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 m0
 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)

i0
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