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LECTURE 18
- More on Self Inductance
- Calculation of Self-Inductance for
Simple Cases
- RL Circuits
- Energy in Magnetic Fields
8/16/12 2
Self Inductance
- The magnetic field produced by the
current in the loop shown is
proportional to that current.
I
- The flux is also proportional to the current.
- Define the constant of proportionality between flux
and current to be the inductance, L
8/16/12 3
Self Inductance
- Archetypal inductor is a long solenoid, just as a pair of
parallel plates is the archetypal capacitor
d
A
- - - - -
+ + + +
r << l
l
r N turns
d << A
8/16/12 4
RL Circuits
R
I
a
b
L
I
- At t =0, the switch is closed and
the current I starts to flow.
Initially, an inductor acts to oppose changes in current through
it. A long time later, it acts like an ordinary connecting wire.
8/16/12 5
RL Circuits
R
I
a
b
L
I
- Find the current as a function of
time.
I =
ε
R
1 − e
− Rt / L
ε
R
1 − e
− t / τ RL
- What about potential differences?
8/16/12 6
RL Circuits ( on)
Current
Max = / R
63% Max at t = L / R
L / R
t
I
2 L / R
0
0 1 2 3 4
0
1
t/RC
Q f( x)
x
/ R
V
L
0
t
"
0 1 2 3 4
1 1
f( x)
0 x 4
Voltage on L
Max = / R
37% Max at t = L / R
I =
ε
R
1 − e
− Rt / L
V L
= L
dI
dt
= ε e
− Rt / L
8/16/12 7
RL Circuits
R
I
a
b
L
I
increase for
larger L?
decrease for
larger R?
8/16/12 8
RL Circuits
R
I
a
b
L
I
After the switch has been in
position for a long time, redefined
to be t=0, it is moved to position b.
8/16/12 13
Where is the Energy Stored?
- Claim: (without proof) energy is stored in the magnetic
field itself (just as in the capacitor / electric field case).
- To calculate this energy density, consider the uniform field
generated by a long solenoid:
- The inductance L is:
- Energy U :
- The energy density is found by dividing U by the volume
containing the field:
l
r
N turns
B = μ 0
N
l
I
L = μ
0
N
2
l
π r
2
U =
1
2
LI
2
=
1
2
μ 0
N
2
l
π r
2
⎛
⎝
⎜
⎞
⎠
⎟
I
2
=
1
2
π r
2
l
B
2
μ 0
u =
U
π r
2
l
B
2
0
8/16/12 14
Mutual Inductance
1 2
2 1
and
dI dI
M M
dt dt
ε = − ε = −
A changing current in a
coil induces an emf in an
adjacent coil. The
coupling between the
coils is described by
mutual inductance M.
M depends only on geometry of the coils (size,
shape, number of turns, orientation, separation
between the coils).
8/16/12 15
Mutual Inductance Applications Superconductors
Resistivity versus temperature
for an ordinary metal
Resistivity versus temperature
for ‘superconductor’
infinite
mobility!
- Cool it down
- Move magnet
What will happen?
dt
d
emf
mag
Current in the loop will produce its own B to compensate
for any changes in magnetic flux.
Magnetic Flux Through a Superconducting Ring
8/16/12 18
Superconductors
Resistivity ρ = 0 for temperature T < T
c
A type 1 superconductor is a perfect
diamagnetic material with χ
m
B =
B
app
m
Meissner effect (1939)
B field becomes =0 because
superconducting currents on
the surface of the
superconductor create a
magnetic field that cancel the
applied one for T<T c
8/16/12 19
Magnetic levitation:
Repulsion between
the permanent
Magnetic field
producing the
applied field and the
magnetic field
produced by the
currents induced in
the superconductor
magnetic levitation of a type 1 superconductor
DEMO
Superconductor: YBa
2
Cu
3
O
7
Magnet: NdFeB
