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Lecture 13: June 19th 2009
Physics for Scientists and Engineers II
Physics for Scientists and Engineers II , Summer Semester 2009
1
Ampere’s Law
Magnetic field lines are tangentia l to a circle surroundin g the current.
B ds
I
B
a
I
Top View
0 I
 B  d s  B  d s  2 r 2 r   0 I
Physics for Scientists and Engineers II , Summer Semester 2009
2
Ampere’s Law
Picking a different path.
Top View
r2
r1
I
ds
ds
B
ds
 Bds 

left semicircle
 Bds 
Bds 
straightsection
Physics for Scientists and Engineers II , Summer Semester 2009

right semicircle
Bds 
 Bds
straightsec tion
3
Ampere’s Law
 Bds 

 Bds 
Bds 
left semicircle
straightsec tion
 r1
0 I
0 I
d
s

0

2 r1 0
2 r2
I I
 0  0  0 I

2

right semicircle
 Bds
Bds 
straightsec tion
 r2
ds 0
0
2
The line integral
 B  d s around any closed path equals  I , where
0
I is the total steady current passing through any surface bounded
by the closed path.
 Bds   I
0
Physics for Scientists and Engineers II , Summer Semester 2009
4
Ampere’s Law
Top View
Picking a different path.
r1
I
r2
B
ds
ds
B
ds
 Bds 

inner semicircle
 Bds 
Bds 
straightsec tion
Physics for Scientists and Engineers II , Summer Semester 2009

right semicircle
Bds 
 Bds
straightsec tion
5
Ampere’s Law
 Bds 

inner semicircle


 Bds 
Bds 
straightsec tion
 r1
0 I
0 I
d
s

0

2 r1 0
2 r2
0 I 0 I
2

The line integral
2

right semicircle
 Bds
Bds 
straightsec tion
 r2
ds 0
0
0
 B  d s around any closed path equals  I , where
0
I is the total steady current passing through any surface bounded
by the closed path.
 Bds   I
0
Physics for Scientists and Engineers II , Summer Semester 2009
6
Example Application of Ampere’s Law
I2
I1
Inner conductor : r  r1
Insulator (air) : r1  r  r2
Outer conductor : r2  r  r3
Outside : r3  r
Inner conductor : r  r1
I1
I1
2
 B  d s  0 I inside  B 2 r   0  r12  r  B   0 2 r12 r
Insulator : r1  r  r2
 B  d s  0 I inside  B 2 r   0 I1  B   0
Physics for Scientists and Engineers II , Summer Semester 2009
I1
2 r1
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Example Application of Ampere’s Law
Outer conductor : r2  r  r3
2


r 2   r2 
 B  d s   0 I inside  B 2 r   0  I1  I 2  r32   r2 2 
 0 
 r 2   r2 2 
B
I1  I 2

2 r 
 r3 2   r2 2 
Outside : r3  r
 Bds   I
0 inside
 B 2 r   0 I1  I 2 
 0  I1  I 2 
B
2 r
Physics for Scientists and Engineers II , Summer Semester 2009
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Preventing Pitfalls
ds
 Bds  0
 Bds  0
I
but not necessaril y that B  0
In fact, we know that B  0 along the blue path. Rather,
B is perpendicu lar to the path (coming out of the page).
Physics for Scientists and Engineers II , Summer Semester 2009
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Preventing Pitfalls
ds
I
 Bds  μ I
0
along the red path.
 Conclusion : You already need to know from symmetry etc. the direction
of the magnetic field and that the magnitude of B  constant along the path to
draw the right conclusion s from Ampere' s laws.
Physics for Scientists and Engineers II , Summer Semester 2009
10
A Long Solenoid (Wire wound in the form of a helix)
Long solenoids produce reasonably uniform magnetic fields in their " interior".
I
 Bds  0
Doesn' t mean that B  0 !!!!
 Bds  μ I
0
 B  0 outside the coil, but it is small.
Physics for Scientists and Engineers II , Summer Semester 2009
11
A Long Solenoid (Wire wound in the form of a helix)
Make the assumption that Bexternal  Binternal
 Bds  μ
Binternal
Bexternal
Physics for Scientists and Engineers II , Summer Semester 2009
0
NI
(N  number of turns inside the blue loop)
N
 B L  μ 0 N I  B  μ0
I  μ0 n I
L
N
(n 
is the number of turns per unit length)
L
L
12
Problem 33 in the book
J s (out of paper in y - direction)
is a linear current density.
Show that B is parallel to the sheet,
J
perpendicu lar to J s and B  μ0 s .
2
z
x
L
Physics for Scientists and Engineers II , Summer Semester 2009
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Problem 33 in the book
J s (out of paper in y - direction)
is a linear current density.
Symmetry shows that B is parallel to the sheet and
perpendicu lar to J s .
z
x
L
 Bds  μ
0
I
 2 B L  μ0 I  B  μ0
Physics for Scientists and Engineers II , Summer Semester 2009
1 I
1
 μ0 J s
2 L 2
14
Gauss’s Law in Magnetism
Magnetic flux thoug h surface element d A : d B  B  d A
Magnetic flux thoug h a surface :  B   B  d A
Units of magnetic flux : 1 Tm 2  1Wb (weber).
Gauss' s law in magnetism : The magnetic flux throu gh any closed surface
is always zero :
 B  dA  0
Physics for Scientists and Engineers II , Summer Semester 2009
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Gauss’s Law Comparison
 E  dA 
qinside
0
Electric flux through closed surface
is proportional to the amount of electric
charge inside (electric monopoles).
Physics for Scientists and Engineers II , Summer Semester 2009
 B  dA  0
Isolated magnetic monopoles have
never been found.
16
Magnetism in Matter
We now know how to build “electromagnets” (using electric current through a wire).
We also found that a simple current loop produces a magnetic field / has a magnetic
dipole moment.
How about the “current” produced by an electron running around a nucleus?
Let’s use a classical model (electron is a point charge orbiting around a positively
charged nucleus.
L
r
+
I
Orbital angular momentum of electron
direction of motion of electron

The tiny current loop produces a magnetic moment
Physics for Scientists and Engineers II , Summer Semester 2009
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Magnetism in Matter
q e e
ev
I  

t T 2 2 r
L
r
+
I
-

 ev  2 1
 r  e v r
  IA  
2
 2 r 
L
L  me v r  v r 
me
1 L  e 
 L
 e
 
2 me  2me 
(   L)
L = “orbital angular momentum”
Physics for Scientists and Engineers II , Summer Semester 2009
18
Quantization
Quantum Physics : Orbital angular momentum L is " quantized"
h
(means : it only occurs in multiples of  
 1.05 10 34 Js h  Plank' s constant).
2π
 smallest nonzero value of  :
 e 
e
 L  2

2me
 2me 
  
(the factor 2 has to do with " total" orbital angular momentum, which is different from
that only in one direction. ...this is beyond the scope of this class).
Physics for Scientists and Engineers II , Summer Semester 2009
19
Spin
Spin : An intrinsic property of electrons that contribute s to the total magnetic moment of an electron.
Magnitude of spin angular momentum : S 
Magnetic moment due to spin :  spin 
3

2
e
J
 9.27 10  24
2 me
T
e
  B is called the Bohr magneton
2 me
 total   orbital   spin
For atoms where all electrons are " paired"   spin  0
For atoms with uneven number of electrons an unpaired electron must exist   spin  0
The nucleus of an atom also has a magnetic moment due to it' s protons and electrons.
 nucleus   electron
Physics for Scientists and Engineers II , Summer Semester 2009
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