Copyright © 2008 Pearson Education Inc., publishing as Pearson Addison-Wesley PowerPoint ® Lectures for University Physics, Twelfth Edition – Hugh D. Young and Roger A. Freedman Lectures by James Pazun Chapter 28 Sources of Magnetic Field
Copyright © 2008 Pearson Education Inc., publishing as Pearson Addison-Wesley
PowerPoint® Lectures for
University Physics, Twelfth Edition
– Hugh D. Young and Roger A. Freedman
Lectures by James Pazun
Chapter 28
Sources of MagneticField
Copyright © 2008 Pearson Education Inc., publishing as Pearson Addison-Wesley
Topics for Chapter 28
• Magnetic field generated by a moving charge
• Magnetic field generated by a current-carrying
conductor, current loop and infinitely long,
straight conductor.
• Magnetic force between current-carrying
conductors
Intermission
• Ampere’s Law
• iClicker questions
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Magnetic field generated by a moving charge
We know that two charges moving in the same direction attracts,
while two charges moving in opposite directions repel.
Let’s work backward to deduce the magnetic field generated by
q1.
+ v1q1
+ v2q2
F2
r F 2 = q2
r v 2
r B 1( )
r B 1 points out of the page.
+ v1q1
+v2q2
F2
r F 2 = q2
r v 2
r B 1( )
r B 1 points out of the page.
=>The magnetic field lines form circles around q1
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The magnetic field of a moving charge
r B =
μo
4
q
r v ˆ r ( )r2 (for apoint charge moving at constant
r v )
μo = Universal constant = 4 x10 7 Tesla meter
Ampere
(associates with magnetic field)
o = 8.854x10 12 C2
N m2
(associates with electric field)
c1
oμo
= 3x108 m
s (= speed of light)
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Example
Follow Example 28.1
Find the ratio of FE/FB
1
1
2
2
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Magnetic field of a current element
qr v Id
r l (current segment)
dB =μo
4
I d
r l ˆ r ( )r2
The total magnetic generated by
the entire current loop is :
B =μo
4
I dr l ˆ r ( )r2
This is NOT an easy integral; we'll
consider special cases.
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Magnetic field of a straight current-carrying conductor
dB =μo
4
I d
r l ˆ r ( )r2
dr l = dyˆ j
r r = xˆ i yˆ j ˆ r =
xˆ i yˆ j
x 2 + y 2
dr l ˆ r =
(dy)x
x 2 + y 2( ˆ j ˆ i ) =
(dy)x
x 2 + y 2
ˆ k ( )
B =μo
4
I dr l ˆ r ( )r2
=μoI
4
(dy)x
(x 2 + y 2)3 / 2ˆ k ( )
a
a
=μoI
4
2a
x x 2 + a2
B =μoI
2 x
for a
This is a simple case to integrate
because dB contributed from all the
segments point in the same direction.
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Fields generated by multiple long wires
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Magnetic field of a circular current loop
We will focus on a special case where point P is at the center of the circle.
Again, this is a simple case where dB from all the segments points in the samedirection (x -direction)
dB =μo
4
I d
r l ˆ r ( )r2
dr l ˆ r = dlˆ i
r2 = a2 (same for all segments)
B =μo
4
I dr l ˆ r ( )r2
=μoI
4 a2
( dl)ˆ i
=μoI
4 a2
(2 a)ˆ i
=μoI
2a
i
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Forces and parallel conductors
• Two parallel currents in the same
direction attract.
• Assuming “infinitely long wires”,
we’ll calculate the force per unit
length.
Magnetic field generated by I1
at the location of I2 is given by :r B 1 =
μoI12 r
(direction indicated in figure)
Force per length on I2 is given by :r F /l =
r I 2
r B 1 =
μoI1I2
2 r (direction indicated in figure)
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Intermission
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Guass’s Law for magnetic field
Gaussian
surface
(r B • ˆ n )dA =μo enclosed "magnetic charge"( ) = 0
since "magnetic charges" always occurs in pair;
north and south poles.
See figure, B - field lines enter and exit the
Gaussian surface, consistent with (r B • ˆ n )dA =0.
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Ampere’s Law
Amperian loop can
be any shape
r B • d
r l = positive
r B • d
r l = negative
No current enclosed.
r B • d
r l = μoIenclosed
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Ampere’s Law - Examples
• Another right-hand-rule todetermine whether thecurrent is “positive” or“negative” in application ofAmpere’s Law
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Use Ampere’s Law to find magnetic field generated by current with cylindricalsymmetry
• An infinitely long cylinder carrying a current I.
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Use Ampere’s Law to find magnetic field of a(infinitely long) solenoid
• A helical winding of wire on a cylinder.
• Refer to Example 28.9 and Figures 28.22–28.24.
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Use Ampere’ Law to find magnetic field of a toroidalsolenoid
• A doughnut-shaped solenoid.
• Refer to Example 28.10 and Figure 28.25.
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Magnetic materials
• Consider Figure 28.27at right.
• Consider Figure 28.28below.
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Magnetic materials II
• Consider Figure 28.29 below.
• Follow Example 28.12.