Transcript Physics 2102 Spring 2002 Lecture 8

Physics 2102
Jonathan Dowling
Physics 2102
Lecture 16
Ampere’s law
André Marie Ampère
(1775 – 1836)
Ampere’s law:
Remember Gauss’ law?
Given an arbitrary closed surface, the electric flux through it is
proportional to the charge enclosed by the surface.
q
q
Flux=0!


Surface
  q
E  dA 
0
Gauss’ law for Magnetism:
No isolated magnetic poles! The magnetic flux through any closed
“Gaussian surface” will be ZERO. This is one of the four
“Maxwell’s equations”.
B

d
A

0

  q
E

d
A


0
Ampere’s law:
a Second Gauss’ law.

B  d s  0 i
i4
loop
The circulation of B
(the integral of B scalar
ds) along an imaginary
closed loop is proportional
to the net amount of current
traversing the loop.
i2
i3
i1
ds
 
 B  d s  0 (i1  i2  i3 )
loop
Thumb rule for sign; ignore i4
As was the case for Gauss’ law, if you have a lot of symmetry,
knowing the circulation of B allows you to know B.
Sample Problem
• Two square conducting loops carry currents
of 5.0 and 3.0 A as shown in Fig. 30-60.
What’s the value of ∫B∙ds through each of
the paths shown?
Ampere’s Law: Example 1
• Infinitely long straight wire
with current i.
• Symmetry: magnetic field
consists of circular loops
centered around wire.
• So: choose a circular loop C
-- B is tangential to the loop
everywhere!
• Angle between B and ds = 0.
C
(Go around loop in same
direction as B field lines!)
R
 
B

d
s


i
0

C
Bds

B
(
2

R
)


i
0

 0i
B
2R
Ampere’s Law: Example 2
i
• Infinitely long cylindrical
wire of finite radius R
carries a total current i with
uniform current density
• Compute the magnetic field
at a distance r from
cylinder axis for:
– r < a (inside the wire)
– r > a (outside the wire)
r
Current into
page, circular
field lines
R
 
 B  ds  0i
C
Ampere’s Law: Example 2 (cont)
 
B

d
s


i
0

C
Current into
page, field
tangent to the
closed
amperian loop
r
B(2r )  0ienclosed
 0ienclosed
B
2r
2
i
r
2
2
ienclosed  J (r )  2 r  i 2
R
R
 0ir
For r>R, i =i, so
For r < R
B
2
B= i/2R
2R
enc
0
Solenoids
 
 B  ds  0ienc
 
 B  ds  0  B h  0  0
ienc  iN h  i ( N / L)h  inh
 
 B  ds  0ienc  Bh  0inh  B  0in
Magnetic Field of a Magnetic Dipole
A circular loop or a coil currying electrical current is a magnetic
dipole, with magnetic dipole moment of magnitude =NiA.
Since the coil curries a current, it produces a magnetic field, that can
be calculated using Biot-Savart’s law:



0
0 

B( z ) 

2
2 3/ 2
2 ( R  z )
2 z 3
All loops in the figure have radius r or 2r. Which of these
arrangements produce the largest magnetic field at the
point indicated?
Sample Problem
Calculate the magnitude and direction of the
resultant force acting on the loop.