Download Phys132 Lecture 5 - University of Connecticut

Physics 1502: Lecture 22
Today’s Agenda
• Announcements:
– RL - RV - RLC circuits
• Homework 06: due next Wednesday …
• Induction / AC current
Induction
Self-Inductance, RL Circuits
a
I
I
R
XXX
XXXX
XX
b
L

long
solenoid
Energy and
energy density
 on
/R
L/R
2L/R
 off
/R
I
L/R
2L/R
I
0
t
0

0
VL
VL
0
-
t
t
Charging
C
RC
Discharging
2RC
C
q
0
RC
2RC
q
0
t
t
0
/R
I
I
0
- /R
t
t
Mutual Inductance
• Suppose you have two coils
with multiple turns close to
each other, as shown in this
cross-section
Coil 1
Coil 2
B
• We can define mutual inductance
M12 of coil 2 with respect to coil 1
as:
It can be shown that :
N1
N2
Inductors in Series
• What is the combined (equivalent)
inductance of two inductors in series,
as shown ?
a
a
L1
Note: the induced EMF of two inductors
now adds:
L2
b
Since:
And:
Leq
b
Inductors in parallel
• What is the combined (equivalent)
inductance of two inductors in
parallel, as shown ?
Note: the induced EMF between
points a and be is the same !
a
L1
a
L2
b
Also, it must be:
We can define:
And finally:
Leq
b
LC Circuits
• Consider the LC and RC
series circuits shown:
C R
C
L
• Suppose that the circuits are
formed at t=0 with the
capacitor C charged to a value Q. Claim is that there is a
qualitative difference in the time development of the currents
produced in these two cases. Why??
•
Consider from point of view of energy!
• In the RC circuit, any current developed will cause energy to be
dissipated in the resistor.
• In the LC circuit, there is NO mechanism for energy dissipation;
energy can be stored both in the capacitor and the inductor!
RC/LC Circuits
i
Q+++
---
i
Q+++
---
C R
0
i
0
1
t
L
LC:
current oscillates
RC:
current decays exponentially
-i
C
0
t
LC Oscillations
(qualitative)
+ +
- -
C
L

C

C
L

L

-
-
+ +
C
L
Energy transfer in a
resistanceless,
nonradiating LC
circuit.
The capacitor has a
charge Qmax at t = 0,
the instant at which
the switch is closed.
The mechanical
analog of this circuit is
a block–spring
system.
LC Oscillations
(quantitative)
• What do we need to do to turn our
qualitative knowledge into quantitative
knowledge?
• What is the frequency w of the
oscillations (when R=0)?
– (it gets more complicated when R
finite…and R is always finite)
+ +
- -
C
L
LC Oscillations
(quantitative)
i
• Begin with the loop rule:
Q
+ +
- -
C
L
• Guess solution: (just harmonic oscillator!)
remember:
where:
• w0 determined from equation
• f, Q0 determined from initial conditions
• Procedure: differentiate above form for Q and substitute into
loop equation to find w0.
Review: LC Oscillations
i
• Guess solution: (just harmonic oscillator!)
Q
where:
+ +
- -
C
L
• w0 determined from equation
• f, Q0 determined from initial conditions
which we could have determined
from the mass on a spring result:
1
The energy in LC circuit conserved !
When the capacitor is fully charged:
When the current is at maximum (Io):
The maximum energy stored in the
capacitor and in the inductor are the same:
At any time:
Lecture 22, ACT 1
• At t=0 the capacitor has charge Q0; the resulting
oscillations have frequency w0. The maximum
current in the circuit during these oscillations has
value I0 .
– What is the relation between w and w2 , the
1A frequency of oscillations when0 the initial
charge
= 2Q0 ?
(a) w2 = 1/2 w0
(b) w2 = w0
t=0
+ +
Q  Q0
- -
C
(c) w2 = 2 w0
L
Lecture 22, ACT 1
t=0
• At t=0 the capacitor has charge Q0; the
resulting oscillations have frequency
w0. The maximum current in the circuit
during these oscillations has value I0 .
+ +
Q  Q0
- -
C
1B • What is the relation between I0 and I2 , the maximum
current in the circuit when the initial charge = 2Q0 ?
(a) I2 = I0
(b) I2 = 2 I0
(c) I2 = 4 I0
L
Summary of E&M
• J. 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
I 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!
• Gauss’ Law:
• Symmetry: both E and B obey the same kind of equation
(the difference is that magnetic charge does not exist!)
• Ampere’s Law and Faraday’s Law:
!
• If Ampere’s Law were correct, the right hand side of Faraday’s
Law should be equal to zero -- since no magnetic current.
• Therefore(?), maybe there is a problem with Ampere’s Law.
• In fact, Maxwell proposes a modification of Ampere’s Law by
adding another term (the “displacement” current) to the right
hand side of the equation! ie
Displacement current
Remember:
Iin
FE
Iout
changing electric
flux
Maxwell’s Displacement Current
• Can we understand why this “displacement current” has the
form it does?
• Consider applying Ampere’s
Law to the current shown in
the diagram.
• If the surface is chosen as
1, 2 or 4, the enclosed
current = I
• If the surface is chosen as
3, the enclosed current = 0!
(ie there is no current
between the plates of the
capacitor)
circuit
Big Idea: The Electric field between the plates changes in time.
“displacement current” ID = 0 (dfE/dt) = the real current I in the
wire.
Maxwell’s Equations
• These equations describe all of Electricity and
Magnetism.
• They are consistent with modern ideas such as
relativity.
• They even describe light