Announcements: RL  RV  RLC circuits Homework 06: due next Wednesday … Maxwell’s equations / AC current. Physics 1502: Lecture 23 Today’s Agenda. X X X X X X X X X. Induction. SelfInductance, RL Circuits. long solenoid. Energy and energy density. In series (like resistors). a.
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RL  RV  RLC circuits
Homework 06: due next Wednesday …
Maxwell’s equations / AC current
Physics 1502: Lecture 23Today’s Agendaa
a
a
a
L1
L2
Leq
Leq
L1
L2
b
b
b
b
And finally:
And:
Inductors in Series and ParallelJ. C. Maxwell (~1860) summarized all of the work on electric and magnetic fields into four equations, all of which you now know.
However, he realized that the equations of electricity & magnetism as then known (and now known by you) have an inconsistency related to the conservation of charge!
Gauss’ Law
Faraday’s Law
Gauss’ Law
For Magnetism
Ampere’s Law
Summary of E&MI don’t expect you to see that these equations are inconsistent with conservation of charge, but you should see a lack of symmetry here!
!
Ampere’s Law is the Culprit!Can we understand why this “displacement current” has the form it does?
circuit
Maxwell’s Displacement CurrentBig Idea: The Electric field between the plates changes in time.
“displacement current” ID = e0 (dfE/dt)= the real current I in the wire.
S the form it does?
S
S
N
N
N
End View
Side View
Power ProductionAn Application of Faraday’s LawAdd resistance to circuit: the form it does?
R
C
L
e
RLC  CircuitsAnswer: the form it does?Yes, if we can supply energy at the rate the resistor dissipates it! How? A sinusoidally varying emf (AC generator) will sustain sinusoidal current oscillations!
R
+
+
C
L


We could solve this equation in the same manner we did for the LCR damped circuit. Rather than slog through the algebra, we will take a different approach which uses phasors.
R
C
L
e
~
AC CircuitsSeries LCRBefore introducing phasors, per se, begin by considering simple circuits with one element (R,C, or L) in addition to the driving emf.
R
i
R
e
~
m
m / R
Note: this is always, always, true… always.
0
0
m
m / R
t
0
t
0
x
eR CircuitÞ
Voltage across R in phase with current through R
Consider a simple AC circuit with a purely resistive load. For this circuit the voltage source is e = 10V sin (2p50(Hz)t) and R = 5 W. What is the average current in the circuit?
R
e
~
A) 10 A
B) 5 A
C) 2 A
D) √2 A
E) 0 A
Lecture 23, ACT 1aConsider a simple AC circuit with a purely resistive load. For this circuit the voltage source is e = 10V sin (2p50(Hz)t) and R = 5 W. What is the average power in the circuit?
R
e
~
A) 0 W
B) 20 W
C) 10 W
D) 10 √2 W
Chapter 28, ACT 1bAverage values for I,V are not that helpful (they are zero). For this circuit the voltage source is
Thus we introduce the idea of the Root of the Mean Squared.
In general,
RMS ValuesSo Average Power is,
Voltage For this circuit the voltage source is across C lags current through C by onequarter cycle (90°).
C
e
~
Cm
m
Is this always true?
YES
0
0
m
Cm
0
t
t
0
x
Þ
Þ
A circuit consisting of capacitor For this circuit the voltage source is C and voltage source e is constructed as shown. The graph shows the voltage presented to the capacitor as a function of time.
Which of the following graphs best represents the time dependence of the current i in the circuit?
e
t
i
i
i
(c)
(a)
(b)
t
t
t
Lecture 23 , ACT 2Voltage For this circuit the voltage source is across L leadscurrent through L by onequarter cycle (90°).
L
e
~
m
m L
0
0
Yes, yes, but how to remember?
m
m L
0
t
t
0
x
x
Þ
Þ
A For this circuit the voltage source is phasor is a vector whose magnitude is the maximum value of a quantity (eg V or I) and which rotates counterclockwise in a 2d plane with angular velocity w. Recall uniform circular motion:
y
y
Þ
PhasorsÞ
Þ
The projections of r (on the vertical y axis) execute sinusoidal oscillation.
w
x
Suppose: For this circuit the voltage source is
i
w
i
0
wt
t
w
i
i
0
wt
w
i
i
wt
0
ß
Phasors for L,C,RA series LCR circuit driven by emf For this circuit the voltage source is e = e0sinwt produces a current i=imsin(wtf). The phasor diagram for the current at t=0 is shown to the right.
At which of the following times is VC, the magnitude of the voltage across the capacitor, a maximum?
t=0
f
i
i
t=0
t=tc
t=tb
i
i
(c)
(a)
(b)
Lecture 23, ACT 3Here all unknowns, For this circuit the voltage source is (im,f) , must be found from the loop eqn; the initial conditions have been taken care of by taking the emf to be: e = em sinwt.
R
C
L
e
~
From these equations, we can draw the phasor diagram to the right.
R
C
L
e
~
Þ
w
Phasors: LCRÞ
w right.
R
C
L
e
~
The unknowns (im,f) can now be solved for graphically since the vector sum of the voltages VL + VC + VR must sum to the driving emf e.
Phasors: LCROhms >
y right.
x
imXL
em
f
imR
Z
 XLXC

imXC
R
“Full Phasor Diagram”
“ Impedance Triangle”
Phasors:TipsFrom this diagram, we can also create a triangle which allows us to calculate the impedance Z: