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```Physics 2112
Unit 14
Today’s Concept:
What Causes Magnetic Fields
 0 I d s  rˆ
dB 
2
4 r
Unit 14, Slide 1
Compare to Electric Fields

1 qrˆ12
E
2
4 o r12

  0 qv  rˆ12
B
2
4 r12
0  4 107 Tm/ A
v out of the
screen
In the same direction as r12
Perpendicular to r12
Unit 14, Slide 2
Biot-Savart Law

 0 qv  rˆ12
B
2
4 r12
But remember from
previous slides

  0 I d s  rˆ
dB 
2
4 r
B field from one
moving charge
I  qnAvavg
B field from tiny

ds of current
carrying wire.
Unit 14, Slide 3
Example 14.1 (Infinite wire of current)
What is the magnetic field a distance yo away
from a infinitely long wire of current I?
 Conceptual Plan
Use Biot-Savart Law
 Strategic Analysis
Done in prelecture in detail
 0 I d s  rˆ
Integrate dB 
4 r 2
(Similar to E field for infinite line of charge)
Unit 14, Slide 4
Main Idea
rˆ
Q
f

ds
f

ds

B

 0 I d s  rˆ  0 I dx * sin( )

2
2
2
4

( x  yo )
4 r
Unit 14, Slide 5
Magnitude:
0 I
B
2 r
B
Current I OUT
r
•
0  4 10 Tm/ A
7
Remember:

E
 o 2 r
Unit 14, Slide 6
Example 14.2 (B field from hexagon)
120o
b
A current, I, flows clockwise
through a hexagonal loop of
wire. The perpendicular
distance between each side
and the center of the loop is b.
What is the magnetic field in
the center of the loop?
Q
f

ds
rˆ
f

ds
Unit 14, Slide 7
Example 14.3 (From Loop)
A current, I, flows clockwise
through a circular loop of wire.
The loop has a radius a.
P
yo
a
What is the magnetic field at a
point P a distance yo above
the plane of the loop in the
center?
Q
x
BcosQ
Q
x
Unit 14, Slide 8
Force Between Current-Carrying Wires
. . . . . . . . . . .
. . . . . . . . . . .
. . . . . . . . . . . . . .
.....................
X X X X X X X X X X X X X X X X X X X
X X X X X X X X X X X X X
X
X
X
X
X
X
X
X
X
X
X
F
X
X
X
X
X
X
F12  I 2 L 
X
X
X
 0 I1
I1 B  2 r
X
X
I2
 
F12  I2 L  B
o
I1
2d
Unit 14, Slide 9
Force Between Current-Carrying Wires
I towards
us
•
B
•
d
F

 
F12  I2 L  B
F12  I 2 L 
o
I1
2d
Another I towards us
Conclusion: Currents in same direction attract!
I towards
us
•
d
B
 F
Another I away from us
Conclusion: Currents in opposite direction repel!
Unit 14, Slide 10
Example 14.4 (Two Current Carrying Wires)
50cm
I2 = 10A
10cm
I1 = 5A
Two current carrying wires a 10cm apart for a length for
50cm. Wire 1 carries 5A and Wire 2 carries 10A with both
current to the left.
What is the magnitude and direction of the force on wire 2
due to wire 1?
Unit 14, Slide 11
CheckPoint 1A
X
B
F
What is the direction of the force on wire 2 due to wire 1?
A) Up
B) Down C) Into Screen D) Out of screen E) Zero
Unit 14, Slide 12
CheckPoint 1B
What is the direction of the torque on wire 2 due to wire 1?
A) Up
B) Down C) Into Screen D) Out of screen E) Zero
Uniform force at every segment of wire
Unit 14, Slide 13
CheckPoint 3A
What is the direction of the force on wire 2
due to wire 1?
A) Up
B) Down C) Into Screen
D) Out of screen E) Zero
Unit 14, Slide 14
CheckPoint 3B
What is the direction of the torque on wire 2
due to wire 1?
A) Up
B) Down C) Into Screen
D) Out of screen E) Zero
LET’S DRAW A PICTURE!
Unit 14, Slide 15
Checkpoint 2: Force on a loop
A current carrying loop of width a and
length b is placed near a current carrying
wire.
How does the net force on the loop
compare to the net force on a single wire
segment of length a carrying the same
amount of current placed at the same
distance from the wire?
A.
B.
C.
D.
E.
the forces are in opposite directions
the net forces are the same.
the net force on the loop is greater than the net force on the wire segment
the net force on the loop is smaller than the net force on the wire segment
there is no net force on the loop
Checkpoint question
Current flows in a loop as shown in
the diagram at the right. The direction
is such that someone standing at
point a and looking toward point b
would see the current flow clockwise.
What is the orientation of the
magnetic field produced by the loop at
points a and b on the axis?
(A)
(B)
(C)
(D)
B on axis from Current Loop
I
Resulting B Field
Current in Wire
Electricity & Magnetism Lecture 14, Slide 18
Biot-Savart Works, but need to do numerically
See Simulation!
Unit 14, Slide 19
Two Current Loops
Two identical loops are hung next to each other. Current
flows in the same direction in both.
The loops will:
A) Attract each other
B) Repel each other
C) There is no force between them
Two ways to see this:
1) Like currents attract
2) Look like bar magnets
N
S
N
S
Unit 14, Slide 20
Right Hand Rule Review
1. ANY CROSS PRODUCT

 
F  qv  B
 
  r F

 
F  IL  B

  B
 0 I d s  rˆ
dB 
4 r 2
2. Direction of Magnetic Moment
Fingers: Current in Loop
Thumb: Magnetic Moment
3. Direction of Magnetic Field from Wire
Fingers: Magnetic Field
Thumb: Current
Unit 14, Slide 21
Example 14.2
y
Two parallel horizontal wires are
located in the vertical (x,y) plane as
shown. Each wire carries a current
of I  1A flowing in the directions
shown.
y
I1  1A
.
4cm
4cm
I2  1A
Front view
What is the B field at point P?
x
3cm
z
P
Side view
 Conceptual Analysis
Each wire creates a magnetic field at P
B from infinite wire: B  0I / 2r
Total magnetic field at P obtained from superposition
 Strategic Analysis
Calculate B at P from each wire separately
Total B = vector sum of individual B fields
Unit 14, Slide 22
Example 14.5 (Curved Loop of Wire)
If I = 6A, what is the
magnitude of the
magnetic field at point
P?
20cm
Conceptual Plan
Use Biot-Savart Law
Strategic Analysis
 0 I d s  rˆ
Integrate dB 
both loops
2
4 r
Note straight sections cancel out.
P
12cm
Good News!!!!!
Remember how we used Gauss’
Law to avoid doing integral in E
field?
We got similar law for B fields!
Unit 14, Slide 24
```