AMPERE’S LAW (22.9 in text)
Ampere’s law: relates magnetic field around a closed path to current through a
surface bounded by that path

 Useful for calculating B only in case of high symmetry
o Similar tool for magnetic fields that Gauss’s law is for electric fields
Plan:
 1st: state Ampere’s Law
 2nd: explain what it means
 3rd: justify by showing that it gives previous result for field around a long wire
WHAT IS AMPERE’S LAW?
 
 B  ds  0 I through loop
around
closed
loop
WHAT DOES THIS MEAN?
 Look at a closed path (any shape) around a current I
 At each point around path, can take scalar product of


B with small segment of path ds
 
B
 result:  ds

 
 
B

d
s

means sum B  ds for all segments around
B
ds
ds
Iout of screen/page
ds
B
closed loop
the loop
 Ampere’s law says that resulting sum equals  0 I where I is the
TOTAL current “threading” through the closed loop.
B
AMPERE’S LAW:
 
 B  ds  0 I through loop
around
closed
loop
 True for ANY loop (called Amperian loop) around a steady current BUT:
 Only helpful for calculating field for special (symmetric) cases
o i.e. ones for which B can be factored out of the integral
 Cases where Ampere’s law might be useful:

o B is known constant over part of the path (by symmetry)
 


o B parallel to ds so that B  ds  B ds for part of the path
 


o B perpendicular to ds so that B  ds  0 for part of the path

o B  0 for part of the path
Ampere’s Law says
around
loop
for ANY loop around a given current.
 
 B  ds is the same for loop 1 and
 Example:
loop 2
2
1
I
Ampere’s Law says
 Example:
 
 B  ds  0 I
 
 B  ds  0
around
loop
for ANY loop if no current threads it
 
 B  ds  0 for this loop.
 
 Since B  ds  0 for the two straight
segments, this relates field at two
different distances from wire
I
Now look at long, current-carrying wire: use loop, radius r, centred on wire
 No end effects so magnetic field lines must be circles centred on wire
o no component parallel to wire (symmetry)
 
B
  ds  B ds for every segment of the loop
o SO:
 
 B  ds  B
around
loop
 because
 ds  B 2 r
around
loop
 ds  circumference 2 r
around
loop
 Ampere’s Law says
 
 B  ds  0 I
around
loop
o SO: B  2 r   0 I
0 I
B

 Rearrange to get previous long wire result:
2 r
B
ds
ds
Iout of screen/page
ds
B
B
TOROID – a wire wound around a ring
 For this case, 7 turns around ring
B

 To find B at distance r from centre
o Draw Amperian loop of radius r
r
I

 B is tangent to Amperian loop everywhere
 
B
o so:   ds  B  ds  B 2  r
I
 Amount of current “threading” through Amperian loop is I through  N turns I  7 I
 
 Ampere’s law says  B  ds   0 I through
o SO: B 2 r   0 N turns I
 Result: Btoroid 
 0 N turns I 7 0 I

2 r
2 r
o Note: field is NOT uniform inside the toroid
Amperian
Loop
SOLENOID – Important device – wire wound into a long helix

magnetic field lines run up inside solenoid and
return outside
I
o for a real solenoid, the field is weaker at the
ends – end effects

o for an “infinite solenoid”, B is uniform inside
and the field is zero outside
I

B FIELD INSIDE AN INFINITE SOLENOID (with current I )

By symmetry: field inside is uniform, field outside is zero

Draw Amperian loop through side
o Contains N turns in length l
 


o B  ds  0 for sides 2 and 4 ( B  ds )

o B  0 on side 3

B
Iin
Iout
 
So only contribution to  B  ds is from side 1
 
 
 B  ds  side 1B  ds  Bside 1ds  Bl
o around
2
1
3
4
loop

Current “threading” the Amperian loop of length l is NI

So Ampere’s law says:

Result for field inside a solenoid with n=N/L turns per unit length:
Bl   0 N I
B 0 n I
l
MAGNETISM IN MATERIALS

Unpaired electrons in atoms have magnetic moments
o They act as if they are spinning BUT really a relativistic effect
Paramagnetism

Materials with atoms having unpaired electrons
o Some salts (i.e. cerous magnesium nitrate)
o Alloys (PrNi5)
o Oxygen
o Some organometallic compounds

Not magnetic in isolation but in a magnetic field, alignment of unpaired
electrons slightly biased
o Magnetization proportional to applied field
o Magnetization inversely proportional to temperature
 Thermal energy randomizes spin alignment
o Can be attracted into a magnetic field
FERROMAGNETS
 Materials in which unpaired electron spins in small volumes (domains) orient
spontaneously below a transition temperature (TCurie)
o In absence of applied field, magnetization of
domains randomly oriented
 no net magnetization
Bapplied
o In external magnetic field, domains with
magnetization oriented with field grow at
expense of others
 Material develops net magnetization
o If domain walls difficult to move, sample stays magnetized when field
removed
 Permanent magnet
 Examples: Fe (Tc=1043 K), Co (Tc=1388 K), Ni (Tc=627 K), etc.