Physics 2112
Unit 15
Today’s Concept:
Ampere’s Law
 
B

d



I
o
enclosed

Unit 15, Slide 1
Ampere’s Law
We know for an infinite current
carrying wire
o I
B
2R
But what is 2R? Circumference of circle!
B  2R   o I
 
B

d



I
o

Electricity & Magnetism Lecture 15, Slide 2
Ampere’s Law
Any closed loop
 
B

d



I
o
ENCL

Current enclosed
by that closed
loop
Electricity & Magnetism Lecture 15, Slide 3
Checkpoint 1A
Two loops are placed near identical current carrying wires as
shown in Case 1 and Case 2 below.
For which loop is ∫B·dl greater?
A. Case 1
B. Case 2
C. The integral is the same for
both
Electricity & Magnetism Lecture 15, Slide 4
Checkpoint 1B
Two loops are placed near identical current carrying wires as
shown in Case 1 and Case 2 below.
For which loop is ∫B·dl greater?
A. Case 1
B. Case 2
C. The integral is the same for
both
Electricity & Magnetism Lecture 15, Slide 5
Checkpoint 1C
Two loops are placed near current carrying wires as shown in
Case 1 and Case 2 below. In both cases the direction of the
current in the two wires are opposite to each other.
For which loop is ∫B·dl greater?
A. Case 1
B. Case 2
C. The integral is the same for
both
Electricity & Magnetism Lecture 15, Slide 6
Example 15.1 (B field from a thick wire)
A wire with a radius of r=1cm has a
uniform current of 1A flowing
through it.
What is the B field 1 meter from the
center of the wire?
What is the B field 0.5cm from the
center of the wire?
Unit 15, Slide 7
Example 15.1 (graph)
X
X X X
X X X X X
A wire with a radius
of r=1cm has a
uniform current of 1A
flowing through it.
X
X X X
X X X X X
B
r
Unit 15, Slide 8
CheckPoint 2A
An infinitelyX long hollow
conducting tube carries
current I in the direction
shown.
What is the direction of the magnetic
field inside the tube?
A. clockwise
B. counterclockwise
C. radially inward to the center
D. radially outward from the center
E. the magnetic field is zero
Electricity & Magnetism Lecture 15, Slide 9
Why is that?
X
X
Electricity & Magnetism Lecture 15, Slide 10
Example 15.3 (Pipe of current)
y
An infinitely long cylindrical shell
carries a uniformly distributed current
of 5A out of the screen. The inner
radius is a=4cm and outer radius=8cm
a
I
x
b
What is B at r = 2cm?
What is B at r = 6cm?
What is B at r = 16cm?
Electricity & Magnetism Lecture 15, Slide 11
Example 15.3 (Pipe of current)
 Conceptual Analysis
Complete cylindrical symmetry (can only
depend on r)
 can use Ampere’s law to calculate B
 Strategic Analysis
Calculate B for the three regions separately:
1) r < a
2) a < r < b
3) r > b
 
 B  d   o I enc
For circular path concentric with shell.
B  d  o I enc
Electricity & Magnetism Lecture 15, Slide 12
Example 15.2 (Two thick wires)
A
X
D
B
C
Two thick cables both of
radius R and length L carry
a current, I, side-by-side as
shown to the left. The left
cable has the current into
the screen and the right
cable has the current out of
the screen.
What is the magnetic field at points A, B, C and D?
- A is at the center of the left cable
- B is a distance R/2 from the center of the left cable
- C is at the point where the cables touch.
- D is the bottom of the left cable
Unit 15, Slide 13
The Plan
A
X
B
C
What is the magnetic field at
points A, B, C and D?
D
 Conceptual Analysis
use Ampere’s law to calculate B from each cable separately
 Strategic Analysis
Note direction of B from each cable at each point.
Add B like vectors
Unit 15, Slide 14
Sign of the dot product
I into
screen
 
B

d


I enc
Electricity & Magnetism Lecture 15, Slide 15
Ampere’s Law
 
 B  d   o Ienc
dl
B
dl
B
dl
B
Electricity & Magnetism Lecture 15, Slide 16
Sign of the dot product
 
 B  d   o Ienc
dl
dl B
B
B
dl
Electricity & Magnetism Lecture 15, Slide 17
Sign of the dot product
 
 B  d   o Ienc
dl
B
B
dl
B dl
Electricity & Magnetism Lecture 15, Slide 18
B Field in Solenoid
~cancel out
X
X
.
X
X
.
.
.
add up
Unit 15, Slide 19
B Field in Solenoid
X
X
.
X
X
.
.
X
.
X
.
X
X
.
.
.
L
 
 B  d   o I enc
0  0  0  Bx * L   o * n * L * I
Bx   o nI
n = turns per unit length
(Ideal Solenoid L>>>r)
Unit 15, Slide 20
Solenoid   Bar Magnet
Unit 15, Slide 21
CheckPoint 2B
A current carrying wire is wrapped around cardboard tube
as shown below.
In which direction does the
magnetic field point inside the
tube?
A.
B.
C.
D.
E.
F.
left
right
up
down
out of the screen
into the screen
Electricity & Magnetism Lecture 15, Slide 22
Example 15.3 (B from Solenoid)
A long thin solenoid consists of
500 turns of wire carrying 1.5A. It
has a length of 250cm and a
radius of 1cm.
What is the magnetic field in the exact center of the coil?
What is the contribution to this field from the 250th coil
(the center coil)?
Unit 15, Slide 23
Make Sense?
X
X
.
X
X
.
.
X
.
X
.
X
X
.
.
.
Note:
•
B250/Btot = 0.024
•
So closest coil contributes over 10 tens more than
average coil. Make sense?
(1/500 = 0.002)
Unit 15, Slide 24
When does “Loop”  “Solenoid”?
Electricity & Magnetism Lecture 15, Slide 25
A hint…..
 
B

d
A

0

surface
 
B

d
l


I
o ENCL

loop
 
Qenc
E

d
A

0



o
surface
 
E

d
l

0

loop
Unit 15, Slide 26