Download −The magnetic field −When a field is generated in a volume of space

−The magnetic field
−When a field is generated in a volume of space it
means that there is a change in energy of that volume,
and furthermore that there is an energy gradient so that a
force is produced.
− The force can be detected by
1. The acceleration of an electric charge moving in the
field
2.The force on a current-carrying conductor
3.Then torque on a magnetic dipole
4.A reorientation of spins on electrons within certain
types of atoms.
− What cause magnetic field
1.Electrical charge in motion – an electrical current
flowing in a conductor
2.Permanent magnet- there are the orbital motions
and spins of electrons
The magnetic field exerts a force on both
(1) Current-carrying conductors
(2) Permanent magnets
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− Definition of magnetic field strength H
the magnetic field H
𝐜𝐨𝐧𝐧𝐞𝐜𝐭𝐢𝐨𝐧
The generating electrical current
−the unit of magnetic field strength:
ampere
meter
(In terms of the generating current)
− An infinitely long solenoid containing n turns per
meter of coil and carrying a current of 1/n amperes.
−A current of 1 ampere passing through a straight 1 meter
length of conductor generates a tangential field strength of
¼ 𝜋 ampere/meter at a radial distance of l meter.
− The Biot-savart law
(It is a statement experimental observation rather than a theoretical
prediction)
− Enables us to calculate the magnetic field H generated
by an electrical current.
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Solenoid
A current-carrying conductor
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− The law gives the field contribution generated by a
current flowing in an elementary length of conductor
1
4πr2
δ H=
iδ ℓ × u
i: the current flowing in a conductor
δ ℓ :an elemental length of a conductor
r: the radius distance
u : a unit vector along the radial direction
δ H : the contribution to magnetic field at r due to iδℓ
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－Field due to a long conductor:
Determine the H at some point P distant a meters from an
infinitely long conductor carrying a i amps.

 
1
δH 
iδ   u
4 πr 2

1
δH 
iδsin(90  α)
2
4 πr
 δcosα  r  δα
   
A  B  A B cosθ
 
A  B  â n ABsinθ
 rcosα  a 
r  δα
aδα 

 δ 

a 
cosα cos 2 α  r 

cosα 

 icosα  δα
δH 
4πa
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( For steady current. Biot-savart law is equivalent to
Ampere’s Circuital law)
ex. Calculate The field at a distance of 10 cm from the
conductor when it carries a current of 0.1 A

i
cos d
2 4a
H   2

i A

2a m


1 A
if a  0.1 m and i  0.1A, H 
or H  0.159 A
m
2 m
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－ Field patterns around current-carrying conductors
－The field circulates around a single current-carrying conductor
in a direction given by the right-hand corkscrew rule.
－ If we look along the conductor in the direction of the
conventional current, the magnetic field circulates in a clockwise
direction.
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8
-
In a bar magnet, the field emerges from one end of the magnet
“north pole” (N) of a magnet as a source of magnetic field H.
While a “south pole” (S) behaves as a field sink.
The line of force leave the N pole and return at the S outside the
magnet. (Whether such poles have any real existence is debatable)
The strength of the magnetic field is proportional to the density
of the line of force.
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-
Notice that H produced by a bar magnet ≠ that of a solenoid.
In particular, the magnetic field lines within the bar magnet
run in the opposite direction to the field lines within the
solenoid.
- It can be explained because the bar magnet has a magnetization
M, while the solenoid does not, and this M leads to the
generation of a magnetic dipole which acts as a source and sink
for magnetic field.
Ampere’s circuital law
( How can we calculate the strength of a magnetic field generated
by an electrical current ?)
- The magnetic field generated by an electrical circuit.
(According to Ampere)
the shape of the circuit ( conduction path )
Depended on
the current carried
- By assuming that each circuit is made up of an infinite number
of current elements each contributing to the field, and by
summing or integrating these contribution at a point to determine
the field.
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line vector
Interacting along a closed path around the conductor at a distance r
 

 H  d   2πrH  i

i
H
2πr
According above equations, Ampere’s law = Biot-savart law.
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Magnetic induction (B)
(How does a medium respond to magnetic field?)
－ Magnetic induction B, sometime call the flux density.
－ When a magnetic field H has been generated in a medium (in
accordance with Ampere’s law), the response of the medium
is its magnetic induction B. (All media will respond with
some induction.)
－ Permeability of medium: the relation between magnetic
induction (B) and magnetic field (H)
B unit: Webers
meter 2


