1 Engineering Electromagnetics Essentials Chapter 4 Basic concepts of static magnetic fields.
static magnetic fields
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Transcript of static magnetic fields
Static magnetic fields• Charge in motion yields a current I• I = j area• j is a vector -- current density --
amperes/meter2
• Ampere’s circuital law B dl = o Ienc
•1 Tesla = 104 Gauss
•B at equator 1 Gauss
current ==> magnetic field
Henries104 7o
jB o
dsjdsB o
dsjdlB o
r2IB enco
Iar
r2B 2
2o
r a a
IIarI 2
2
enc
a r
B
r2IB enco
Ir2
B o
r a a
IIIenc
a r
B
B dl = o Ienc
• due to one wire • Ienc = I• B dl = B [2 r]• B = o {1 / 2 r} I
• due to other wire• B = o {1 / 2 r} I
• superposition
•Ienc = N I
B dl = B [2 L ]
•B = o N I / 2L
•B = o N I / L -- in center / top and bottom
L
we know that • B = 0vector potential A
we know that • [ x vector] = 0we can now specify the vectorlet vector be A such that B = x A
William Thomson shows that Neumann's
electromagnetic potential A is in fact the
vector potential from which may be
obtained via B = x A.
vector potential A
we also know x B = µo jB = x A
x x A = • A) - A
- A A = - µo j is similar to Poisson’s
equation but we have to solve three PDE’s
A and j are in the same direction!!
'dv
''
4O
rrrjrA
z'dz'zI'dv' urj
22 'zr
z22
22O
LrL
LrLln4
I urA
R' rrRz’
r
dz’2 L
I
A
z
Rz’
r
dz’2 L
I
B
z
rArB
urBr
A z
urB
22o
rL
Lr2I
urBr2I
Llim o
Biot-Savart integral
'dv'
'4 2
'O
rrurjrB rr
Rz’
r
dz’2 L
I
B
z
'dv'
'4 2
'O
rrurjrB rr
'dzr'z
r'zI4 2/322
rzz
O uuu
u
22O
rL
Lr2I
Rz’
r
dz’2 L
I
B
z
ur2I
Llim o
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zrO uuu
2/322
2O
zza
aI2
B
'dv'
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rrurjrB rr
large hadron collider
Earth’s magnetic field protects us
Inductance L
currentflux magneticL
current
)area(xdensityflux magnetic
I
)area(xLNIo
Coaxial cable
abcurrentflux magneticL
I
(length) x drr2Ib
ao
(length) x abln
2o
Inductance of a microstrip
z
d
w
I
I
B
currentflux magneticL
d0
0z0 dy
wIdz
I1
zw
d0
x
y
z
An induced electric current flows in a direction such that the current opposes the change that induced it. This law was deduced in 1834 by the Russian physicist Heinrich Friedrich Emil Lenz (1804-65).
Faraday’s law dt
tdtV m
ΔsBtm
either B or s individually change in time or they both
change in time together
dt
tdILtV
zudxdyds
T
BeatVT
t2
zuB Tt
eB
dt
dtV m
dt
BeadtV
Tt2
x y
z
dsBm
dsBm
zudxdydszuB B
dt
dtV m
dt
tsinABdtV
x y
z tsinAAs
tcosABtV
x y
z dsBm
dt
dtV m
cosBLWm
dtd
ddtV m
tsinBLWtV R
tVtI
t
dsBdlE
dsEdlE x
Faraday’s law
apply Stoke’s theorem
t
dsB
tx
BE
t
0x
BE
0 B
wire carrying current I
I
2w
L u
V
Luwx2
IV o
22o
wxILwuV
Luwx2
Io
Bewley’s book• trick questions• not every motion
generates a voltage
• uniform B & v• substitution of
circuit• Vgen = 0!
XB
XB
1 2
cu
V12= 0
XB
1 2
cu
V12= Bcu
XB
1 2
cu
V12= Bcu
XB
V12= Bcu1 2
cu
XB
1 2
cu
V12= -Bcu
V12= Bcu
XB
1 2
cu
V12= -Bcu
V12= Bcu
I1 dl1
1 B1
I2 dl2
1B2
I1 dl1 I2 dl2
1B2 B1
BvdF dQ
dvv BvdF
BIdldF
I1 dl1 I2 dl2
1B2 B1
dsdlBjdF