Download Lecture 9 ELEC 3105

Magnetostatics The Basics
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• Stationary charge:
• vq = 0
•E0
B=0
A stationary charge
produces an electric
field only.
•Moving charge:
• vq  0 and vq = constant A uniformly moving
•E0
•Accelerating charge:
B0
charge produces an
electric and magnetic
field.
A accelerating
• vq  0 and aq  0
charge produces
an electric and
•E0
B  0 magnetic field and
a radiating
• Radiating field
electromagnetic
field.
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
B
Units and definitions
1T  10 G
4
Tesla
SI unit

H
Gauss
Wb
1T  1 2
m
Magnetic field vector
Magnetic induction
Magnetic flux density
Magnetic field strength
Weber


B  H
3
Permeability
  r o
Permeability of free space
Relative permeability for a medium
H 
 o  4 10  
m
7
Exact constant
Permeability of the medium
H 
Wb 
  
m
m
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5
What happens when we cycle the applied magnetic field.
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7
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Magnetostatics 1st Postulate
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POSTULATE
1 FOR THE MAGNETIC FIELD

A current element I d immersed in a magnetic


field B will experience a force dF given by:
  
dF  I  Bd
Units of Newtons {N}
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: dF  IBd sin  
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B
E
12
postulate 1 for the magnetic field
Consider a straight segment 
Net force on the segment
F   dF  IB

Right hand rule for direction
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Magnetic force on a moving charge
postulate 1 for the magnetic field
d

I

J

v
dA
q
Current density:


J  v
Volume charge density
Current through cross section dA:
 


Id   JdAd  v dAd  qv
 
I  JdA
Where dAd is an element of volume enclosing charge q
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Magnetic force on a moving charge
postulate 1 for the magnetic
field

I

J
d

v
q
Modify force equation:
  
dF  I  Bd

 
F  qv  B
dA


Id   qv

 
dF  qv  B
Net force on charge q
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Magnetic force on a moving charge
postulate 1 for the magnetic field
d

v
dA
q
Lorentz force
B
F
qv

 
F  qv  B
Often used to define the magnetic field.
Force, charge, and charge velocity are
measurable.
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Magnetic force on a moving charge

 
F  qv  B
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


Force at right angle
to v and B vectors
Force proportional
to v
Can do no work on a
charge
 
dW   F  dr

v dt
 
qv  B



Force along electric
field lines
Force independent
of v
Can do work
  
dW  qv  B  v dt
dW  0
0 always
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Hall effect
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Hall effect
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3-D view of block

I

J

B
q

v

Fmag
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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3-D view of block

B
z

B  Bzˆ

I

J
q

v

Fmag
y
x

 
Fmag  qv  B

v  vyˆ

Fmag  qvBxˆ
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Negative charge build up on
this surface
Look onto this surface
from above
Top view of block
z
- - - - - - - - - - - -

I

J
q

B

B  Bzˆ

v

Fmag
y
+ + + + + + + + + +
x

v  vyˆ
Positive charge build up on
this surface
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Top view of block
Accumulation of charge
continues until induced
electric force equals
magnetic force.
- - - - - - - - - - - -

FE
q

B

Fmag
+ + + + + + + + + +
Voltage difference across
charge distribution
VHall

v

I


FE  Fmag
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VHall  ?????
3-D view of block
Current density

B

 I
J
wt
w
q
t
Current

Fmag


I  qNv

v

I

J
get v
with N density of carriers in the material.
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VHall  ?????

B
3-D view of block
w
Magnetic force
q

Fmag
t
Fmag
Jwt
v
qN
Velocity of the moving charge
Jwt

B
N

v

I

J
Fmag
Jwt
 qvB  q
B
qN
Simplified magnetic force on the charge
moving at velocity v
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Accumulation of charge
continues until induced
electric force equals
magnetic force.
Top view of block
- - - - - - - - - - - -
w

FE
q

B

Fmag
+ + + + + + + + + +
Voltage difference across
charge distribution
VHall  EH w 

E  V


FE  Fmag

v

I
with
Fmag 
Jwt
B
N
and

qVHall
FE  qEH 
w
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Fmag

qV
Jwt
F

B E  qEH  Hall
w
N
In the steady state


FE  Fmag
- - - - - - - - - - - -

FE
w
q

B
then

v

Fmag

I
Jwt
JBw 2t
VHall  w
B
qN
qN
+ + + + + + + + + +
VHall
Jwt
JBw 2t
VHall  w
B
qN
qN
VHall  w
I
B
qN
B
qNVHall
wI
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Magnetostatics 2nd Postulate
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POSTULATE
2 FOR THE MAGNETIC FIELD

I d
A current element
produces a magnetic

field B which at a distance R is given by:

  o I  Rˆ
dB 
d
2
4 R

dB
Units of {T, G, Wb/m2}
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POSTULATE 2 FOR THE MAGNETIC FIELD
Postulate 2 implies that the magnetic field is
everywhere normal to the element of length d
 o Id
dB 
sin  
2
4 R
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Magnitude of the magnetic field

  o I  Rˆ
dB 
d
2
4 R
1
dB  2
R
Id  Qmag
 o Id
dB 
sin  
2
4 R
Similar to:
1
dE  2
R
Conceptually similar to a magnetic
charge.
Magnetic charges have not yet been found.
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For a closed path made up of current elements
Id

  o I d   Rˆ
Br  
4 C R 2
 
R  r  r

R
ˆ
R
R
Biot-Savard Law
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

I d  Jdvvol
Magnetic field produced by extended conductor
 
I   J  dA
S

  o J  Rˆ
Br  
dAd
2

4 v R

I

J

  o J  Rˆ
Br  
dvvol
2

4 v R
dA
 
R  r  r

R
ˆ
R
R
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Magnetic field produced by single moving charge conductor

v
q

R

B


I d  qv
   o qv  Rˆ
Br  
4 R 2
 
R  r  r

R
ˆ
R
R
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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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


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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Is it possible then that a moving
charge generate a magnetic force
on a second moving charge?

Answer “YES”
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

Id   qv
Recall for a moving charge that the following substitution is possible:




 o I 2 d  2  I1d 1  Rˆ12
F12 
4
R122
THEN
BECOMES

q1v1





 o q2 v2  q1v1  Rˆ12
F12 
4
R122
F

F
R12


q2v2
 

 o q1q2 v2  v1  Rˆ12
F12 
4
R122

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o
2
Fm 
q
q
v
1 2
4R 2

q1v1
4o R
Fm
  o ov 2
Fe
Fm
Fe
Fe 
1
q1q2
1

Fm
Fe
2
o

q2v2
c
o
2
Fm v
 2
Fe c
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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.

q1v1
Fm
Fe
2
Fm
Fe

q2v2
v
Fm  2 Fe
c
maximum
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