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Faraday’s lawFaraday: A transient current is induced in a circuit if
• A steady current flowing in an adjacent circuit is switched off• An adjacent circuit with a steady current is moved• A permanent magnet is moved into or out of the circuit
No current flows unless
• the current in the adjacent circuit changes or • there is relative motion of circuits
• Faraday related the transient current flow to a changing magnetic flux
1830) (circaLaw sFaraday' dt
d.dforce motive Electro
.dflux Magnetic
E
SB
S
Faraday’s law
Total or convective derivative:
0 B vvB
vBBvBvvBvB
BvBBB
AvAA
v
v
...
identity. ... x x
field inductionmagnetic x x t
dt
d
t
dt
d
f field scalar f t
f
dt
df
derivative total t
dt
d
.
.
. )
t
x
T(x,t)
∂T(x,t)/∂t
dx/dt . ∂T(x,t)/∂x
∂T(x,t)/∂t + dx/dt. ∂T(x,t)/∂x
Faraday’s lawConsider two situations:
(1) Source of B field contributing to f is moving
(2) Surface/enclosing contour on which f is measured is moving
Which situation applies depends on observer’s rest frame
Situation (1)
Rest frame of measured circuit(unprimed frame)
B is changing on S becausesource circuit is moving at v
SS
SS
SB
SBE
BB
.dt
.ddt
d.d
t
dt
d
Sv
Faraday’s law
Situation (2)
Rest frame of source circuit (primed frame)
B’ is changing because measured circuit is moving at v
Theorem Stokes'by '.d'x '.dx '
'.dx 'x ''.ddt
d
rest at is source since 0 t
' x 'x
dt
'd
''
''
''
CC
SS
SS
B vvB
SvBSB
BvB
B
S’v
Faraday’s law
Situation (2)Rest frame of source circuit (primed frame)B’ is changing because measured circuit is moving at v
'
''
''
''
C
CC
SS
SS
''.d 'q
' q charge on force 'x q '
Theorem Stokes'by '.d'x '.dx '
'.dx 'x ''.ddt
d
rest at is source since 0 t
' x 'x
dt
'd
EEF
B vF
B vvB
SvBSB
BvB
B
S’v
Lenz’s Law
dt
d.d
E
Minus sign in Faraday’s law is incorporation of Lenz’s Law which states
The direction of any magnetic induction effect is such as to oppose the cause of the effect
It ensures that there is no runaway induction (via positive feedback) or non-conservation of energy
Consider a magnetic North Pole moving towards/away from a conducting loop
SN v
dSB
Bind
SN v
dSB
Bind
B.dS < 0Flux magnitude increasesdf/dt < 0
B.dS < 0Flux magnitude decreasesdf/dt > 0
Motional EMF
Charges in conductor, moving at constant velocity vperpendicular to B field, experience Lorentz force, F = q v x B. Charges move until field established which balances F/q.No steady current established.
B
v
F = q(vxB)
-
+
Completing a circuit does not produce a steady current either
B
v
F = q(vxB)
-
+
-
+
Motional EMFemf in rod length L moving through B field, sliding on fixed U shaped wire
Charge continues to flow while rod continues to move
resistance R R
vBL Current
vBL.d.d q
1emf
B
A
B
A
I
EF
Law. sLenz' field.magnetic external the opposes
current inducedby produced field that implies sign
.dflux Magnetic
1830) (circaLaw sFaraday' dt
d.d
vLdt
d(area) since BvL
dt
d
.dflux Magnetic
S
S
SB
E
SB
I
F = q(vxB)
+ +
- -
B
v
emf induced in circuit equals minus rate of change of magnetic flux through circuit
Faraday’s Law in differential form
- potential scalarby generated is field of part This 0x NB
form aldifferenti inLaw sFaraday' t
-x
.dx t
.dx .ddt
d
Theorem Stokes' .dx .d
.ddt
d.d
notEE
BE
SEB
SESB
SEE
SBE
0
S
SS
SC
SC
. particular a is choice One fields. and
same the produce which, of choices ealternativ are There
potential vector of curl as field Representx
waythis field general Represent t
--
tx-
t
x-
t-x
- be cannot This zero!y Identicall x-
Law sFaraday' t
-x
potential of gradient as drepresente fieldtic Electrosta -
gaugeBE
A
AB A B
EA
E
A ABE
E
BE
E
0
Electric vector potential
Inductance
H or mH~typically More
0.5H turns 5000 radius, 5cm withcoil cm 50
) V/(As1 1H henry 1
inductance (self) LRN
dt
d
dt
dLRN
dt
demf
L length in loops current NL linking''flux .NL R .N
flux magnetic loops of er.area.numbB
1-
22o
22o
NL
2o
solenoidNL
L
L
L
L
II
I
Self-Inductance in solenoid
Faraday’s Law applied to solenoid with changing magnetic flux implies an emf
Area of cross section = pR2
N loops (turns) per unit length
B
IL
InductanceWork done by emf in LR series circuit
t
0
t
00
t
0
t
0
RL
dtR2
1
dtRddt' R dt'
ddt'V W
sdifference potential R Vdt
dV
'II
'II'I'I
II
II
22
2I
L
LL
LVo
L
R
First term is energy stored in inductor B fieldSecond term is heat dissipated by resistorsolenoid inductance L = moN2pr2L solenoid field B = moNIW = ½ LI2 = ½ moN2pr2L I2 = ½ (moN I)2 pr2L/mo = ½ B2 volume/mo
E
dv
dW
B
dv
dW
dvE W dvBW
2electric
2magnetic
2electric
2magnetic
22
22
1
o
o
o
o
VV
elastic exchange
of field energy
Inductance
LCR series circuit driven by sinusoidal emf
m
k
C
1 at Resonance
t)cos(m
Fxxx t)cos(
VQ
C
1Q
RQ
t)cos(Fkxxmxm t)cos(VC
QRQQ
Q RQ VC
Q VQV
R VC
Q V
dt
dV
oo
o2o
o
oo
RCL
RCL
L
LLL
L
L
L
I
II
Vo
L
C R
elastic exchange of kinetic and potential energy
Displacement currentAmpere’s Law
currentssteady -non for t
1
1
0.
0..
j
Bj
BjjB
o
oo
Problem!
Steady current implies constant charge density so Ampere’s law consistent with the continuity equation for steady currents (only).
Ampere’s law inconsistent with the continuity equation (conservation of charge) when charge density is time dependent.
Continuity equation
Displacement current
Add term to LHS such that taking Div makes LHS also identically equal to zero:
The extra term is in the bracket
extended Ampere’s Law (Maxwell 1862)
0..
0..
jj
?j
B?j
or
1
o
jE
E
EE
...
..
ttt oo
oo
t
t
ooo
oo
EjB
BE
j
1
Displacement current in vacuum (see later)
Displacement current
Relative magnitude of displacement and conduction currents
magnitude of order same currents if rangeray - xhard in isFrequency
s rad 10~10
8.854x10
Cu for m10~ty conductivielectric
t)(s t)cos(
1-198
12
1-1-8
j
E
Ej
in EE
EE
EjB
oo
t
t
t
o
oo
σ σ
Maxwell Equations in Vacuum
Maxwell equations in vacuum
Law s Ampere'Extended
Law sFaraday'
monopolesmagnetic of Absence .
Law Gauss' .o
t
t
ooo
EjB
BE
B
E
0