-Savart Law (no new physics).
– I’m not going to ask you to become proficient
in using this law to calculate magnetic fields,
but you need to be aware of the basic content
in the law. It is the analog of Gauss’ Law for
Electric Fields. But note also an analogy to
Kirchoff’s loop rule
Ampere’s Law
The line integral of the magnetic
field around any closed loop in
space is proportional to the
current passing through the
loop.
r
r
µ
=
B
d
s
I
0
through
Arbitrary loop
in space
Ampère’s Law Simple Example
Infinite straight wire.
r
r
µ
B
d
s
=
I
0
through
B is the same magnitude
everywhere on the circle and
tangent to the circle, so the
integral is trivial:
µ
=
B
2
d
I
0
µ
I
0
B
=
2
d
Identical to the result obtained
from the Biot-Savart law.
Ampère’s Law Example
r
r
B
d
s
What is around the closed curve below?
4 wires, going in/out of the page.
Be careful to use the
2.0A
right-hand rule to get
X
7.0A
the signs on the
currents.
1.0A
X
X
7.0A
r
r
( )
7
6
B
d
s
=
4
10
×
2
7
+
1
=
5
.
0
×
10
Tm
Field of a Solenoid
Intense, almost uniform field inside
Weak field outside
MRI scanner machine, built with a
superconducting solenoid magnet.
Field of an Ideal Solenoid
• Very long and narrow. To a good
approximation we can say that the field
inside is uniform, while outside it is zero.
r
r
µ
µ
=
=
B
d
s
I
NI
0
through
0
µ
BL
=
NI
0
µ
NI
0
µ
B
=
=
nI
0
L
We will use this formula in many
problems, both because it is simple
(uniform field) and because solenoids
are widely used in practice.
:s_thumbup::s_thumbup: