1 Magnetostatics The Basics. 2 Stationary charge: Stationary charge: v q = 0 E 0B = 0 Moving...
Transcript of 1 Magnetostatics The Basics. 2 Stationary charge: Stationary charge: v q = 0 E 0B = 0 Moving...
ELEC 3105 BASIC EM AND POWER ENGINEERING
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Magnetostatics The Basics
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Magnetostatics• Stationary charge:
• vq = 0
• E 0 B = 0
•Moving charge:
• vq 0 and vq = constant
• E 0 B 0•Accelerating charge:
• vq 0 and aq 0
• E 0 B 0• Radiating field
A uniformly moving charge produces an electric and magnetic field.
A stationary charge produces an electric field only.
A accelerating charge produces an electric and magnetic field and a radiating electromagnetic field.
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Magnetostatics
Units and definitions
Magnetic field strength
Magnetic field vector
Magnetic induction
Magnetic flux densityGT 4101
Tesla Gauss
SI unit
211
m
WbT
Weber
B
H
HB
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Magnetostatics
Permeability
or Permeability of free space
Relative permeability for a medium
Permeability of the medium
m
Ho
7104
m
Wb
m
HExact constant
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ferromagnetic
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Ferromagnetic materials WHY?
What happens when we cycle the applied magnetic field.
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PERMITTIVITY AND PERMEABILITY
ELEC 3105 BASIC EM AND POWER ENGINEERING
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Magnetostatics 1st Postulate
Magnetostatics
POSTULATE 1 FOR THE MAGNETIC FIELD
A current element immersed in a magnetic field will experience a force given by:
dI
B
Fd
dBIFd
Units of Newtons {N}
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Magnetostatics
POSTULATE 1 FOR THE MAGNETIC FIELD
A current element experiences a force which is at right angles to the plane formed by the current element and magnetic field direction magnitude: sinIBddF
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BE
Magnetostatics vs Electrostatics
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Magnetostatics postulate 1 for the magnetic field
IBdFF
Consider a straight segment
Net force on the segment
Right hand rule for direction
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postulate 1 for the magnetic field
Magnetic force on a moving charge
Current density:
Current through cross section dA:
J
I
qv
d
dA
vJ
Volume charge density
dAJI
vqdAdvdAdJId
Where is an element of volume enclosing charge q dAd
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postulate 1 for the magnetic field
Magnetic force on a moving charge
Modify force equation:
Net force on charge q
J
I
q
v
d
dA
vqId
dBIFd
BvqFd
BvqF
X
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postulate 1 for the magnetic field
Magnetic force on a moving charge
Lorentz force
Often used to define the magnetic field. Force, charge, and charge velocity are measurable.
q
v
d
dA
BvqF
qv
FB
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Magnetic force on a moving charge BvqF
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Magnetic Electric
Force at right angle to v and B vectors
Force proportional to v
Can do no work on a charge
Force along electric field lines
Force independent of v
Can do work
rdFdW
dtv
Bvq
dtvBvqdW
0 always
0dW
Electric AND Magnetic
ELEC 3105 BASIC EM AND POWER ENGINEERING
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Hall effect
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Hall effect
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Hall Effect
J
I
q
v
magF
B3-D view of block
When a conductor that carries a current is placed in a uniform magnetic field, an electrostatic field appears whose direction is perpendicular both to the magnetic field and to the current. The electric field here is known as the Hall field and reaches equilibrium in the order of 10-14 s. The electric field is characterized also by the Hall voltage across the faces of the conductor.
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Hall Effect
J
I
q
v
x
y
z
zBB ˆ
magF
B
yvv ˆ
BvqFmag
xqvBFmag ˆ
3-D view of block
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Hall Effect
J
I
q
v
x
y
z
zBB ˆ
magF
B
yvv ˆ
Top view of blockLook onto this surfacefrom above
- - - - - - - - - - - -
+ + + + + + + + + +
Negative charge build up on this surface
Positive charge build up on this surface
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Hall Effect
I
q
v
magF
B
Top view of block
Accumulation of charge continues until induced electric force equals magnetic force.- - - - - - - - - - - -
+ + + + + + + + + +
Voltage difference across charge distribution
EF
magE FF
HallV
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Hall Effect
J
I
q
v
magF
B3-D view of block
w
t
wt
IJ
Current density
?????HallV
vqNI
Current with N density of carriers in the material.
get v
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Hall Effect
J
I
q
v
magF
B3-D view of block
w
t
BqN
JwtqqvBFmag
Velocity of the moving charge
?????HallV
qN
Jwtv
Magnetic force
Simplified magnetic force on the charge moving at velocity vB
N
JwtFmag
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Hall Effect
I
q
v
magF
B
Top view of blockAccumulation of charge continues until induced electric force equals magnetic force.
