Transcript Document
Lectures 18-19 (Ch. 30)
Inductance and Self-inductunce
1.
2.
3.
4.
5.
6.
Mutual inductunce
Tesla coil
Inductors and self-inductance
Toroid and long solenoid
Inductors in series and parallel
Energy stored in the inductor,
energy density
7. LR circuit
8. LC circuit
9. LCR circuit
Mutual inductance
2 N2
d 2
Virce verse: if current in coil 2 is
changing, the changing flux through
coil 1 induces emf in coil 1.
1 N1
dt
N 2 2 M 21 i1
M
21
dt
N 1 1 M 12 i 2
N 2 2
M 12
i1
2 M 21
d 1
N 1 1
i2
di 1
1 M 12
dt
di 2
dt
M 12 M 12 M
M
N 1 1
i2
N 2 2
i1
Units of M
M
,[M ]
i
1H
1Wb
1 H ( henry )
1A
1T 1m
2
Vs
s
1A
A
1 Ns
1Vs
F q v B 1T
2
Cm
m
E
F
q
V
m
,[E ]
N
C
V
m
1H s
Typical magnitudes: 1μH-1mH
Joseph Henry (1797-1878)
Examples where mutual inductance is useful
Tesla coil
N 2 2
1 .r2 r1 , M
B1 0 n1 i1
i1
, 2 B1 A1
0 N 1i1
l1
0 A1 N 1 N 2
M
l1
why M
Estimate.
l1 0 . 5 m , A1 10 cm
M
4 10
7
Tm 10
3
2
10
2
3
0 A1 N 1 N 2
l2
m , N 1 1000 , N 2 10 , i 2 2 10
2
3
m 10 10
25 H
0 . 5 mA
1 M
di 2
25 10
6
H 2 10
dt
6
A
50 V
s
With ferromagne tic core
M
A1 N 1 N 2
Nikola Tesla (1856 –1943)
[B]=1T to his honor
take K
l1
?
, K m0
1000 M 25 mH , 50 kV
6
A
s
t
2 .if r2 r1 2 B1 A 2
M
0 A2 N 1 N 2
l1
3 .if r2 r and is an angle between axises
M
0 A2 N 1 N 2
cos
l1
Example: M=?
Mutual inductance may induce unwanted emf in
nearby circuits. Coaxial cables are used to avoid it.
Self-inductance
N
d
, N Li
dt
L
di
dt
,L
N
i
Thin Toroid
L
N
, BA
i
Thin solenoid
with
approximately
equal inner
and outer
radius.
B
Ni
2 r
N A
2
L
2 r
Long solenoid
L
N
i
B max
, max B max A
Ni
l
N A
2
L max
l
Example. Toroidal solenoid with a rectangular area.
L
N
, d Bhdr
i
B
Ni
2 r
,
N h
2
L
2
ln
b
a
Nih
2
b
a
dr
r
Nih
2
ln
b
a
Inductors in circuits
L
di
dt
V ab L
di
dt
Energy stored in inductor
Pin IV ab iL
di
dt
dW Pin dt Lidi
I
W
Lidi
2
0
UL
Compare to
UC
Q
2
2C
CV
2
LI
2
2
2
LI
QV
2
2
Magnetic energy density
uB
UL
UL
volume
B
,L
N AI
uB
the result
2 r
2
turns out to be
correct for a
general case
2
2 r 2 2 rA
NI
2 r
I
B 2 r
N
N AB ( 2 r )
2
uB
Let’s consider
a thin toroidal
solenoid, but
2
2
uB
2 rA
N A
2
LI
UL
2
2
2 r 2 2 rA N
2
B
2
2
2
Energy is stored in B
inside the inductor
Compare to:
uE
Energy is stored in E
inside the capacitor
E
2
2
Example. Find U of a toroidal solenoid with rectangular area
1 .U L
LI
2
N h
2
,L
2
2
I N h ln
2
2
b
a
UL
4
u
2 .U L
B
dV
volune
uB
B
2
2
,B
NI
2 r
dV h 2 rdr
I N h ln
2
UL
2
b
a
4
