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GENERAL PHYSICS II
Electromagnetism
&
Thermal Physics
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Chapter X
Magnetic Field
§1. Magnetic interaction and magnetic field
§2. Magnetic forces on a moving charged particle
and on a current-carrying conductor
§3. Magnetic field of a current – magnetic field calculations
§4. Amper’s law and application
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§1. Magnetic interaction and magnetic field
1.1 Magnetic phenomena:
 Some history:
 Magnetic effects from natural magnets have been known for a long
time. Recorded observations from the Greeks more than 2500 years
ago.
 The word magnetism comes from the Greek word for a certain type of
stone (lodestone) containing iron oxide found in Magnesia, a district in
northern Greece.
 Properties of lodestones: could exert forces on similar stones and
could impart this property (magnetize) to a piece of iron it touched.
 Bar magnet: a bar-shaped permanent magnet. It has two poles: N and S
Like poles repel; Unlike poles attract.
We say that the magnets can interact each with other. This kind of
interaction differs from electric interactions, and is called magnetic
interaction
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field lines (direction and density)
NS
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Magnetic Field Lines of
a bar magnet
Electric Field Lines
of an Electric Dipole
NS
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Magnetic Monopoles ?
 Perhaps there exist magnetic charges, just like electric charges. Such an
entity would be called a magnetic monopole (having + or - magnetic
charge).
 How can you isolate this magnetic charge?
Try cutting a bar magnet in half:
• Many searches for magnetic monopoles no monopoles have
ever been found !
NS
Even an individual
electron has a
magnetic “dipole”!
N NS S
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Source of Magnetic Fields?
 What is the source of magnetic fields, if not magnetic charge?
 Answer: electric charge in motion!
 e.g., current in wire surrounding cylinder (solenoid) produces very
similar field to that of bar magnet.
 Therefore, understanding source of field generated by bar magnet
lies in understanding currents at atomic level within bulk matter

and on a current-carrying conductor:
2.1 Magnetic force on a moving charge:
BvqF




Magnetic Force:
(Lorentz force)
• In the formula B is measured in Tesla (T): 1T = 1 N / A.m
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Example 1:
 Two protons each move at speed v (as
shown in the diagram) in a region of
space which contains a constant B field
in the -z-direction. Ignore the interaction
between the two protons.
 What is the relation between the
magnitudes of the forces on the two
protons?
(a) F
1
< F
2
(b) F
1
= F
2
(c) F
1

at speed v (as shown in the diagram)
in a region of space which contains a
constant B field in the -z-direction.
Ignore the interaction between the two
protons.
 What is the relation between the
magnitudes of the forces on the two
protons?
(a) F
1
< F
2
(b) F
1
= F
2
(c) F
1
> F
2
A
• The magnetic force is given by:

θqvBFBvqF sin



• In both cases the angle between v and B is 90!!
Therefore F
1

B
• To determine the direction of the force, we use the corkscrew rule
(or right-hand rule).
• As shown in the diagram, F
2x
< 0.
BvqF




F
1
F
2
B
x
y
z
1
2
v
v
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(a) decreases (b) increases
(c) stays the same
C
B
x
y

2
BR
V
m
q

m
q
Vv 2
2

2
2







RB
m
q
v
and
Example 2: Determine the ratio of charge to mass for an electron ?
e
-
V
‘gun’

dq
I 
)(
Bvq


 Force on each charge =
 Total force =
 Current =
N
S
The case for a straight length of wire L carrying
a current I, the force on it is:
BLIF




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2.3 Magnetic force and torque on a current loop:
 Consider loop in magnetic field as on
right: If field is to plane of loop, the
net force on loop is 0!
• If plane of loop is not to field, there will
be a non-zero torque on the loop!
B
x
.
F
F

since the wire is parallel to B.
bc: F
bc
= ILB RHR: I is up, B is to the right, so F points into the screen.
By symmetry:
da bc
F F
 
n et ab b c cd d a
0F F F F F    
    

Example:
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Calculation of Torque:
• Suppose the loop has width w (the side
we see) and length L (into the screen).
The torque is given by:
 Note: if loop B, sin= 0  = 0
maximum occurs when loop parallel to B
Frτ





AIB sin

F = IBL
where

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• We can define the magnetic dipole moment of a current loop as
follows:
+ direction: to plane of the loop in
the direction the thumb of right hand
points if fingers curl in the direction of
current.
+ magnitude:

AI
• Torque on loop can then be rewritten as:
• Note: if loop consists of N turns,

= NAI

AIB sin


Bμτ



B
x
.


F

F