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W06D2
Magnetic Forces and Sources of
Magnetic Fields
W06D2 Magnetic Force on Current Carrying Wire, Sources
of Magnetic Fields: Biot-Savart Law
Reading Course Notes: Sections 8.3, 9.1-9.2
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Announcements
Review Week 6 Tuesday from 9-11 pm in 26-152
PS 6 due Week 6 Tuesday at 9 pm in boxes outside 32-082 or
26-152
W06D3 PS05: Calculating Magnetic Fields and Magnetic Force
Reading Course Notes: Sections 8.9, 8.10, 9.10.1, 9.11.1-.3,
9.11.7-.8
Exam 2 Thursday March 21 7:30 - 9:30 pm
Conflict Exam 2 March 22 9-11 am or 10-12 noon
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Outline
Magnetic Force on Current Carrying Wire
Sources of Magnetic Fields
Biot-Savart Law
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Magnetic Force on Current-Carrying Wire
Current is moving charges, and we know that
moving charges feel a force in a magnetic field
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Magnetic Force on Current-Carrying Wire
dFmag  dqv  B
d s dq
dqv  dq

d s  Id s
dt dt
dFmag  Id s  B
Fmag 

Id s  B
wire
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Magnetic Force on Current-Carrying Wire
If the wire is a uniform magnetic field then
Fmag


   Id s   B
 wire

If the wire is also straight then
Fmag  I (L  B)
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Demonstration:
Wire in a Magnetic Field (Jumping
Wire) G8
dFmag  Id s  B
http://tsgphysics.mit.edu/front/?page=demo.php&letnum=G%208&show=0
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Demonstration:
Series or Parallel Current Carrying
Wires G9
http://tsgphysics.mit.edu/front/?page=demo.php&letnum=G%209&show=0
You tube video of experiment showing parallel wires attracting
http://youtu.be/rU7QOukFjTo
or repelling:
http://youtu.be/bnbKdbXofEI
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Group Problem: Current Loop
A conducting rod of uniform
mass per length l is
suspended by two flexible
wires in a uniform magnetic
field of magnitude B which
points out of the page and a
uniform gravitational field of
magnitude g pointing down. If
the tension in the wires is zero,
what is the magnitude and
direction of the current in the
rod?
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Sources of Magnetic Fields
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What Creates Magnetic Fields:
Moving Charges
http://youtu.be/JmqX1GrMYnU
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Electric Field Of Point Charge
An electric charge produces an electric field:
1
q
ˆ
E
r
2
4 o r
r̂
: unit vector directed from the charged object
to the field point P
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Magnetic Field Of Moving Charge
Moving charge with velocity v produces magnetic field:
P
o q v x rˆ
B
2
4 r
r̂ : unit vector directed
r̂
from charged object
to P
0  4  10 T  m  A
7
1
permeability of free space
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Currents are Sources of Magnetic Field
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Demonstration:
Field Generated by Wire G12
http://tsgphysics.mit.edu/front/?page=demo.php&letnum=G%2012&show=0
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Continuous Moving Charge
Distributions:
Currents & Biot-Savart
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From Charges to Currents?
dB  dq v
[meter]
 [coulomb]
[sec]
v
[coulomb]

