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Lecture 16: June 29th 2009
Physics for Scientists and Engineers II
Physics for Scientists and Engineers II , Summer Semester 2009
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Examples: Chapter 31 Problem 48
r2  0.150m
r1  0.100m
0.500m
3.00 
6.00 
5.00 
0.500m
0.500m
B increases at a rate of 100T/s. What is the current in each
resistor?
Physics for Scientists and Engineers II , Summer Semester 2009
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Examples: Chapter 31 Problem 48
I1
I2
r2  0.150m
r1  0.100m
0.500m
3.00 
6.00 
I1
 ind ,1
I3
5.00 
 ind , 2
0.500m
I2
0.500m
Lenz' s law suggests that the currents I1 and I 2 run as shown. The direction of I3
is an initial guess. The loop directions are indicated as blue arrows. The induced
voltages are represente d as batteries with thei r polarity based on Lenz' s law.
Physics for Scientists and Engineers II , Summer Semester 2009
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Examples: Chapter 31 Problem 48
Junction rule :
I1  I 2  I 3  0
Loop 1 :  ind ,1  6.00I1  3.00I 3  0
Loop 2 :  ind , 2  5.00I 2  3.00I 3  0
 ind ,1 
d B1
dB
T
2
 r12
  0.100m  100  3.14V
dt
dt
s
 ind , 2 
d B 2
dB
T
2
 r22
  0.150m  100  7.07 V
dt
dt
s
Physics for Scientists and Engineers II , Summer Semester 2009
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Examples: Chapter 31 Problem 48
Junction rule :
I1  I 2  I 3  0  I1  I 2  I 3
(1)
Loop 1 : 3.14V  6.00I1  3.00I 3  0
(2)
Loop 2 : 7.07V  5.00I 2  3.00I 3  0
(3)
(1) into (3): 3.14V  6.00I 2  I 3   3.00I 3  0
 3.14V  6.00I 2  9.00I 3  0
(4)
(4) + 3 *(3): 21.4V  21.00I 2  0
 I 2  1.02 A
(5) into (3):
(5)
7.07 V  5.00 1.02 A  3.00I 3  0
 I3 
1.97V
 0.657 A
3.00
Physics for Scientists and Engineers II , Summer Semester 2009
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Chapter 32: Inductance
2a

R
w
SW
D (D  w)
Imagine you suddenly close the switch (SW) on the right side.
Physics for Scientists and Engineers II , Summer Semester 2009
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The switch-on process in “slow motion”
I t  (is increasing during switch - on process)
R

Bt  (is increasing during switch - on process)
A current will start to flow in the indicated direction (until it reaches a maximum).
Before the switch is closed there was no magnetic field created by the wires of this loop.
As the current increases, an increasing magnetic field builds up in the interior of this loop
due to the current in the wires.
The magnetic flux through the loop increases.
An emf is induced in the loop according to Faraday’s law.
(Think of this as a slow-motion observation of the switch-on process. The current doesn’t go
from zero to maximum in no time. It takes a finite amount of time to reach maximum).
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Polarity of induced emf
I t 
R
 B t 
 ind

The polarity of the induced emf ( ind ) is such that it' s effect opposes what caused
the induction (Lenz' s law).  Polarity is opposite to that of the battery ( ).
This process is called " self - induction" , because the changing flux and the resultant
induced emf arise from the circuit itself.
Physics for Scientists and Engineers II , Summer Semester 2009
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Flux calculation
I t 
dr
r
w r
 ind

Calculatin g the flux throu gh the loop as a function of current :
 Neglect th e magnetic field created by vertica l wires at left and right end (D  w).
d B 
0 I
0 I
 I1
1 
 Ddr
dA 
dA  0  
2 r
2 w  r 
2  r w  r  
w a
B 

a
0 I  1
 D  wa
1 
 
 Ddr  0 ln 
I
2  r w  r  

a


  ind  
 D  w  a  dI
d B t 
d   D  wa 
   0 ln 
 I    0 ln 

dt
dt  
a

a

 

 dt
Physics for Scientists and Engineers II , Summer Semester 2009
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Inductance
 ind
0 D  w  a  dI
d B t 


ln 

dt

 a  dt
This parameter combination is purely geometric.
Let’s replace it with “L”. L is also called the
“Inductance” of a certain conductor configuration
(like this particular wire loop). Other geometries
result in different inductances.
 ind   L
dI
dt

