Download PHYS_3342_111011

Magnetic field around a straight wire
For the
fieldmagnitude
:
0I  sin dx
B

2

4  r
a
ad
[r 
; x  acot; dx 2 ] 
sin
sin 
0I 
0I

sin

d


4a 0
2a
(where
ais thedistance
fromthewire)
Magnetic Field of Two Wires
Field at points on the x-axis
to the right of point (3)

I
0
B

;
1
2

(
xd
)

I
0
B

;
2
2

(
xd
)

I
d
0
B

B

B

t
o
t
a
l
2 1
2 2

(
x

d)
Magnetic field outside of a conductor pair falls off more rapidly
Magnetic field of a circular arc
For the field magnitude at O :
0 I
B
4R 2
0 I
 ds  4R 2 R
0 I

4R

Magnetic Field of a Circular Current Loop
For field on the axis :
 I cos  ds
B ( x )  Bx ( x )  0  2
4 x  R 2
0 I
R
2R

2 3/ 2
2
4 ( x  R )

0 IR 2

2( x 2  R 2 )3 / 2 2 ( x 2  R 2 )3 / 2
[ x  R]

Falls off just as the electric field of
the electric dipole
Magnetic Field on the Axis of a Coil
Bx 
Bx 
0 NIR 2
2( x 2  R 2 )3/ 2
0 
2 ( x  R )
2
2 3/ 2
;
0 
  NIA
0 
2 x 3
The magnetic field of a (small) loop behaves “on the outside” like the electric field
of the electric dipole of the same orientation – that’s why “magnetic dipole”.
Magnetic force between two parallel conductors with currents
Magnetic field from conductor 2:
0 I
B2 
2 r
Magnetic force on conductor 1:
'

II
F1  I ' LB2  0 L
2 r
Absolutely the same magnitude is
for the magnetic force on conductor 2
but F1   F2
FB 0 II '

L
2 r
Currents in the same direction attract
Currents in opposite directions repel
Definition of 1 Ampere :
Identical current in two wires
separated by 1 m
is 1 Ampere
when the force per 1 meter
is 2 10 7 N/m
Example: Two straight, parallel, superconducting wires 4.5 mm apart carry
15,000 A current each in opposite directions
Should we carry about the mechanical strength of the wires?
F 0 II '

 104 N / m
L 2 r
Ampere’s Law

Circulation of B around a closed loop is 0 times
the total current through the surface bounded by the loop


 B d l 
 B dl  B  dl 
0 I
(2 r )  0 I
2 r


 B d l 
b
d
a
c
 B dl  B1  dl  ( B2 ) dl 
0 I
I
(r1 )  0 (r2 )  0
2 r1
2 r2
General Statement


 B d l  0 Iencl
 (Ampere's Law)
Magnetic fields add as vectors, currents – as scalars
Just as with the integral form of Gauss’s law, the integral form of
Ampere’s law is powerful to use in symmetric situations
Magnetic field around and inside a straight w ire
0 I 0
For path 1 : B  (2r )  0 I 0  B 
2r
0 I 0 r
r2
For path 2 : B  (2r )  0 I 0 2  B 
R
2R 2
Magnetic Field of a Solenoid
Wire wound around a long cylinder
produces uniform longitudinal field in
the interior and almost no field outside
For the path in an ideal solenoid:
BL  0nIL  B  0nI
(n turns of the coil per unit length)
Field of a toroidal solenoid
Magnetic field of a toroid :
For any path outside, the total current is zero
For the path inside :
B(2r )   0 NI
B
 0 NI
for total N loops of wire
2r
Magnetic Field of a Sheet of Current
The field is parallel to the plane
(still perpendicular to the current)
For the path: 2 Bl  0 J sl
B
0 J s
for current J s per unit length
2
Independent of distance from the plane
just as the electric field of the charged sheet
The field of a magnetic “capacitor”
BR  0 J s
BP  Bs  0
Magnetic materials
When materials are placed in a
magnetic field, they get
magnetized.
In majority of materials, the
magnetic effects are small. Some
however show strong responses.
The small magnetism is of two kinds:
• Diamagnetics are repelled from magnetic fields
• Paramagnetics are attracted towards magnetic fields
This is unlike the electric effect in matter, which always causes dielectrics
to be attracted.
The Bohr Magnetron
Magnetic effects have to do with microscopic currents
(magnetic moments) at the atomic level such as the
orbital motion of electrons:
e
ev
Current I 

T
2 r
e
e
Magnetic moment μ  I   r 2  ( )mvr  ( ) L
2m
2m
The angular momentum is quantized
h
L
n; n  integer number
2
h=6.626  10-34 J  s  Planck's constant
Fundamental unit of magnetic moment
=
e  h

2m  2
eh


 Bohr magnetron

4

m

B  9.274  1024 J / T
There is also magnetic moment associated with
eh
electron spin: spin 
=B
4 m
Magnetization
Magnetization of a substance M is its magnetic moment per unit volume
(similar to polarization in case of dielectrics in electric fields)

M

 total
V
Total magnetic field at a point is a sum
B  B 0  0M
All equations can be adapted by replacing 0  K m 0
Small magnetic effects are linear:
m  Km  1
   0 for diamagnetics
Magnetic susceptibility 
   0 for paramagnetics
• Diamagnetism occurs in substances where magnetic moments
inside atoms all cancel out, the net magnetic moment of the
atom is zero. The induced magnetic moment is directed
opposite to the applied field. Diamagnetism is weakly
dependent on T.
• Diamagnetic (induced atomic moment) effect is overcome in
paramagnetic materials, whose atoms have uncompensated
magnetic moments. These moments align with the applied field
to enhance the latter. Temperature T wants to destroy
alignment, hence a strong (1/T) dependence.
B
M=C   Curie's Law
T
Magnetic effects are a completely quantum-mechanical phenomenon,
although some classical physics arguments can be made.
Example: Magnetic dipoles in a paramagnetic material
Nitric oxide (NO) is a paramagnetic compound. Its molecules have maximum magnetic
moment of ~ B . In a magnetic field B=1.5 Tesla, compare the interaction energy of the
magnetic moments with the field to the average translational kinetic energy of the molecules
at T=300 K.
U max   B B  1.4  1023 J  8.7  105 eV
3
K  kT  6.2  1021 J  0.039 eV
2
Ferromagnetism
• In ferromagnetic materials,
in addition to atoms having
uncompensated magnetic
moments, these moments
strongly interact between
themselves.
• Strongly nonlinear
behavior with remnant
magnetization left when the
applied field is lifted.
Permeability Km is much
larger, ~1,000 to 100,000
Alignment of magnetic
domains in applied field