F araday 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 ID: 163912
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Slide1
Faraday’s law
Faraday: A transient current is induced in a circuit ifA steady current flowing in an adjacent circuit is switched offAn adjacent circuit with a steady current is movedA permanent magnet is moved into or out of the circuitNo current flows unless the current in the adjacent circuit changes or there is relative motion of circuitsFaraday related the transient current flow to a changing magnetic fluxSlide2
Faraday’s law
Total or convective derivative:
)
t
x
T(
x,t
)
∂T(
x,t
)/∂t
dx/
dt
. ∂T(
x,t
)/∂x
∂T(
x,t
)/
∂t +
dx/
dt.
∂T(
x,t
)/∂xSlide3
Faraday’s law
Consider two situations:(1) Source of B field contributing to f is moving(2) Surface/enclosing contour on which f is measured is movingWhich situation applies depends on observer’s rest frameSituation (1)Rest frame of measured circuit(unprimed frame)B is changing on S becausesource circuit is moving at v
S
vSlide4
Faraday’s law
Situation (2)Rest frame of source circuit (primed frame)B’ is changing because measured circuit is moving at v
S’
vSlide5
Faraday’s law
Situation (2)Rest frame of source circuit (primed frame)B’ is changing because measured circuit is moving at v
S’
vSlide6
Lenz’s Law
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 effectIt 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
v
d
S
B
B
ind
S
N
v
d
S
B
B
ind
B
.d
S
< 0
Flux magnitude increases
d
f
/
dt
< 0
B
.d
S
< 0
Flux magnitude decreases
d
f
/
dt
> 0Slide7
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(
v
x
B
)
-
+
Completing a circuit does not produce a steady current either
B
v
F
= q(
v
x
B
)
-
+
-
+Slide8
Motional EMF
emf in rod length L moving through
B field, sliding on fixed U shaped wire
Charge continues to flow
while rod continues to move
I
F
= q(
v
x
B
)
+
+
-
-
B
v
emf induced in circuit equals minus rate of change of magnetic flux through circuitSlide9
Faraday’s Law in differential formSlide10
Electric vector potentialSlide11
Inductance
Self-Inductance in
solenoid
Faraday’s Law applied to solenoid with
changing
magnetic flux implies an
emf
Area of cross section =
p
R
2
N loops (turns) per unit length
B
I
LSlide12
Inductance
Work done by emf in LR series circuit
V
o
L
R
First term is energy stored in inductor B field
Second term is heat dissipated by resistor
solenoid inductance
L
=
m
o
N
2
p
r
2
L
solenoid field B =
m
o
N
I
W = ½
L
I
2
= ½
m
o
N
2
p
r
2
L
I
2
= ½ (
m
o
N
I
)
2
p
r
2
L/
m
o
= ½ B
2
volume
/
m
o Slide13
elastic exchange
of field energy
Inductance
LCR series circuit driven by sinusoidal emf
V
o
L
C
R
elastic exchange of kinetic and potential energySlide14
Displacement current
Ampere’s Law
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 equationSlide15
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)
Displacement current
in vacuum (see later)Slide16
Displacement current
Relative magnitude of displacement and conduction currentsSlide17
Maxwell Equations in Vacuum
Maxwell equations in vacuum