L
 ind
dI
The units of inductance are
Physics for Scientists and Engineers II , Summer Semester 2009
dt
1
Vs
 1 H (henry )
A
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Kirchhoff’s loop rule
I t 
 ind

 B t 
R
dI
 RI  0 (a differenti al equation)
dt
Solve using a change of variables :
 L
x
x

R
I
 dx  dI
t
dx
R


x x 0 L dt
0

 x
 x
R
ln     t
L
 x0 
Physics for Scientists and Engineers II , Summer Semester 2009


R
I 
L dx
0
R dt
 x  x0 e

L dI
0
R dt
dx
R
  dt
x
L
R
 t
L
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Kirchhoff’s loop rule
dI
 RI  0 (a differenti al equation)
dt
Solved using a change of variables :
 L
x  x0 e


R
R
 t
L
I 
where

R
e
R
 t
L
x

R

 I and x 0 
 I

1  e
R 
R
 t
L

R

R
I 
 I t  0  
L dI
0
R dt

R






1  e


R

L
where  
is the " time constant of the RL circuit".
R
I

R
 t
L
t

 
  1  e 
 R


Physics for Scientists and Engineers II , Summer Semester 2009
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Measuring the time constant

R
0.632
I(t)

R
I
L

R
Physics for Scientists and Engineers II , Summer Semester 2009

1  e
R 
R
 t
L




t
13
Circuit Diagram
L
+ -

R
Physics for Scientists and Engineers II , Summer Semester 2009
Symbolically represent the
inductance of the whole loop
with one circuit element.
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Kirchhoff loop rule during switch-on process
L
I
+ -

R
dI
dt
Sign convention for counting the voltage drop :
Voltage drop across the inductor :  ind   L
dI
dt
dI
If loop direction is in the opposite direction of current :  L
dt
If loop direction is in the same direction as current :
Physics for Scientists and Engineers II , Summer Semester 2009
L
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Kirchhoff loop rule during switch-on process
L
dI
  L  IR  0
dt
I

R
Physics for Scientists and Engineers II , Summer Semester 2009
R
 t
L
1  e
R 
(as shown before)
I
+ -





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Kirchhoff loop rule during switch-off process
L
L
dI
 IR  0
dt
dI
R

  dt
I
L
I
t
dI
R
     dt
I
L
Ii
0

I
I
+ -

R
If the current wa s on for a long time
before switching it off :
 Ii 

R
 I

R
e

dI
 IR  0
dt
L
I
R
 ln     t
L
 Ii 
 I  Iie
R
 t
L
 Iie

t

t

Physics for Scientists and Engineers II , Summer Semester 2009
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Measuring the time constant from switch-off process

R
0.368
I(t)

R
L

R
Physics for Scientists and Engineers II , Summer Semester 2009
t
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Making sense of the time constant

L
R
A large inductance L  large induced emf  slow increase in current.
A large resistance R  a small final current  final current is reached faster.
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Back to flux and inductance calculations
The basic process of getting L :
1) Assuming a current I runs through t he inductor, calculate the flux
throug h the inductive loop (inductor) due to that current.
2) Usually :  B  I   B  C I (where C is some constant depending on the geometry.)
3)  ind  
d B t 
dI
dI
 C
 L
dt
dt
dt
 LC
B
I
(so, L is in fact that constant C....)
(where  B is the flux throu gh the inductor due to the current I in the inductor)
4) For loops with N turns

LN B
I
(where  B is the flux throu gh one turn of the inductor due to the current I in the inductor)
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