Faraday`s Law

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Faraday’s law
Faraday: 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
Magnetic
flux  
 B .d S
S
Electro motive force  
 E .d   
d
dt
Faraday' s Law (circa 1830)
Faraday’s law
T(x,t)
Total or convective derivative:
d

dt
df

t

dt
dA

dt
dB
dt

f
t
 v .
 v . f
A
t
B
t
∂T(x,t)/∂t
dx/dt) . ∂T(x,t)/∂x
total derivative
∂T(x,t)/∂t +
dx/dt. ∂T(x,t)/∂x
scalar field f
 v . A
  x B x v  magnetic
t
induction
field B
 x B x v   B  .v   v  .B    v . B  B . v
B . v
  .v   . B  0
identity
x
Faraday’s law
Consider two situations:
(1) Source of B field contributing to  is moving
(2) Surface/enclosing contour on which  is measured is moving
Which situation applies depends on observer’s rest frame
S
v
Situation (1)
Rest frame of measured circuit
(unprimed frame)
B is changing on S because
source circuit is moving at v
dB
dt
S

B
S
t
   E .d   
d
dt
 B .d S   
S
S
B
t
.d S
S’
v
Faraday’s law
Situation (2)
Rest frame of source circuit (primed frame)
B’ is changing because measured circuit is moving at v
dB ' S '
dt
 
  x B ' x v 
d
dt
t
 0 since source is at rest
 B '.d S '     x B ' x v .d S '
S'
   B ' x v .d  ' 
C'
B ' S '
S'
  v x B ' .d  '
C'
by Stokes' Theorem
v
S’
Faraday’s law
Situation (2)
Rest frame of source circuit (primed frame)
B’ is changing because measured circuit is moving at v
dB ' S '
dt
 
  x B ' x v 
d
dt
t
 0 since source is at rest
 B '.d S '     x B ' x v .d S '
S'
   B ' x v .d  ' 
C'
B ' S '
S'
  v x B ' .d  '
by Stokes' Theorem
C'
F '  q v x B ' force on charge q
F'
q
 E'
 
 E '.d  '
C'
 E .d   
d
Lenz’s Law
dt
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
S
N
B
S
N
v
B
dS
Bind
B.dS < 0
Flux magnitude increases
d/dt < 0
v
dS
Bind
B.dS < 0
Flux magnitude decreases
d/dt > 0
Motional EMF
B
-
Charges in conductor, moving at constant velocity v
perpendicular to B field, experience Lorentz force, F = q v x B.
Charges move until field established which balances F/q.
No steady current established.
v
B
+
v
-
F = q(vxB)
+
-
+
F = q(vxB)
Completing a circuit does not produce a steady current either
Motional EMF
emf in rod length L moving through B field, sliding on fixed U shaped wire
emf 
1
B
B
B
F .d    E .d   vBL

q
A
Current
A
I
vBL
R  resistance
-
R
Magnetic
flux  
 B .d S
+
I
S
d
 BvL
since
d(area)
dt
 E .d   
Magnetic
Faraday' s Law (circa 1830)
dt
flux  
 B .d S
S
 sign implies that field produced
opposes
the external
magnetic
+
F = q(vxB)
 vL
dt
d
v
-
Charge continues to flow
while rod continues to move
emf induced in circuit equals
minus rate of change of magnetic
flux through circuit
by induced current
field. Lenz' s Law.
Faraday’s Law in differential form
 E .d   
C
d
dt
 B .d S
S
 E .d     x E .d S
C

Stokes' Theorem
S
d
dt
 B .d S    x E .d S
S
S
 B



x
E
.d S  0
   t

S
 xE-
B
t
Faraday' s Law in differenti al form
NB  x E  0 This part of E field is not generated
by scalar potential
- 
Electric vector potential
E  - 
 xE -
Electrosta tic field represente d as gradient
B
Faraday' s Law
t
-  x   0
 xE -
B
E  -  B  xA
t
A
t
of potential
Identicall y zero! This E cannot be -  
-
  x A 
t
 - x
A
t
Represent
general E field this way
Represent
B field as curl of vector potential
There are alternativ e choices of A ,  which produce
E and B fields. One choice is a particular
gauge .
A
the same
Inductance
Self-Inductance in solenoid
Faraday’s Law applied to solenoid with changing magnetic flux implies an emf
 NL  B solenoid .area.numb
er of loops  magnetic
  o N I .  R .NL  flux ' linking' NL current
2
d  NL
emf  
dt
L   oN  R L
2
2
   oN  R L
2
2
dI
 L
dt
flux
loops in length L
dI
dt
(self) inductance
1 henry  1H  1 V/(As
-1
)
50 cm coil with 5cm radius, 5000 turns L  0.5H
More typically
B
L ~ mH or  H
L
I
Area of cross section = R2
N loops (turns) per unit length
Inductance
Work done by emf in LR series circuit
VL   L
dI
V
dt
t
R
 I R potential
t
Vo
difference s
t
I
dI


2
W   I V dt'   I   L
 R  dt'  L  I' d I'  R  I dt '
dt'

0
0 
0
0

1
2
R
t
L
L I  R  I dt '
2
2
0
First term is energy stored in inductor B field
Second term is heat dissipated by resistor
solenoid inductance L = oN2r2L solenoid field B = oNI
W = ½ LI2 = ½ oN2r2L I2 = ½ (oN I)2 r2L/o = ½ B2 volume/o
W magnetic 
dW magnetic
dv
1
2o

B
 B dv
2
W
electric

V
2
2o
dW electric
dv
o
2

 E dv
2
V
 oE
2
2
Inductance
Vo
LCR series circuit driven by sinusoidal emf
VL   L
dI
V
dt

VL   L Q
V
C
C

Q
V
C

Q
R
V
C
  Q
 R  Q  V cos(  t)
LQ
o
C
  R Q
  1 Q  V o cos(  t)
Q
L
LC
L
Resonance
at  o 
1
LC
R
 IR
C
R
Q
R
 I
Q
L
m x  m  x  kx  Fo cos(  t)
x   x   o2 x 
o 
Fo
m
cos(  t)
k
m
elastic exchange
of field energy
elastic exchange of kinetic and potential energy
Displacement current
  B  o j  j 
 .j 
1
o
 .j  
1
o
 B
 .  B   0

t
Ampere’s Law
 0 for non - steady currents
Problem!
Continuity equation
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.
Displacement current
j  ?  
Add term to LHS such that
taking Div makes LHS also
identically equal to zero:
 .E 


t

o
o
 B
 . j   .?   0
or  . j   . j  0
  o  .E  
 o  .E  
1
Displacement current in vacuum (see later)

 E 
  .  o
   . j
t
t 

The extra term is in the bracket
extended Ampere’s Law (Maxwell 1862)
1
 E 
j    

 B

o
t   o

  B  o j   o  o
E
t
Displacement current
Relative magnitude of displacement and conduction currents
E 

  B  o  j   o


t


E  E o cos(  t)
jσ E
o
E
t
   E o s in (  t)
ty ~ 10  m
8
σ electric conductivi
-1
-1

E
 t  8.854x10
8
j
Frequency
10
 12

for Cu
~ 10
19
 rad s
-1
is in hard x - ray range if currents
same order of magnitude
Maxwell Equations in Vacuum
Maxwell equations in vacuum
 .E 

Gauss' Law
o
 .B  0
 E  
Absence
B
monopoles
Faraday' s Law
t
  B  o j   o  o
of magnetic
E
t
Extended
Ampere' 
s Law

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