B  f ( F)
magnetic induction
 Tesla
(The Weber is the amount of magnetic flux)
the force on a moving electric charge or
electric current
－ A magnetic induction B of 1 Tesla generates a force of 1
Newton per meter on conductor carrying a current of 1
Ampere perpendicular to the direction of the induction
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
magnetic field H
magnetic induction 

(B)
magnetizat ion M of the medium

H 
 There are two contributions to magnetic induction  


M





In free space B  0 H
 In many medium, B is a linear function of H 


In
particular
in
free
space


μ 0 : the permeability of free space (universal constant)
H:A
0 
V  s  volt  sec ond 
 amp 
B : tesla  2 



m  meter 
m  meter 2 
B
H
V s
2
m

A
m

 If the value of B in free space 


 is known, then H in free space 
 is immediatel y known from 


 this relationship



V  s  volt  second 


m  A  meter  amp 
H  heneries 


m  meter 
0  4  107 H m

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 ferromagne ts 


ferrimagne
ts


 nor is it even a single - valued function 


 of H


－ However
in other media, B is no longer a linear

function of H .
1.In paramagnet s and diamagnets μ is constant over a considerab le range 



B  μH of values of H

 2.In ferromagne t μ varies rapidly with H



is not necessaril y a constant
－
 A
A field H  m  gives rise to magnetic induction

B (tesla) in a medium with permeability μ H m .
 
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Magnetic flux (Φ)
-Whenever a magnetic field is present in free space, there
will be a magnetic flux (Φ) .
- Unit: weber
-The weber is the amount of magnetic flux which when reduced
uniformly to zero in one second produces an e.m.f. of one volt in a
one-turn coil of conductor through which the flux passes.
-The amount of flux generated by a given field strength depends
on the properties of the medium and varies from one medium to
another.
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Force per unit length on a current-carrying conductor in a magnetic field
(The unit of magnetic induction has been defined in terms of the force
exerted on a current-carrying conductor. This will now be generalized to
obtain the force F on a current-carrying conductor in a magnetic
induction B)


B  f (F )
The force exerted on current-carrying conductor
The force per meter on a conductor carrying a current i in the direction
of the unit vector l caused by a magnetic induction B.
  
F  il  B
i : conductor carrying a current
l : current direction
 

- In free space, F  oil  H
 o
F
i1i2
2a
i1
a
i2 同相：相互吸引
異相：相互排斥
If two long wires are arranged parallel at a distance of a meter
apart and carry currents of i1 and i2 amps the force per meter
exerted by one wire on the other is:
F 
o
i1i2
2a
Electromagnetic induction
-
(can the magnetic field generate an electrical current or
voltage in return?)
-
When the magnetic flux linking an electric circuit changes
an e.m.f. is induced and this phenomenon is called
electromagnetic induction.
-
Faraday’s law: the voltage induced in an electrical circuit is
proportional to the rate of change of magnetic flux linking
the circuit.
d
d
) N  : magnetic flux
is equal to the induced e.m.f.
dt
dt
Where  is the magnetic flux passing through a coil of N turns
V  (
and d
dt
is the rate of change of flux
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Lenz’s law
- The induced voltage is in a direction which oppose the flux change
producing it.
• (電動勢所產生的電流往反抗磁通量變化的方向流動)
- Magnetic flux: ϕ (weber)
  (since the magnetic induction is the flux density)
•
B 
A

dB
V   NA
dt
•
Important result
(an electrical current can be generated by a time-depend)
-
Ex.
What is the voltage induced in a 50 turn coil area 1 cm2 when the
magnetic induction linking it changes uniformly from 3 test to zero
in 0.01 seconds?

dB
(50)(1104 )(3)
V   NA

 1.5 volts
dt
0.01
The magnetic dipole
•
(What is the most elementary unit of magnetism)
•
A circular loop of a conductor carrying an electric current, which can
generate a magnetic field.
-
A circular current loop can be considered the most elementary unit
of magnetism.
-
If a current loop has area A and carries a current i, then its magnetic
dipole moment is m=IA.
-
The units of magnetic moment: A‧m2 (amp ‧meter2)
B tries align the dipole, so that the moment m lies parallel to the induction
• The torque on a magnetic dipole of
moment m in a magnetic induction B
 
is then simply   m  B
• In free space
 
  o m  H

(-This mean that B tries to align the dipole so that
the moment m lies parallel to the induction)
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-
The energy of the dipole moment m in the presence of a magnetic
induction (If no frictional forces are operating, thework done by

the turning force will be conserved) E   m  B


In free space E   o m  H
The field produced by a current loop is identical in form to the field
produced by calculation from two hypothetical magnetic poles of
strength separated by a distance l.
N
M
i
A
≡
+P
l
S
-P
P: magnetic
pole strength
l: distance