1 chapter 28. 2 gravitational and electric fields we expect that b will 1. have a 1/r 2 dependence...
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Chapter 28
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2
Gravitational and Electric Fields
rr
mGg ˆ
2
rr
qkE ˆ
2
We expect that B will1. have a 1/r2 dependence2. Directed along the radius
Recall F= iLxB and F=qE and F=mg Based on the pattern
rr
iB ˆ
2
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3
Broken Symmetry
But those other fields diverge and B does not
Based on our experience with bar magnets, B must return to the magnet
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Biot-Savart Law
Point P
X
X X
X
X
X
B into page
i ds
1. B goes into the page2. i ds and r are both
perpendicular to B3. B proportional to i ds x r4. But needs to be 1/r2 so
32 4
ˆ
4 r
rsdiBdor
r
rsdiBd oo
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A new constant, 0
Called the permeability of free spaceValue = 4 x 10-7 T*m/AAnd yes, 0/ 410-7 T*m/A
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Modified Biot-Savart
If a charged particle has a constant velocity, v, then I can modify Biot-Savart:
2
ˆ
4 r
rvqB o
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Solving Biot-Savart Problems
Biot-Savart problems are typically one of geometry
You must integrate about the limits of the current loop while evaluating the cross-product
Sometimes, ds and r are parallel, which eliminates the contribution. You should examine the problem to see where this is true.
Sometimes, ds and r are perpendicular, which forces the cross-product to its maximum value.
Mostly, the ds and r are related only through sin . This means that you may have to create an integrand over the angle, not necessarily the length of the current loop.
2
ˆ
4 r
rsdiBd o
In conclusion, Biot-Savart problems are strongly dependent on creating the appropriate geometry over which to integrate.
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Magnetic Field Due to a Long, Straight Wire
22 Rsr
Point P
ds
s
R
r
i
Direction of current
X
X
XX
X
Direction of B
22)sin(sin
Rs
R
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Integrating
R
i
Rs
s
R
iB
Rs
Rdsi
Rs
ds
Rs
RiB
r
dsidBdBB
oo
oo
o
22
22
sin
22
02
122
0 23
220
2222
02
0
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Magnetic Field due to Current of a Circular Arc of Wire
R
dsr
B out of page
r always perpendicular to ds so cross-product is ds
R
iB
dR
iB
RddswhereR
dsiB
r
dsidBB
o
o
o
o
4
4
4
sin
4
0
2
2
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A Complete Loop
We can take the previous result and evaluate it at =2
The result is the same as your text on pg. 1077 Eq. 28-17
R
i
R
iB oo
24
2
aRwherea
iB o
2
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Force Between Two Long Wires with Parallel Currents
zd
iB aoa ˆ
2
ia
ib
d
L
XX
XX
Force on b caused by the magnetic field of a
Fa on b =ibL x Ba
d
Lii
d
iLiF baoaobbona
22
By RH rule, Fa on b is towards a
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By Symmetry
Fb on a is equal and opposite to Fa on b
So the two wires with parallel currents are attracted to one another
If I reverse the current on b (anti-parallel), then the forces generated by the wires will repel one another.
Parallel currents attract; anti-parallel currents repel.
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Derivation of Ampere’s Law
IRR
IRBdsBsdB o
o )2(2
)2(
Consider a distance R from a wire carrying current I. Now consider all of the points which are a distance R from the wire. They form a circle of circumference 2R. Now we evaluate the closed loop integral at this point.
So the circle that we made is called an “Amperian Loop”.
enclosedoIsdB
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Ampere’s Law
Ampere’s Law says that the magnitude of B is proportional to the net current enclosed within the Amperian Loop.
Amperian loops do not have to be circles. They could be rectangles but a circle is usually more convenient to integrate. The only rule is that they have to be closed!
enclosedoIsdB
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What is the direction of the magnetic field?
We have an agreed convention:Curl your fingers around the Amperian loop
with your fingers curling in the direction of the integration. If the current is in the same direction as your thumb, then current is assigned a positive value. If your thumb is opposite the direction of the current, then the current is assigned a negative value.
