phys 270 – suppl. #8 dennis papadopoulos february 19, 2010 1
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PHYS 270 – SUPPL. #8
DENNIS PAPADOPOULOSFEBRUARY 19, 2010
1
dILdt
/V Q C
Biot Savart
Bback
back
dN
dt
B dA B
B I
dI
dt
Inductors - Inductance
Define the Self Inductance L as the proportionality constant in
VL LdI
dt
Units ? /
B
B
ddIL Ndt dt
L N I
CALCULATION OF INDUCTANCE OF A SOLENOID
m Ncoil NoNIl
A
L m
I
oN2
lA
Inductance of a solenoid
The inductance of a solenoid having N turns, length l and cross-section area A is
T-m2 /A =Henry =HDepends only on geometry
ENERGY STORAGE IN INDUCTOR
WHERE?
C
QU
Eu
c
oE
2
2
1
2
2
2
2
2
1
2
LIU
Bu
c
oB
WHY?
2
0 2
1LIIdILU
dt
dIILVI
dt
dUP
I
L
LL
)2/)((
THAT 2
oL BAlU
SHOW
Follow eqs. 34.39 to 34.42 of the book
The potential difference across an inductor
The potential difference across an inductor with an inductance of L and carrying a current I is
€
I = Io exp[−t /(L /R)]
€
VC + ΔVL = 0
Q
C− L
dI
dt= 0
I = −dQ
dt
Q
C+ L
d2Q
dt 2= 0
d2Q
dt 2+
1
LCQ = 0
ωo2 ≡1/LC
d2Q
dt 2+ωo
2Q = 0
2
2
2
22
2
/
0
( ) sin cos
( 0)
( 0) 0
, 0
( ) cos
( ) sin
o
o
o o
o
o
o o
o o
d xm kx
dt
k m
d xx
dtx t A t B t
x t x
v t
A x B
x t x t
v t t
€
Correction
A = 0,B = xo
LC CIRCUIT
)sin()sin(
)cos()(
/1
01
0
0
0
max
2
2
2
2
tItQI
tQtQ
LC
QLCdt
Qd
dt
QdL
C
Q
dt
dQI
dt
dIL
C
Q
VV
o
o
LC
ENERGY STORAGE OSCILLATION
The current in an LC circuit
The current in an LC circuit where the initial charge on the capacitor is Q0 is
The oscillation frequency is given by
INDUCTOR VOLTAGE
dt
dILV
IRV
dt
dIL
dt
d
L
R
m
EXAMPLE 34.12 The length of an inductor
QUESTION:
EXAMPLE 34.13 Large voltage across an inductor
QUESTION:
EXAMPLE 34.13 Large voltage across an inductor
LC CIRCUIT
)sin()sin(
)cos()(
/1
01
0
0
0
max
2
2
2
2
tItQI
tQtQ
LC
QLCdt
Qd
dt
QdL
C
Q
dt
dQI
dt
dIL
C
Q
VV
o
o
LC
ENERGY STORAGE OSCILLATION
The current in an LC circuit
The current in an LC circuit where the initial charge on the capacitor is Q0 is
The oscillation frequency is given by
EXAMPLE 34.12 The length of an inductor
QUESTION:
EXAMPLE 34.13 Large voltage across an inductor
QUESTION:
EXAMPLE 34.13 Large voltage across an inductor