1
Lecture 24
Ampère’s law
Ampère s law
Circulation around wire
Draw an imaginary loop around a straight infinite wire and compute
⋅
∫
G
G
v
B dl
G
G
is perpendicular to
B
r
B
→
dl
→
θ
⋅
=
G
G
B dl
Brd
rd
θ
I
r
→
dr
π
μ
μ
θ
θ
μ
π
π
⋅
=
=
=
∫
∫
∫
G
G
v
v
2
0
0
0
0
2
2
I
I
B dl
d
d
I
μ
π
=
0
2
I
B
r
μ
θ
π
=
0
2
I
d
Ampère’s law
This result turns out to be true for ANY loop around ANY current.
We will not prove it in the general case. It is partially done in the
book.
μ
⋅
=
∫
G
G
v
0
enclosed
B dl
I
Line integral
Current outside the loop does not make a contribution:
I
⋅
=
∫
G
G
v
Here
0
B dl
Exercise: Prove it for the infinite straight wire.
Calculating
E
and
B
fields
πε
=
G
2
0
1
ˆ
4
q
E
r
r
Coulomb Law
μ
π
×
=
G
G
0
2
ˆ
4
v
r
B
q
r
BiotSavart Law
Always true, can always use, but requires superposition:
ε
⋅
=
∫
G
G
enclosed
closed
0
surface
q
E
dA
Gauss Law
μ
⋅
=
∫
G
G
v
0
enclosed
B dl
I
Ampere’s Law
Always true. Useful to get
E
or
B
when charge/current
distributions are symmetric
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2
Direction of the Amperian loop
Same righthand rule as in the
B
field handytrick:
1)
choose
a direction for
positive
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 Spring '07
 Johnson
 Physics, Current, Magnetic Field, dl, amperian loop

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