ln
b
a
LR circuit, storing energy in the inductor
di
iR L
0
dt
di
(i
dt
)
R
R L
di
(i
)
dt
L
,
R
R
(i
ln(
)
R ) t
R
ε
L
i I (1 e
t
), I
R
L L
di
dt
e
t
Energy conservation law
iR L
di
0 , i i R Li
2
dt
Li
di
dt
1 dLi
2
dt
i i R
dU
2
2
dU
i I (1 e
L L
di
dt
L
dt
t
), I
e
,U L
Li
2
2
Power output of the battery =power dissipated in the resistor +
the rate at which the energy is stored in inductor
L
General solution
,
dt
dt
di
Initial conditions (t=0)
i0
R
L
t
Steady state (t→∞)
i I
R
L 0
LR circuit, delivering energy from inductor
iR L
di
0,
dt
ε
ε
i
L
I
ln
di
i
i
i
t
I
I
t
L L
iR L
di
0, L
dt
idi
dt
i
dt
dt
,
dt
L
L
R
, i Ie
di
R
IL
t
e
,
t
RIe
t
e
i R 0
2
dt
2
d ( Li / 2 )
di
i R ,
2
dU
dt
L
i R
2
The rate of energy decrease in inductor is equal
to the power input to the resistor.
t
Oscillations in LC circuit
Oscillations in LC circuit
q
2
di
dq d q
q
L
0, i
, 2
0,
C
dt
dt dt
LC
q q 0 ,
2
2
1
LC
q Q cos( t )
i Q sin( t )
Qand are defined by the initial conditions
Compare to mechanical oscillator
m x kx ,
F
x x 0 ,
2
0
x
x X cos( t )
k
m
v X sin( t )
Xand are defined
x q, v i
kx
2
2
q
2
,
2C
m L, k
mv
2
2
1
C
Li
2
2
,
k
m
1
LC
by initial
conditions
General solution
q q 0 ,
2
2
1
LC
Qand are defined by initial conditions
1 .q ( t 0 ) q 0 , i ( t 0 ) 0
q
i (t 0 ) 0 0
T
q Q cos t q ( t 0 ) Q q 0
q q 0 cos t
2
t
i
i q 0 sin t
T
2
t
2 .q ( t 0 ) 0 , i ( t 0 ) i 0
i
general solution
q Q cos( t )
i Q sin( t )
T
2
t
q ( t 0 ) Q cos 0 / 2
q Q cos( t / 2 ) Q sin t
q
i Q sin( t / 2 ) Q cos t
i ( t 0 ) Q i0 Q
i i 0 cos t
q
i0
sin t
i0
T
2
t
: q ( t 0 ) q 0 , i ( t 0 ) i0
3 . Arbitrary initial conditions
q Q cos( t )
i Q sin( t )
q 0 Q cos
i 0 Q sin
i0
tan ar tan
q0
i0
q0
( q 0 ) i0
2
( q 0 ) i 0 ( Q ) Q
2
2
2
2
2
2
2
q0
i0
2
2
Energy conservation law
q
di
L
C
0, i
dt
q dq
dt
Li
C dt
d
dq
(
q
2
2
2C
0
dt
)
dt 2 C
q
di
d
dt
Li
(
Li
2
)0
2
2
const
2
q0
2
2C
Li 0
2
2
Q
2
2C
LI
2
2
UC+UL=const
UL
UC
UL
UC
T/2
T
2
t
-Q
Q q
Example. In LC circuit C=0.4 mF, L=0.09H.
The initial charge on the capacitor is 0.005mC and the initial current is zero.
Find: (a) Maximum charge in the capacitor (b) Maximum energy stored in the
inductor; (c) the charge at the moment t=T/4, where T is a period of oscillations.
1 .q ( t 0 ) q 0 , i ( t 0 ) 0
i (t 0 ) 0 0
q Q cos t q ( t 0 ) Q q 0 0 . 005 mC
2 .U max
L
U max
C
Q
2
2C
q0
2
3 . 12 10
2C
3 ) q q 0 cos( T / 4 ) q 0 cos( / 2 ) 0
6
J
Example. In LC circuit C=250 ϻF, L=60mH.
