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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
d 2
dt
N 2  2  M 21i1
 2  N2
N 2 2
M 21 
i1
 2   M 21
di1
dt
Virce verse: if current in coil 2 is
changing, the changing flux through
coil 1 induces emf in coil 1.
d1
 1   N1
dt
N11  M 12i2
N11
M 12 
i2
di2
 1   M 12
dt
M 12  M 12  M
N11 N 2  2
M

i2
i1
Units of M

1Wb
M  ,[M ] 
 1H (henry )
i
1A
1T 1m 2 Vs
1H 

 s
1A
A

 
1Ns 1Vs
F  qv  B  1T 
 2
Cm m
F V
N V
E   ,[E]  
q m
C m
1H  s
Typical magnitudes: 1μH-1mH
Joseph Henry (1797-1878)
Examples where mutual inductance is useful
Tesla coil
1.r2  r1 , M 
B1   0 n1i1 
M 
3
 0 N1i1
l1
 0 A1 N1 N 2
why M 
Estimate.
N 2 2
,  2  B1 A1
i1
l1
 0 A1 N1 N 2
l2
?
l1  0.5m, A1  10cm  10 m , N1  1000, N 2  10, i2  2 106
2
2
4 10 7 Tm10 3 m 210310
M
 25H
0.5mA
di
A
 1   M 2  25 10 6 H 2 106  50V
dt
s
With ferromagnetic core
 A1 N1 N 2
M
,   K m 0
l1
Nikola Tesla (1856 –1943)
[B]=1T to his honor
take K  1000  M  25mH ,   50kV
A
t
s
2.if r2  r1   2  B1 A2
M 
 0 A2 N1 N 2
l1
3.if r2  r and  is an angle between axises
M 
 0 A2 N1 N 2
l1
cos 
Example: M=?
Mutual inductance may induce unwanted emf in
nearby circuits. Coaxial cables are used to avoid it.
Self-inductance
d
  N
, N  Li
dt
di
N
  L , L 
dt
i
Thin Toroid
Thin solenoid
with
approximately
equal inner
and outer
radius.
N
L
,   BA
i
Ni
B
2r
N 2 A
L
2r
Long solenoid
N
L
,  max  Bmax A
i
Ni
Bmax 
l
N 2 A
Lmax 
l
Example. Toroidal solenoid with a rectangular area.
N
L
, d  Bhdr
i
Ni
Nih b dr Nih b
B
, 

ln

2r
2 a r
2
a
N 2 h b
L
ln
2
a
Inductors in circuits
di
  L
dt
di
Vab    L
dt
Energy stored in inductor
di
Pin  IVab  iL
dt
dW  Pin dt  Lidi
I
LI 2
W   Lidi 
2
0
LI 2
UL 
2
Compare to
Q 2 CV 2 QV
UC 


2C
2
2
Magnetic energy density
UL
UL
uB 

volume 2rA
LI 2
N 2 A
UL 
,L 
2
2r
N 2 AI 2
uB 
2r 2  2rA
Let’s consider
a thin toroidal
solenoid, but
the result
turns out to be
correct for a
general case
NI
B 2r
B
I 
2r
N
N 2 AB 2 (2r ) 2
uB 

2
2
2r 2  2rA N
B2
uB 
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
LI 2
N 2 h b
1.U L 
,L 
ln
2
2
a
b
I 2 N 2 h ln
a
UL 
4
2.U L 
u
B
dV
volune
B2
NI
uB 
,B 
2
2r
dV  h 2rdr
b
2
2
I N h ln
a
UL 
4
LR circuit, storing energy in the inductor
di
0
dt
di
 R
 (i  )
dt
R L
di
dt
L
  , 


R
(i  )
R
  iR  L

(i  )
t
R
ln(
)

ε


R

L
t
i  I (1  e  ), I 

R
t

di
 L   L  e 
dt
Energy conservation law
di
di
2
  iR  L  0, i  i R  Li ,
dt
dt
di 1 dLi 2 dU L
Li 2
Li 

,U L 
dt 2 dt
dt
2
Power output of the battery =power dissipated in the resistor
dU L
2
i  i R 
the rate at which the energy is stored in inductor
dt
General solution

t
i  I (1  e ), I 


R
t

di
 L   L  e 
dt
Initial conditions (t=0)
i0
 L  
Steady state (t→∞)
iI 
L  0

R
+
LR circuit, delivering energy from inductor
ε
ε
L
di
di
R
 iR  L  0,  i
dt
dt
L
i
i
di
dt
L
I i  I  ,  R
t
di
idi 2
 iR  L  0, L
i R  0
dt
dt
d ( Li 2 / 2) 2
dU L

 i R, 
 i2R
dt
dt

i
t
ln   , i  Ie  ,
I

t
t
t
t


di IL 
 L   L  e  RIe   e 
dt 
The rate of energy decrease in inductor is equal
to the power input to the resistor.
Oscillations in LC circuit
Oscillations in LC circuit
q
di
dq d 2 q q
 L  0, i   , 2 
 0,
C
dt
dt dt
LC
1
q   q  0,  
LC
2
2
q  Q cos(t   )
i  Q sin( t   )
Qand are defined by the initial conditions
Compare to mechanical oscillator
mx   kx,
F
k
x   x  0,  
m
2
0
x
x  X cos(t   )
v  X sin( t   )
Xand are defined by initial conditions
x  q, v  i
kx2
q 2 mv2
Li 2

,

2
2C 2
2
1
k
m  L, k  ,

C m
1
LC
General solution
1
q   q  0,  
LC
2
2
Qand are defined by initial conditions
1.q(t  0)  q0 , i (t  0)  0
q
i (t  0)  0    0
T
q  Q cos t  q(t  0)  Q  q0
q  q0 cos t
i  q0 sin t
2

t
i
T
2

t
2.q (t  0)  0, i (t  0)  i0
i
general solution
q  Q cos(t   )
i  Q sin( t   )
q (t  0)  Q cos   0     / 2
q  Q cos(t   / 2)  Q sin t
i  Q sin( t   / 2)  Q cos t
i (t  0)  Q  i0  Q 
i  i0 cos t
q
i0

sin t
i0

T
2

q
T
2 t

t
3. Arbitrary initial conditions : q (t  0)  q0 , i (t  0)  i0
q  Q cos(t   )
i  Q sin( t   )
q0  Q cos 
i0  Q sin 
i0
i0
  tan     ar tan
q0
q 0
(q0 )  i0  (Q)  Q 
2
2
2
2
(q0 )  i0
2

2
2
 q0 
2
i0
2
2
Energy conservation law
q
di
dq
 L  0, i  
C
dt
dt
q dq
di

 Li  0
C dt
dt
d q2
d Li 2
( ) (
)0
dt 2C dt 2
2
2
q0
Li0
q 2 Li 2
Q 2 LI 2

 const 



2C
2
2C
2
2C
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)  q0 , i (t  0)  0
i (t  0)  0    0
q  Q cos t  q(t  0)  Q  q0  0.005mC
2
q0
Q2
2.U max L  U max C 

 3.12 10 6 J
2C
2C
3)q  q0 cos(T / 4)  q0 cos( / 2)  0
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
Q
I
1)V  , Q   i (t  0) LC
C

i (t  0) LC
L
V
 i (t  0)
 2.4mV , t  T / 4
C
C
di q
2)V  L   4mV
dt C
T
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 (
3
)  i1 (
2  


3
b) q ( )  0.4mC

2
c)
q0

2C
Lef i0
2
T
3
)  i1 (T  )  0.1A, i2 ( )  0.1A
2

2
 2.25  10  2 J ,
1
1 1
 
Lef L1 L2
q
di1
di1
d )  L1
 q  CL1
 108C
C
dt
dt
LCR circuit
q
di
dq
 L  iR  0, i  
C
dt
dt
q



Lq  Rq   0
C
q  2q  02 q  0,  
q ~ e t 
R
, 0 
2L
2  2  02  0
1
LC
Characteristic equation
1, 2     (02   2 ) ,  2  02   2
2
1
R
4L
2
2
2
2
  0  0   
 2 R 
LC 4 L
C
Critical damping
4L
a) Underdamped oscillations: 02   2 
 R2
C
1, 2    i
q(t )  Q(t ) cos(t   )
i(t )  Q(t )[ cos(t   )   sin( t   )],
Q(t )  Qe t
4L
 R2
b) Critically damped oscillations: 02   2 
C
4
L
02   2 
 R2
C
1, 2  
q (t )  (C1  C2t )e 1t
i (t )  [ (C1  C2t )  C2 ]e 2t
2
2
c) Overdamped oscillations: 0   
1, 2 are real numbers
q (t )  C1e 1t  C2 e 2t
i (t )  1C1e 1t  2C2 e 2t
if   0  1  0, 2  2
4L
 R2
C
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
25Ω 1
2
50V

10µF
r
10mH
A
2)U dis

50V
1)i  I (1  e ), I  
 2A
R 25
L 10  2 H
 
 0.4ms  0.5s 
R
25
i (t  0.5s )  I  2 A
LI 2
 UL 
 20mJ
2

3)
Induced oscillations in LRC circuit, resonance
q
di
 L  Ri   cos t  0
C
dt
dq
1
R

i   , 0 
, 
,f  ,
dt
2L
L
LC
~
2
q  2q  0 q  f cos t
q  Q cos(t   )
q  Q sin( t   )
q   2Q cos(t   )
Q
(0   2 )Q cos(t   )  2Q sin( t   )  f cos(t     ) 
2
 f [cos(t   ) cos   sin( t   ) sin  ]
(0   2 )Q  f cos 
2


 2Q  f sin 
f2
2
Q 
,
tan


2
2
[(0   2 ) 2  (2 ) 2 ]
( 2  0 )
2
0

At the resonance condition:
  0

an amplitude greatly insreases
0
