Transcript Lecture22

Ampere’s Law:Symmetry
 B  ds   I
0
•Ampere’s law because easy (just as gauss’s law did) when
we have symmetry and a uniform field
•From a infinitely long wire:
Radial symmetric, so
B(2pr)=0I
(same expression as from Biot-Savart law)
Quiz: Ampere’s Law
•Consider three wires with
current flowing in/out as shown
•Consider three different loops
surrounding the wires
X
Y
Which of the loops has the largest and
smallest integrals of the magnetic
field around the loops drawn?
A) X > Y > Z
C) Y > Z > X
B) X > Z > Y
D) Y > X > Z
2A
3A
1A
Z
Ampere’s Law: Wire
•We can use ampere’s law on problems difficult to solve with
the Biot-Savart law
We know that the current is uniformly distributed within the
wire, so Ienc=I(pr2)/(pR2)
B(2pr)=0I(pr2)/(pR2)
B=0I(r)/(2pR2)
Inside the field is proportional to the distance from the center
Outside the field falls over off
Magnetic Field From a
Plane of Current
What is the magnetic field?
•By symmetry, magnetic field
will be the same on both sides
L
•Draw a loop crossing the
boundary to use Ampere’s Law
N wires in
length L
 B  ds   I   NI
 B  ds  BL  BL  0  0
0 l
0
B
 0 NI
2L
Current I in
each wire
Magnetic Field of an Ideal Solenoid
length l
N turns
Length L
•If the length is much greater than the radius,
•Magnetic field far away is zero
•Magnetic field everywhere outside is small
•Draw a path for Ampere’s law that
crosses inside the solenoid
 B  ds   I
0 l
R
Current I
l
I l  NI
L
=Bl
B
 0 NI
L
Current Coil
•A single circular loop has an axial field
B( z ) 
0iR 2
•Use Biot-Savart law to derive
2( R 2  z 2 )3 / 2
B( z ) 
0iR 2
2( z 3 )
•Far away from the loop
Current Coil as a Dipole

0iR 2
B( z ) 
2( z 3 )


0iA
0 
B( z ) 

3
2p ( z ) 2p ( z 3 )
•Far away from the loop
Rewrite in different form
•So we can yet again consider a current-loop
as a magnetic dipole
Magnetic Flux
•Magnetic Flux is the amount of magnetic field flowing through
a surface
•It is the magnetic field times the area, when the field is
perpendicular to the surface
•It is zero if the magnetic field is parallel to the surface
•Normally denoted by symbol B.
•Units are T·m2, also known as a Weber (Wb)
Magnetic
Field
B = BA
Magnetic
Field
B = 0
Magnetic Flux
•When magnetic field is at an angle, only the part perpendicular
to the surface counts (does this ring a bell? Remember Gauss’s
B
law? )
•Multiply by cos 
Bn

•For a non-constant magnetic
field, or a curvy surface, you
have to integrate over the surface
B = BnA= BA cos 
 B   B  dA   B cos  dA
Quiz
A sphere of radius R is placed near a very long straight wire
that carries a steady current I. The total magnetic flux
passing through the sphere is:
A) oI
B) oI/(4pR2)
C) 4pR2oI
D) zero
Gauss’s Law
•Magnetic field lines are always loops, never start or end on
anything (no magnetic monopoles)
•Net flux in or out of a region is zero
 B  dA  0
•Gauss’s Law for electic fields
 E   E  dA  4p ke qin
Four closed surfaces are shown. The areas Atop and Abot of
the top and bottom faces and the magnitudes Btop and
Bbot of the uniform magnetic fields through the top and
bottom faces are given. The fields are perpendicular to the
faces and are either inward or outward. Which surface has
the largest magnitude of flux through the curved faces?
Ampere’s Law
 B  ds   I
I
0
•Ampere’s law says that if we take the dot product of the field
and the length element and sum up (i.e. integrate) over a
closed loop, the result is proportional to the current through
the surface
•This is not quite the same as gauss’s law
A Problem with Ampere’s Law
•Consider a parallel plate capacitor that is
being charged
•Try Ampere’s Law on two nearly
identical surfaces/loops
I
 B  ds   I
 B  ds   I
1
0 1
0
2
0 2
 0 I  0
•Magnetic fields cannot be different just because
surfaces are chosen slightly different
•When current flows into a region, but not out,
Ampere’s law must be modified
I
Ampere’s Law Generalized
•When there is a net current flowing into a region, the charge
in the region must be changing, as must the electric field.
•By Gauss’s Law, the electric flux must be changing as well
•Change in electric flux creates magnetic fields, just like currents do
•Displacement current – proportional to time derivative of electric
flux
dE
Id  0
dt
dE
 B  ds  0 I  0 0 dt
Magnetic fields are created both
by currents carried by conductors
(conduction currents) and by
time-varying electric fields.
Generalized Ampere’s Law Tested
•Consider a parallel plate capacitor that is
being charged
•Try Ampere’s modified Law on two
nearly identical surfaces/loops
I
d  E1
 B1  ds  0 I1  0 0 dt
d Q
 0 0    0 I
dt   0 
dE2
 B2  ds  0 I 2  0 0 dt  0 I
I
Comparision of Induction
dE
 B  ds  0 I  0 0 dt
dB
 E  ds   dt
•No magnetic monopole, hence no magnetic current
•Electric fields and magnetic fields induce in opposite fashions
Magnetism in Matter: Types
•Depending on details of the electronic structure on can have
permanent or induced magnetic moments – similar to the case
with dipoles.
Diamagnetic substances: induced magnetic moments counter to the
applied field.
All substances have a diamagnetic response, but it is weak.
Paramagnetic substances: the molecules have permanent moments, but are
weakly coupled and require a field to line them up, and the net moments
are in the direction of the field.
Ferromagnetic substances: the atoms have permanent moments and are
strongly coupled. The atoms can remain aligned in the absence of a field.