Chapter 20
Chapter 20
*RIGHT HAND RULES
*RIGHT HAND RULES
10/4/11
1
10/4/11
2
Chapter 20
Chapter 20
*RIGHT HAND RULES
*RIGHT HAND RULES
3
Right-hand rule number 4
!"
wire with a current in a B field:
!"
L
!"
*Direction of B: Use the right hand rule. Curl
fingers in direction of current flow in loop and
thumb points in the direction of B.
!"
L
3. L is taken in the direction of I.!
!"
L
10/4/11
3
10/4/11
4
RL Circuits
RL Circuits (! on)
Current
I=
!
!
1 " e" Rt / L ) = (1 " e"t /# RL )
(
R
R
Max = !/R
63% Max at t=L/R
Voltage on L
VL = ! e" Rt / L = ! e"t /# RL
Max = !/R
37% Max at t=L/R
Initially, an inductor acts to oppose changes in current through
it. A long time later, it acts like an ordinary connecting wire.!
10/4/11
5
RL Circuits
•! Why does "RL increase for
larger L?
L opposes change in
current & slows down the
rate of change
a
I
6
Sources
I
R
10/4/11
b
!
L
•! Why does "RL decrease for
larger R?
Large R decreases final current “easier charge up
goal”
10/4/11
Large R dissipates energy quicker, speeds up
“discharge of inductor” (speeds up current loss)
7
10/4/11
8
Alternating Current in a Resistor
Alternating Current Generators
(N = 2 for this coil)
VR = Vmax sin(! t)
V
V sin(! t)
I = R = max
R
R
I = I max sin(! t)
! B = NBA cos(" t)
%! B
#=$
= NBA" sin(" t)
%t
# max = NBA"
10/4/11
9
Power Dissipated in a Resistor
10/4/11
Using rms values: summary
2
P = I 2 R = I max
Rsin 2 ! t
Using rms values of current and voltage
allows you to use the familiar dc
formulas, such as V = IR and P = I 2 R.
2
Pmax = I max
R
One ac ampere is said to flow in a circuit
if it produces the same joule heating as
one ampere of dc current under the
same conditions.
Peak value
Average value
Pave
10/4/11
1 2
= I ave
R
2
10
At your house the peak voltage will be
!170 V
11
10/4/11
12
Phasors for R
Capacitors in AC Circuits (Phasors for C)
VR = RI R = Vmax sin ! t
VC =
IR =
Vmax
R
Q
= Vmax sin ! t
C
Q = CVmax sin ! t
sin ! t
•! V lags I by 90
$
#'
I C = ! C" m sin & ! t + )
2(
%
( )
= ! C" m cos ! t
•! V in phase with I#
10/4/11
13
Relationship between Irms & VC,rms
I rms =
VC, rms
XC
10/4/11
14
Inductors in AC Circuits (Phasors for L)
VL = L
1
where XC =
is the capacitive reactance
!C
1. XC is similar to R in I rms =
2. SI unit for XC : ! (ohm)
VR, rms
R
!I L
= Vmax sin " t
!t
.
3. Average power delivered to a capacitor in an ac circuit is zero.
IL =
Vmax
!L
="
$
#'
sin & ! t " )
2(
%
Vmax
!L
cos ! t
V leads I by 90
10/4/11
15
10/4/11
16
Relationship between Irms & Vrms
I rms =
VL, rms
XL
Summary
where X L = ! L is the inductive reactance.
1. X L is similar to R in I rms =
VR, rms
Symbol
R
2. SI unit for X L : " (ohm)
3. Average power delivered to an inductor in an ac circuit is zero.
17
Impedances for L, C, R
XC =
1
!C
$L
R
R
IC
current leads VC
IL
current lags VL
18
xx
00,,
r1
r1
.... r1
r1
nn
11
VR = RI R = Vmax sin ! t
IR =
Vmax
R
sin ! t
Q
= Vmax sin ! t
C
( )
I C = ! C" m cos ! t
00
For high $, XL grows large and L acts like an open switch.
For low $, XL grows small and at DC, L acts like a
conducting wire.
$
#'
IL =
sin & ! t " )
!L
2(
%
XL = !L is inductive
Reactance, in Ohms
19
10/4/11
44
66
t
xx
I
!t
I
f(f( xx)) 00
0
!I L
1.01 1
= Vmax sin " t
!t
Vmax
22
11
is capacitive
r1
x 0 , .. r1
reactance, in Ohms n 11
VL = L
!"
!t
1
XC =
!C
XL = $L is inductive Reactance
I
VR
f(f(xx))000
nn
VC =
I
VR
r1
r1
R is resistance, in Ohms
xx 00,, .. r111
is capacitive reactance
For high $, XC goes to zero, C acts like a wire.
For low $, XC grows larger and at DC, C acts like an open
switch
10/4/11
L
10/4/11
Summary:
R is resistance
1
!C
C
.
10/4/11
Reactance X
V
V
C
C
00
!"
VC
2
4
6
x
VL
I
f( x ) 0
0
!"
I
!t
VL
1.01 1
0
0
2
4
x
6
6.28
20
Extras
Induced current opposes change of in flux
from change in current.
Self Inductance
An induced emf, !L, appears in any
coil in which the current is changing.
This called self induced emf.
X XX X
X XX XX X
dI/dt
X XX X
Loop at right: close switch at b, current starts
to flow. While dI/dt is not equal to zero, !L is
induced.
!L = "
a
#N$ B
#i
= "L
#t
#t
b
Energy Stored in an
Inductor
direction of !L opposes change with time in
current.
U=
1 2
LI
2
Self-Induction: Changing current through a loop induces
an opposing voltage in that same loop.
10/4/11
21
Calculation of Inductance
r
N turns
B=
10/4/11
•! A phasor is a “vector” whose magnitude is the maximum value
of a quantity (eg V or I) and which rotates counterclockwise in a
2-d plane with angular velocity ". Recall uniform circular
motion:
µo NI
!
The projections of r
(on the vertical y axis)
execute sinusoidal
oscillation.
x = r cos ! t
µo N 2 IA
! B = NBA =
!
$ µo N 2 A ' *I
" = #&
% ! )( *t
$ µo N 2 A '
L=&
% ! )(
22
Phasors
*Long solenoid with N turns, radius r, length L:
l
10/4/11
y
y
!#
x
y = r sin ! t
Angular speed:phasors rotate counter clockwise about the origin with an
angular speed of $.
Length: represents the amplitude of the AC quantity
Projection:on the vertical axis represents the value of the AC quantity at time t.
Rotation angle: phase of the AC quantity at time t.
only depends on
geometry
23
10/4/11
24