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Ampere’s Law
Is there something like Gauss’ Law for Electric Fields that
helps us solve for Magnetic Fields in a simpler way for cases
with nice symmetries?
∫
loop
v v
B ⋅ dl = μ0ithru
For any closed imaginary loop where the current is constant,
the above relation is true.
Ampere’s Law
Key point
Ampere’s Law is the intergral around a 1-dimensional closed loop.
Gauss’ Law was the integral over a 2-dimensional closed surface.
v v Q
Φ E = ∫ E • da = inside
ε0
surface
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Clicker Question
∫
loop
v v
B ⋅ dl = μ0ithru
114
Example – B-field from a
long straight wire
i
We need a sign convention for i(thru).
Place the fingers of your right hand around the imaginary loop
and your thumb points in the direction of positive i(thru).
What is i(thru) in the below case where all three wires have 5 A?
A) i(thru) = +15 Amps
X
B) i(thru) = +5 Amps
X
C) i(thru) = -5 Amps
D) None of the above
dz
θ
r
B=?
R
z direction
v
v μ i dL × rˆ
dB = 0
4π r 2
v μ0i dz
sin θ
dB =
4π r 2
v μi
dzR
dB = 0 2
4π ( z + R 2 )3 / 2
v
v
B = ∫ dB =
+∞
μ 0i
∫ 4π
−∞
Recall our
derivation
using
BiotSavart
dzR
( z 2 + R 2 )3/ 2
v μi
B= 0
2πR
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Clicker Question
Now try Ampere’s Law...
Look at the previous wire from the view where the wire current
is coming out of the page.
The Ampere circular loop is drawn in blue.
i
Place an imaginary circular loop of radius R
around the wire.
What is the relation of the vector dl that
one integrates around the loop as shown
and the Magnetic Field vector?
A)Parallel
B)Anti – Parallel
C)Perpendicular
D)Depends where in the loop for dl
v v
B
∫ ⋅ dl
loop
Vector dot product!
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i
dl
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1
Thus, this is a special symmetry case where
all around the loop the two vectors are
parallel.
v v
v v v
B
⋅
d
l
=
|
B
|
d
l
∫
∫ =| B | 2πR
i
v v
B || dl
loop
i
loop
Where R is the radius of the loop.
∫
In fact, also due to symmetry, we know that
the magnitude of the B-field is also constant.
loop
v v
B ⋅ dl = μ 0ithru
v μi
| B |= 0
2πR
v
| B | 2πR = μ 0i
v
| B | constant around the loop
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Clicker Question
A long straight copper wire has radius b and carries a
constant current of magnitude i. The current density of
magnitude J=i/(πb2) is uniform throughout the wire. What is
the current contained in the circular loop £, with radius r < b,
centered on the wire's center as shown?
Ampere’s Law (for constant currents) is always true.
∫
loop
120
v v
B ⋅ dl = μ 0ithru
However, like Gauss’ Law, it is only useful if there is a nice
symmetry for solving the left integral.
b
r
b
3
C) i r 3
b
A) i
B) i
r2
b2
r
D) None of these.
J = constant
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∫
Clicker Question
loop
122
v v
B ⋅ dl = μ0ithru
For r < b:
b
loop
A long straight copper wire has radius b and carries a
constant current of magnitude i. The current density of
magnitude J=i/(πb2) is uniform throughout the wire.
How does the magnitude of the B-field a distance r < b from
the center of the wire depend on r?
r
B) B = constant
D) B ∝ 1/r2
v v
B ⋅ dl = μ0ithru
v
⎛ r2 ⎞
| B | 2πr = μ 0 ⎜⎜ i 2 ⎟⎟
⎝ b ⎠
J = constant
b
A) B ∝ r
C) B ∝ 1/r
E) None of these
∫
r
For r >= b:
J = constant
v
r
| B |= μ0i
2πb 2
v μi
| B |= 0
2πr
|B|
123
b
r
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2
∫
Clicker Question
loop
v v
B ⋅ dl = μ 0ithru
Solenoid
We have determined that for the closed loop as drawn below the
integral on the left is -10 Tesla meters. What is the direction of
the current in the red wire shown below?
A)
B)
C)
D)
E)
Into the Page
Out of the Page
Left
Up
Down
A single wire tightly coiled up into loops.
Since it is a single wire, the current magnitude is the same in
all parts of the coil.
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Demonstration
What is the B-field from a solenoid of N turns and length L?
Length = L
Current
Direction
Number of Turns = N
n = N/L
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Side View
v v
B ⋅ dl = μ0ithru
Zero B Outside
∫
Uniform B Inside
v
| Binside | l = μ 0 (nli )
loop
Clicker Question
# loops in length l
X X X X X X X X
l
128
v
N
| Binside |= μ 0 ni = μ 0 i
L
Three long straight solenoids all of
length L and all with the same
(large) number of closely-packed
turns N, all with the same current i,
have different cross-sections as
shown.
A
B
C
Which solenoid has the largest field |B| at its center?
A) A
B) B
C) C
D) All three have the same magnitude magnetic field
B-Field Inside a Solenoid
129
Answers: The field is the same magnitude and uniform for all
three solenoids. The field within a solenoid is B = μni. This
depends only on the current i and the turns per length n. This
formula does not depend on either the cross-sectional shape
of the solenoid or the position within the solenoid.
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