The initial current is 1.55 mA and the initial charge is zero. 1) Find the
maximum voltage across the capacitor . At which moment of time (closest to an
initial moment) it is reached? 2) What is a voltage across an inductor when a
charge on the capacitor is 1 ϻ C?
q
1)V
Q
Q
,
C
V
I
i ( t 0 ) LC
i ( t 0 ) LC
T
i (t 0 )
C
2 )V L
di
dt
L
C
q
C
4 mV
2 . 4 mV ,
t T /4
2
Example. In LC circuit C=18 ϻF, two inductors are placed in parallel:
L1=L2=1.5H and mutual inductance is negligible.
The initial charge on the capacitor is 0.4mC and the initial current through the
capacitor is 0.2A. Find: (a) the current in each inductor at the instant t=3π/ω,
where ω is an eigen frequency of oscillations; (b) what is the charge at the
same instant? (c) the maximum energy stored in the capacitor;(d) the charge on
the capacitor when the current in each inductor is changing at a rate of 3.4 A/s.
a)
i1 (
b)
q(
q0
c)
3
3
2
) i1 (
d)
C
L1
) i1 (T
T
2
) 0 .1 A , i2 (
) 0 . 4 mC
L ef i 0
2C
q
2
2
2 . 25 10
2
di 1
dt
q CL 1
2
J,
1
L ef
di 1
dt
108 C
1
L1
1
L2
3
) 0 .1 A
LCR circuit
q
di
L
C
iR 0 , i
dt
L q R q
dq
dt
q
0
C
2
q 2 q 0 q 0 ,
q~e
t
2L
1
,0
LC
2 0 0
2
1, 2
2
( 0 ) ,
2
0
2
R
2
0
2
2
1
LC
2
Characteristic equation
0
2
R
2
2
4L
R
2
2
4L
C
Critical damping
a) Underdamped oscillations: 02 2
4L
R
2
C
1, 2 i
q ( t ) Q ( t ) cos( t )
i ( t ) Q ( t )[ cos( t ) sin( t )],
Q ( t ) Qe
t
b) Critically damped oscillations: 02 2
0
2
2
4L
R
4L
R
C
2
C
1, 2
q (t ) (C 1 C 2 t ) e
1t
i ( t ) [ ( C 1 C 2 t ) C 2 ]e
2t
c) Overdamped oscillations: 0
2
1 , 2 are real numbers
q ( t ) C 1e
1 t
i ( t ) 1C 1 e
C 2e
1 t
2t
2C 2e
2t
if 0 1 0 , 2 2
2
4L
C
R
2
2
Example. The capacitor is initially uncharged. The switch starts in the open position
and is then flipped to position 1 for 0.5s. It is then flipped to position 2 and left there.
1) What is a current through the coil at the moment t=0.5s (i.e. just before the switch
was flipped to position 2)?
2) If the resistance is very small, how much electrical energy will be dissipated in it?
3) Sketch a graph showing the reading of the ammeter as a function of time after the
switch is in position 2, assuming that r is small.
t
50 V
10µF
25Ω 1
1) i I (1 e
2
), I
R
50V
L
R
r
10mH
10
2
H
25
A
3)
LI
2
2
20 mJ
25
2A
0 . 4 ms 0 . 5 s
i (t 0 .5 s ) I 2 A
2 )U dis U L
Induced oscillations in LRC circuit, resonance
q
L
C
di
dt
i
dq
dt
~
Ri cos t 0
1
,0
,
LC
R
2L
, f
,
L
q 2 q 0 q f cos t
2
q Q cos( t )
q Q sin( t )
2
q Q cos( t )
Q
( 0 ) Q cos( t ) 2 Q sin( t ) f cos( t )
2
2
f [cos( t ) cos sin( t ) sin ]
2
2
0
2 Q f sin
Q
f
2
0
( 0 ) Q f cos
At the resonance condition:
2
[( 0 ) ( 2 ) ]
0
2
2
2
2
, tan
2
( 0 )
2
2
an amplitude greatly insreases