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Magnetic Field of Long, Straight Wire Revisited
R
IB
IRBRdBsdB
IsdB
o
o
enclosedo
2
)2(2
0
I
R
Integrate from 0 to 2
In this case, the enclosed current is defined to be in the negative direction.
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Differential Version of Ampere’s Law
Recall sdvAdv
enclosedo
enclosedenclosed
enclosedo
JB
so
AdJI
and
IsdBAdB
A current density creates a steady magnetic field which circulates around the current density
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Solenoid
Solenoid is a number of coils packed tightly together.
It solves an incredibly tough problem. Consider: It is easy to
make a uniform E-field. All you need are two parallel plates. But how do you make a uniform field out of a bunch of circles (i.e. a B-field).
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Finding the magnetic field of a solenoid
iNB
iNLBL
so
iNLI
BbecausesdB
sdBsdB
BLsdB
sdBsdBsdBsdBsdB
o
o
enclosed
d
c
a
d
c
b
b
a
a
d
d
c
c
b
b
a
00
0
The solenoid is characterized by the number of turns per unit length, NThe magnetic field outside of the solenoid is so weak that it is consider equal to zero
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Toroid– A circular solenoid
Consider a circular solenoid of radius, R
R
iNB
iNRB
so
iNI
RBsdB
o
o
enclosed
2
)2(
)2(
Current into page
Current out of pageAmperian
Loop
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Bohr Magneton
In the last chapter, we discussed how atoms could be thought of as current loops
I=e/T where T=2r/v v=velocity of the electron r= radius of the electron’s orbit
If =IA then =((ev)/2r)*r2
=evr/2 Recall for an orbiting point, the angular
momentum, L, is equal to mvr So =eL/(2m)
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Quantization of angular momentum
Each orbital is an integer value of Planck’s constant, h divided by 2 =h/ 2L=n
=e /2mCalled the “Bohr Magneton” Represents smallest amount of dipole
moment possible
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Magnetic Materials
This represents any material. The arrows indicate the direction of the individual magnetic moments of the atoms. As you can see, their orientation is random and the vector sum of them is zero.
This represents a permanent magnet. The magnetic moments or dipoles have a net vector sum point to the left.
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Magnetic Materials in an External Magnetic Field
Bexternal
Let’s turn on an external B which points to the left
Before
Possibility 1: The dipoles align with the external field
Possibility 2: The dipoles align to oppose the external field
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The degree to which the magnet moments align or oppose the magnetic field determines their classification
Ferromagnetic—Strongly aligns with the magnetic field
Paramagnetic—Weakly aligns with an external magnetic field
Diamagnetic—Weakly opposes the external magnetic field
For any of these cases, we define the total magnetic field as B=Bexternal + B
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B represents the magnetic field created by the alignment/opposition of the dipoles
Bm*Bexternal
mis called the magnetic “susceptibility” of the material
Paramagnetic materials and their susceptibilities Al 2.2 x 10-5
Cu 1.4 x 10-5
Air 3.6 x 10-7
Diamagnetic materials and their susceptibilities Bi -1.7 x 10-5
Ca -2.2 x 10-5
H2O -9.1 x 10-6
Ferromagnetic materials have susceptibilities from 103 to 106
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--The Permeability of a Material
B=Bexternal + m*Bexternal
B=Bexternal (1+ m Relative permeability, Km is defined as
Km= 1+ m
Actually, we could replace with a new “” called the permeability of the material. = Km
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Hysteresis
Let’s say you had a ferromagnet and you measured its magnetic field
You applied an external B-field and the dipoles are aligned with the field
Now you remove the field but some of the dipoles get “stuck” in their new position.
Now you measure the magnetic field of the ferromagnet and find that it is different.
You repeat the process and yet, you never get back to your original value as more or less of the dipoles stick or unstick in their new positions.
This behavior is called hysteresis.