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Biot-Savart law, Problems for practice
1.
Physics 212
Lecture 14
Biot-Savart Law
0 I ds rˆ
dB
4 r 2
:05 Physics 212 Lecture 14, Slide 1
2.
Main Point 1
First, we introduced the Biot-Savart law, the fundamental law that allows us to calculate the
magnetic field dB that is produced a distance r from a current segment Ids. The key features of
the Biot-Savart law are that the magnitude of the field is proportional to the inverse square of the
distance from the current segment, and the direction is given by a cross product. We applied this
law to determine that the magnetic field produced by an infinite straight wire is proportional to
the current it carries and falls of as 1 over R, the perpendicular distance from the wire. The
direction of the field at any point on a circle of radius R from the wire is tangent to that circle
with its sense determined by a right hand rule. Namely, if you place the thumb of your right hand
in the direction of the current, your fingers will curl in the direction of the B field.
Physics 212 Lecture 14, Slide 2
3.
Main Point 2
Second, we calculated the force between two parallel current-carrying wires by determining the Lorentz force
on the charge carriers in one wire due to the magnetic field produced by the other wire. We determined this
force to be proportional to the product of the currents and inversely proportional to the separation between the
Physics 212 Lecture 14, Slide 3
4.
Main Point 3
Finally, we investigated the magnetic field produced by a circular current carrying loop. We found that the field
was directed along the axis of the loop, parallel to the loop’s magnetic dipole moment vector, and that the field
was maximum at the plane of the loop and decreased with increasing z, the distance from the plane of the loop.
Physics 212 Lecture 14, Slide 4
5.
Biot-Savart Law:
What is it?
Fundamental law for 0 I ds rˆ
determining the direction and
dB
4 r 2
magnitude of the magnetic field
due to an element of current
We can use this law to calculate the magnetic field produced by ANY current distribution
BUT
Easy analytic calculations are possible only for a few distributions:
Axis of Current Loop
Infinite Straight Wire
Plan for Today: Mainly use the results of these calculations!!
GOOD NEWS: Remember Gauss’ Law? NEXT TIME: Introduce Ampere’s Law
Allowed us to calculate E for Allows us to calculate B for
symmetrical charge distributions symmetrical current distributions
:05 Physics 212 Lecture 14, Slide 5
6.
B from infinite
line
I ds rˆ
of current
Integrating dB 0
gives result
4 r2
Magnitude:
0 I B Current I OUT
B r
2 r
0 4 10 7 Tm / A •
r = distance from wire
Direction:
Thumb: on I
Fingers: curl in direction of B I
Current in Wire
Resulting B field
:07 Physics 212 Lecture 14, Slide 6
7.
Checkpoint 1
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
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
:18 Physics 212 Lecture 14, Slide 7
8.
Physics 212 Lecture 14, Slide 8
9.
Checkpoint 2
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. The forces are in opposite directions
A The net forces are the same
B.
B
C.
C The net force on the loop is greater than the net force on the wire segment
D The net force on the loop is smaller than the net force on the wire segment
D.
E There is no net force on the loop
E.
:24 Physics 212 Lecture 14, Slide 9
10.
Physics 212 Lecture 14, Slide 10
11.
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
:21 Physics 212 Lecture 14, Slide 11
12.
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
:24 Physics 212 Lecture 14, Slide 12
13.
Physics 212 Lecture 14, Slide 13
14.
Currents + Charges
A long straight wire is carrying current from left to right. Two
identical charges are moving with equal speed. Compare the force
on charge a moving directly to the right, to the force on charge b
moving up and to the right at the instant shown (i.e. same
distance from the wire).
v
v
(a) • (b) •
r r I
F qv B
a) |Fa |> |Fb|
b) |Fa |= |Fb|
c) |Fa |< |Fb|
Physics 212 Lecture 14, Slide 14
15.
Physics 212 Lecture 14, Slide 15
16.
Adding Magnetic Fields
Two long wires carry opposite current
x
x
What is the direction of the magnetic field above, and midway
between the two wires carrying current – at the point marked “X”?
A) Left B) Right C) Up D) Down E) Zero
:13 Physics 212 Lecture 14, Slide 16
17.
Force between current-carrying wires
0
I towards
F12 I1 I2 L
2 d
us • •
Another I towards us
Conclusion: Currents in same direction_____________
I towards
us
•
Another I away from us
Conclusion: Currents in opposite direction ____________
:16 Physics 212 Lecture 14, Slide 17
18.
Physics 212 Lecture 14, Slide 18
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
Physics 212 Lecture 14, Slide 19
20.
Calculation
Two parallel horizontal wires are located y
y
in the vertical (x,y) plane as shown. Each
wire carries a current of I =1A flowing in I1=1A
the directions shown. .
4cm 3cm
x z P
What is the B field at point P? 4cm
I2=1A
Front view Side view
Conceptual Analysis
Strategic Analysis
:33 Physics 212 Lecture 14, Slide 20
21.
Physics 212 Lecture 14, Slide 21
22.
Physics 212 Lecture 14, Slide 22
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