M.1
Magnetostatics
Magnetic Forces
Biot-Savart Law
Gauss’s Law for Magnetism
Ampere’s Law
Magnetic Properties of Materials
Inductance
BuFm ×= q
2
ˆ
4 RdId RIH ×
=π
0=⋅∇ B
JH =×∇
HB µ=
M.2
Magnetic Forces
The electric field E at a point in space has been defined as the electric force Fe per unit charge acting on a test charge when place at that point.The magnetic flux density B at a point in space is defined in terms of the magnetic force Fm that would be exerted on a charged particle moving with a velocity u.
BuFm ×= q
M.3
Magnetic Forces on Current-Carrying Conductor
A current flowing through a conducting wire consists of charged particles drifting through the material of the wire. Therefore, when a current-carrying wire is place in a magnetic field, it will experience a force.
Demonstration: M5.1 Q1-Q3, M5.2
∫ ×=⇒
×=
cdI
Idd
BlF
BlF
m
m
M.4
Biot-Savart Law
Used for calculating the magnetic field intensity dH due to a current I flowing in an element dl.
The resultant magnetic field intensity H is found by integratingdH for all current elements
mAR
dId /ˆ
4 2
RIH ×=
π
∫×
=l RdI
2
ˆ
4RIH
π
M.5
Gauss’s Law for Magnetism
For electric fields, the net outward flux of the electric flux density D through a closed surface enclosing a net charge Q is equal to Q. The magnetic analogue to a point charge is a magnetic pole, but magnetic poles do not exist in isolation. ---they occur in pairs.
may be called “law of nonexistence of isolated monoploes” or “the law of conservation of magnetic flux” and “Gauss’s law for magnetism”
∫ =⋅⇔
=⋅∇
sd 0
0
sB
B
M.6
Ampere’s Law
The line integral of H around any closed path C is equal to the current I enclosed by the path (in a right-handed sense).
Idc
=⋅⇔
=×∇
∫ lH
JHI
C
dl
Not enclosed
M.7
Example: Magnetic Field of a Long Wire
(a) ar ≤Choose the Amperian contour C1 to be a circular path of radius r1
∫ =⋅1
111CId lH
( )11
2
0 1111
2
ˆˆ1
Hr
drHdC
π
φφφπ
=
⋅=⋅∫ ∫lH
LHS:
The current flowing through the area enclosed by C1 is equal to
IarI
arI
=
= 2
21
2
21
1 ππ
M.8
Example: Magnetic Field of a Long Wire
(b)
21
11 2ˆˆ
aIrH
πφφ ==∴ H
ar >Choose the Amperian contour C2 which encloses all the current I
( )IHr
drHdC
==
⋅=⋅∫ ∫22
2
0 2222
2
ˆˆ2
π
φφφπ
lH
222 2
ˆˆrIHπ
φφ ==∴ H
M.10
Example: Magnetic Field inside a Toroidal Coil
A toroidal coil (also called torus or toroid) is a doughnut-shaped structure with closely spaced turns of wire wrapped around it.
M.11
Example: Magnetic Field inside a Toroidal Coil
Applying the Ampere’s law along the contour C:
( ) ( )
brar
NIH
NIrH
rdHdC
<<−=−=⇒
−=−=
⋅−=⋅∫ ∫
πφφ
π
φφφπ
2ˆˆ
2
ˆˆ2
0
H
lH
M.12
Magnetic Properties of Materials
Magnetization in a material is associated with atomic current loops generated by – Orbital motions of the electrons around the nucleus– Electron spin
The magnetization vector M of a material is defined as the vector sum of the magnetic dipole moments of the atoms contained in a unit volume of the material
( )( )
HHMH
MHMHB
ro
mmo
o
oo
µµχχµ
µµµ
==+=
+=+=
Q1
Magnetic susceptibility
Relative permeability
M.13
Ferromagnetic Materials
Ferromagnetic materials, which include iron, nickel, and cobalt,exhibit strong magnetic properties due to the fact that their magnetic moments tend to align readily along the direction ofthe external magnetic field.
such material remain partially magnetized even after the removal of the external field.
M.15
Ferromagnetic Materials
Typical hysteresis curve for a ferromagnetic material
SaturationResidual flux density
M.16
Summary
Curl equation
Divergence equation(Gauss’s law)
CurrentChargesSource
Static Magnetic fieldStatic Electric field
vρ=⋅∇ D 0=⋅∇ B
JH =×∇0=×∇ E
Qs
=⋅∫ dsD 0=⋅∫s dsB
0=⋅∫c dlE Ic
=⋅∫ dlH
M.17
Inductance
Lij= flux linking due to current in Sicurrent in Si
L11 is called the self-inductance; L12 is called the mutual-inductance
∫ ⋅=Φ2
12 s 21 dsB
ijΦ
1
1212 IL Φ
=
M.18
Example: Inductance of a Coaxial Transmission Line
Due to the current I in the inner conductor, the magnetic field generated inside the coaxial transmission line is given by
rIrIH
IrdH
Idc
πµφ
π
φφφ
φ
π
φ
2ˆ
2
ˆˆ2
0
=⇒
=⇒
=⋅⇒
=⋅
∫∫
B
lH
C
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