[meter]
dq
[sec]
d s dq
dB  dqv  dq

d s  Id s
dt dt
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The Biot-Savart Law
Current element ds of length ds pointing in
direction of current I produces a magnetic field:
0 I ds  rˆ
dB 
2
4 r
http://web.mit.edu/viz/EM/visualizations/magnetostatics/MagneticFieldConfigurations/CurrentElement3d/CurrentElement.htm
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The Right-Hand Rule #2
dir d s  zˆ
dir B( P)  zˆ  rˆ  ˆ
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Concept Question: Biot-Savart
The magnetic field at P points towards the
1.
2.
3.
4.
5.
6.
7.
+x direction
+y direction
+z direction
-x direction
-y direction
-z direction
Field is zero
P18- 20
Concept Question Answer: Biot-Savart
Answer: 6. dir B( P) is in the negative z
direction (into page)
The vertical line segment
contributes nothing to the field at
P (it is parallel to the
displacement). The horizontal
segment makes a field pointing
into the page by the right-hand
rule # 2 (right thumb in direction of
current, fingers curl into page at P.
P18- 21
Concept Question: Bent Wire
The magnetic field at P is equal to the field of:
1.
2.
3.
4.
a semicircle
a semicircle plus the field of a long straight wire
a semicircle minus the field of a long straight wire
none of the above
P18- 22
Concept Question Answer: Bent Wire
Answer: 2. Semicircle + infinite wire
r
All of the wire makes dir B(P) into the page. The
two straight parts, if put together, would make an
infinite wire. The semicircle is added to this to get
the magnetic field
P18- 23
Repeat Demonstration:
Parallel & Anti-Parallel Currents G9
http://tsgphysics.mit.edu/front/?page=demo.php&letnum=G%209&show=0
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Concept Question: Parallel Wires
Consider two parallel current carrying
wires. With the currents running in the
opposite direction, the wires are
1.
2.
3.
4.
5.
attracted (opposites attract?)
repelled (opposites repel?)
pushed another direction
not pushed – no net force
I don’t know
P18- 25
Concept Q. Answer: Parallel Wires
Answer: 1. The wires are repelled
I1 creates a magnetic field into the
page at wire 2. That makes a force
on wire 2 to the right.

F2  I 2 L2  B1

I2 creates a magnetic field into the
page at wire 1. That makes a force
on wire 1 to the left.

F1  I1 L1  B 2

P18- 26
Can we understand why?
Whether they attract or repel can be seen in the
shape of the created B field
http://youtu.be/5nKQjKgS9z0
http://youtu.be/nQX-BM3GCv4
You tube videos of field line animations showing why showing parallel
wires attract: or repel:
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Concept Question: Current Carrying Coils
The above coils have
1. parallel currents that attract
2. parallel currents that repel
3. opposite currents that attract
4. opposite currents that repel
P18- 28
Concept Q. Answer: Current Carrying Coils
Answer: 4. Opposite currents that repel
Look at the field lines at the edge between the
coils. They are jammed in, want to push out.
Also, must be in opposite directions
http://web.mit.edu/viz/EM/visualizations/magnetostatics/ForceOnCurrents/MagneticForceRepel/MagneticForceRepel.htm
P18- 29
Example : Coil of Radius R
Consider a coil with radius R and current I
I
I
P
I
Find the magnetic field B at the center (P)
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Example : Coil of Radius R
Consider a coil with radius R and current I
I
I
P
I
1) Think about it:
• Legs contribute nothing
I parallel to r
• Ring makes field into page
2) Choose a ds
3) Pick your coordinates
4) Write Biot-Savart
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Animation: Magnetic Field Generated by
a Current Loop
http://web.mit.edu/viz/EM/visualizations/magnetostatics/calculatingMagneticFields/RingMagInt/RingMagIntegration.htm
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Example: Coil of Radius R
In the circular part of the coil…
d s  ˆr  | d s  ˆr |  ds
I
I
r̂

ds
I
Biot-Savart:
0 I
dB 
4
0 I

4
0 I

4
d s  rˆ
r2
R d
R2
d
R
0 I ds

4 r 2
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Example : Coil of Radius R
Consider a coil with radius R and current I
I
I


ds
r
0 I
B
2R
I
0 I d
dB 
4 R
B   dB 
2

0
0 I d
4 R
2
0 I
0 I

d 
2 


4 R 0
4 R
direction into page
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Group Problem:
B Field from Coil of Radius R
Consider a coil with radius R and carrying a current I
What is B at point P? WARNING:
This is much
harder than
the previous
problem.
Why??
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