Physics 272 -- Lecture 15
Problem 27-60
Sagitta (arrow) Rule
Phys-272 Lecture 15
Ampere’s Law
r r
B
⋅
d
l
=
µ
I
0
enclosed
∫
A Solenoid
Magentic
Levitation of
High Tc
superconductor
A
B
C
The integral is the same since both enclose the same
amount of current.
A
B
C
Case 2 has no current enclosed.
In which case, is the B field halfway between the
wires the largest ?
(A) Case 1
(B) Case 2
(C) The fields are the same
the wires in case 2 produce a B field coming out of the page at P,
producing a non-zero B field. the wires in case 1 produce opposite
B fields at P, producing a zero B field at P.
A current-carrying wire is wrapped
around a cardboard tube (as shown)
In what direction does the magnetic field
point inside the tube ?
(A) Up
(B) Down
(C) to the right
(D) to the left
using the right hand rule and curling your fingers in the
direction of current, your thumb points in the direction of
the B field, which happens to be to the left.
The direction of the B field from a long current carrying wire is in the
a) Radial direction b) tangential direction c) along the wire d) anti-parallel to the current
The magnitude of the field is proportional to
(a) R
(b) R2
(c) 1/R2
(d) 1/R
Ampere’s Law (Anatomy of the Line Integral)
I into
screen
r r
∫ B ⋅ dl = µ0 I enc
r r
∫ B ⋅ dl = µ0 I enc
Amperes Law
dl
B
dl
B
dl
B
r r
∫ B ⋅ dl = µ0 I enc
Ampere’s Law
dl B
B
dl
B
dl
r r
∫ B ⋅ dl = µ0 I enc
Ampere’s Law
dl
B
B
dl
B dl
Which of the following current
distributions would give rise to the
B . dL distribution on the right.
Hint: Which one will give a line integral of zero on the vertical side ?
A
B
C
Use Ampere’s Law to find the B field of a long solenoid
Assume current I, n turns per unit length
III
II
IV
I
On I, B an dL are parallel
On II and IV, B is perp to dL
On III, B=0
r r
∫ B • dl = Bl
r r
∫ B • dl = Bl = µo (nlI )
By symmetry and Biot-Savart, direction
is along the axis of the solenoid
B = µo nI
Use Ampere’s Law to find the B field of a toroid
Assume N
loops,
current I
A current NI passes through the circular Amperian loop
r r
∫ B •dl = µo NI
r r
∫ B •dl = B(2π r ) = µo NI
µo NI
B=
2π r
y
Example Problem
An infinitely long cylindrical shell with inner radius a
and outer radius b carries a uniformly distributed
current I out of the screen.
a
Sketch |B| as a function of r.
• Conceptual Analysis
–
–
x
b
Complete cylindrical symmetry (can only depend on r)
⇒ can use Ampere’s law to calculate B
B field can only be clockwise, counterclockwise or zero!
r r
∫r B ⋅ dlr = µ0 I enc
B ∫ dl = µ0 I enc For circular path concentric w/ shell
• Strategic Analysis
Calculate B for the three regions separately:
1) r < a
2) a < r < b
3) r > b
I
y
Example Problem
I
r
a
What does |B| look like for r < a ?
r r
∫ B ⋅ dl = µ0 I enc
r
so B = 0
0
(A)
(B)
(C)
b
x
y
Example Problem
I
r
a
What does |B| look like for r > b ?
r r
∫ B ⋅ dl = µ0 I enc
I
(A)
(B)
(C)
b
x
y
Example Problem
dl
r
I
B
a
What does |B| look like for r > b ?
r r
∫ B ⋅ dl = µ0 I
∫ Bdl
B ∫ dl
B 2π r
B 2π r = µ0 I
µ0 I
B=
2π r
b
x
y
Example Problem
I
r
a
What does |B| look like for r > b ?
B=
(A)
µ0 I
2π r
(B)
(C)
b
x
y
Example Problem
I
a
What is the current density j (Amp/m2) in the
conductor?
(A)
I
j= 2
πb
(B)
I
j= 2
2
πb +πa
(C)
b
I
j= 2
2
πb −πa
x
Example Problem
y
I
What is the current density j (Amp/m2) in the
conductor?
j = I/area
area = π b2 − π a 2
I
j= 2
2
πb −πa
a
b
x
y
Example Problem
I
a
What is the current density j (Amp/m2) in the
conductor?
(A)
I
j= 2
πb
(B)
I
j= 2
2
πb +πa
(C)
b
I
j= 2
2
πb −πa
x
y
Example Problem
I
r
a
What does |B| look like for a < r < b ?
(A)
(B)
(C)
b
x
y
Example Problem
I
r
a
What does |B| look like for a < r < b ?
r r
∫ B ⋅ dl = µ0 I enc
2π rB = µ0 j areaencloses
2
areaenc = π r − π a
2
I
2π rB = µ0 (π r − π a ) ×
2
2
π
π
b
−
a
(
)
2
B = µ0
(π r
2
−πa )
I
×
2
2
π
π
2π r
b
a
−
(
)
2
2
Starts at 0 and
increases linearly
b
x
y
Example Problem
I
r
a
What does |B| look like for a < r < b ?
(A)
(B)
(C)
b
x
y
Example Problem
An infinitely long cylindrical shell with inner radius a
and outer radius b carries a uniformly distributed
current I out of the screen.
a
Sketch |B| as a function of r.
I
x
b