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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
Hysteresis and Permanent Magnets
Magnetization value depends on the “history” of applied magnetic field
Example: A ferromagnetic material
A permanent magnet is made of a ferromagnetic material with a M~10 6 A/m
The magnet is in the shape of a cube of side 2 cm. Find magnetic dipole
moment of a magnet. Estimate the magnetic field at a point 10 cm away on the axis
total  MV  8 A  m 2

3
B ~ 0 total

10
T  10 G
3
2 x
Magnetization curve for
soft iron showing
hysteresis
Experiments leading to Faraday’s Law
Electromagnetic Induction – Time-varying magnetic field creates electric field
Changing Magnetic Flux
No current in the electromagnet – B=0 - galvanometer
shows no current.
When magnet is turned on – momentarily current appears
as B increases.
When B reaches steady value – current disappears no matter
how strong B field is.
If we squeeze the coil as to change its area – current appears
but only while we are deforming the coil.
If we rotate the coil, current appears but only while we are
rotating it.
If we start displacing the coil out of the magnetic field –
current appears while the coil is in motion.
If we decrease/increase the number of loops in the coil –
current appears during winding/unwinding of the turns.
If we turn off the magnet – current appears while the
magnetic field is being disappearing
The faster we carry out all those changes
- the greater the current is.
Faraday’s Law quantified


d B

for a single - loop coil
dt
d B
 N
for an N - loop coil
dt
 B  BA cos 
Anything changing magnetic flux
will produce the effect