- - - - - - - - - - - -
+ + + + + + + + + +
Voltage difference across charge distribution
EF
w
qVqEF Hall
HE
wEV HHall
BN
JwtFmag
withw
magE FF
and
VE
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Hall Effect
I
q
v
magF
B
In the steady state
- - - - - - - - - - - -
+ + + + + + + + + +
EF
w
qVqEF Hall
HE
HallV
BN
JwtFmag
w
magE FF
then
qN
tJBwB
qN
JwtwVHall
2
qN
tJBwB
qN
JwtwVHall
2
BqN
IwVHall
wI
qNVB Hall
ELEC 3105 BASIC EM AND POWER ENGINEERING
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Magnetostatics 2nd Postulate
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Magnetostatics
POSTULATE 2 FOR THE MAGNETIC FIELD
A current element produces a magnetic field which at a distance R is given by:
d
R
RIBd o
2
ˆ
4
dI
B
Bd
Units of {T, G, Wb/m2}
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Magnetostatics POSTULATE 2 FOR THE MAGNETIC FIELD
Postulate 2 implies that the magnetic field is everywhere normal to the element of length
sin
4 2R
IddB o
d
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Magnetostatics Magnitude of the magnetic field
d
R
RIBd o
2
ˆ
4
2
1
RdB
Similar to:
sin
4 2R
IddB o
2
1
RdE
magQId Conceptually similar to a magnetic charge.
Magnetic charges have not yet been found.
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Magnetostatics For a closed path made up of current elements
C
o
R
RdIrB
2
ˆ
4
Id
rrR R
RR
ˆ
Biot-Savard Law
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Magnetostatics Magnetic field produced by extended conductor
v
o dAdR
RJrB
2
ˆ
4
voldvJdI
rrR
R
RR
ˆ
J
I
S
AdJI
v
volo dv
R
RJrB
2
ˆ
4
dA
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Magnetostatics Magnetic field produced by single moving charge conductor
2
ˆ
4 R
RvqrB o
vqdI
rrR
R
RR
ˆ
q
v
R
B
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Magnetostatics POSTULATE 1 and 2 FOR THE MAGNETIC FIELD
Magnetic field lines are continuous and close on themselves. There are no magnetic charges for the lines to start or end on. Magnetic forces and magnetic fields are at 90 degrees to their sources.
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Consequence of postulates 1 and 2
From postulate 2: A moving charge produces a magnetic field.
From postulate 1: A magnetic field produces a force on a moving charge.
Is it possible then that a moving charge generate a magnetic force on a second moving charge?
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Consequence of postulates 1 and 2
Is it possible then that a moving charge generate a magnetic force on a second moving charge?
Answer “YES”
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Consequence of postulates 1 and 2
2
12
12112212
ˆ
4 R
RdIdIF o
vqId
Recall for a moving charge that the following substitution is possible:
THEN
BECOMES
2
12
12112212
ˆ
4 R
RvqvqF o
12R 22vq
11vq
FF
2
12
12122112
ˆ
4 R
RvvqqF o
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Compare magnitude of magnetic and electric force
between two moving charges2
2124vqq
RF o
m
22vq
11vq
mFmF
eFeF
2124
1qq
RF
oe
2vF
Foo
e
m
coo
1
2
2
c
v
F
F
e
m
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Compare magnitude of magnetic and electric force
between two moving charges
22vq
11vq
mFmF
eFeF
em Fc
vF
2
2
The magnetic force on these two moving charges can be obtained from the elctric force and the velocity of the two charges. Concepts of relativity come into play.
maximum
Magnetic induction charging
Induction Stove