Today’s Concepts:

A) LC circuits and Oscillation Frequency

B) Energy

C) RLC circuits and Damping

Physics 2112

Unit 19

Electricity & Magnetism Lecture 19, Slide 1

Your Comments

Electricity & Magnetism Lecture 19, Slide 2

differential equations killing me.

Differential equations have taken over my life. I'm not sure I We've

been doing spring stuff in Diff Eq, so this is lining up nicely!

I was never good with springs during 2111 so that pre-lecture just

was rugged. The pre-lecture said without a battery an LC circuit

with a full capacitor would just oscillate back and forth since there is

no energy that can be added, but isn't there heat energy being

released through the wires that will also dampen the oscillation?

For capacitors you told us that huge capacitors were used in car

stereo systems. A similar example for how capacitors and inductors

are used would be nice.

Can you go over the differential equations again?

I find it very convenient that this is pretty much exactly like last

semesters spring problems. I'll need practice, but it seems very

straight forward!

Can we relate flux to snow?

C

I

L Q

LC Circuit

Electricity & Magnetism Lecture 19, Slide 3

+

-

dt

dILVL

+

-

C

QVC

Circuit Equation: 0+dt

dIL

C

Q

dt

dQI

LC

Q

dt

Qd-

2

2

where

Qdt

Qd 2

2

2

-LC

1

)cos()( max + tQtQSolution to this DE:

CL

k

x

m

F = -kx

a

Same thing if we notice that and

Electricity & Magnetism Lecture 19, Slide 4

Qdt

Qd 2

2

2

-LC

1

m

kx

dt

xd 2

2

2

-

Ck

1

Lm

Lm

Just like in 2111

k

x

m

F = -kx

a

Electricity & Magnetism Lecture 19, Slide 5

Also just like in 2111

22

2

1

2

1kxmv

Total Energy constant (when no friction present)

CL

2

2

2

1

2

1

C

QLI

Total Energy constant (when no resistor present)

Unit 19, Slide 6

Prelecture Question

What happens to the frequency ω of the oscillations

when the switch is closed?

A. Increases (faster oscillations)

B. Remains the same

C.Decreases (slower oscillations)

The switch is initially open

and the circuit is

oscillating with a

frequency ω =1/sqrt(LC).

CL

I+ +

--

Q and I Time Dependence

Electricity & Magnetism Lecture 19, Slide 7

Charge and Current “90o out of

phase”

What is the potential difference across the inductor at t = 0?

A) VL = 0

B) VL = Qmax/C

C) VL = Qmax/2C

At time t = 0 the capacitor is fully charged with Qmax and the current through the circuit is 0. L C

The voltage across the capacitor is Qmax/C Kirchhoff's Voltage Rule implies that must also be equal to the voltage across the inductor

since VL VC

CheckPoint 1(A)

Electricity & Magnetism Lecture 19, Slide 8

dI/dt is zero when current is maximum

CheckPoint 1(B)

Electricity & Magnetism Lecture 19, Slide 9

At time t = 0 the capacitor is fully charged with Qmax and the current through the circuit is 0.

L C

What is the potential difference across the inductor at when the current is maximum?

A) VL = 0

B) VL = Qmax/C

C) VL = Qmax/2C

How much energy is stored in the capacitor when the current is a maximum ?

A) U Qmax2/(2C)

B) U Qmax2/(4C)

C) U 0

Total Energy is constant!ULmax ½ LImax

2

UCmax Qmax2/2C

I max when Q 0

CheckPoint 1(C)

Electricity & Magnetism Lecture 19, Slide 10

At time t = 0 the capacitor is fully charged with Qmax and the current through the circuit is 0. L C

The switch is closed for a long time, resulting in a steady current Vb/R through the inductor.

A. Q(t) = Qmaxcos(ωt) B. Q(t) = Qmaxcos(ωt + π/4) C. Q(t) = Qmaxcos(ωt + π/2) D. Q(t) = Qmaxcos(ωt + π)

Prelecture Question

Electricity & Magnetism Lecture 19, Slide 11

At time t = 0, the switch is opened, leaving a simple LC circuit. Which formula best describes the charge on the capacitor as a function of time?

L=0.82H

Example 19.1 (Charged LC circuit)

Unit 19, Slide 12

1 2After being left in position 1 for a long time, at t=0, the switch is flipped to position 2.

All circuit elements are “ideal”.

C=0.01FV=12V

What is the equation for the charge on the upper plate of the capacitor at any given time t?

How long does it take the lower plate of the capacitor to fully discharge and recharge?

R=10W

Example 19.1 (Charged LC circuit)

Unit 19, Slide 13

A) Find the current through the inductor when the switch is in position 1. Energy will be conserved.

B) Find the rate of energy lost in the resistor when the switch is position 1. Energy will be conserved.

C) Find the voltage drop across the inductor when the switch is in position 1. The inductor and the capacitor will be in parallel.

To figure out the equation for the charge on the upper plate of the capacitor at any given time t, I’m going fill in the number in this equation:

How will I figure out Qmax?

)cos()( max + tQtQ

L=0.82H

Example 19.1 (Charged LC circuit)

Unit 19, Slide 14

1 2After being left in position 1 for a long time, at t=0, the switch is flipped to position 2.

All circuit elements are “ideal”.

C=0.01FV=12V

What is the equation for the charge on the upper plate of the capacitor at any given time t?

How long does it take the lower plate of the capacitor to fully discharge and recharge?

R=10W

The capacitor is charged such that the top plate has a charge +Q0 and the bottom plate -Q0. At time t 0, the switch is closed and the circuit oscillates with frequency 500

radians/s.

What is the value of the capacitor C?

A) C = 1 x 10-3 F

B) C = 2 x 10-3 F

C) C = 4 x 10-3 F

L C

L = 4 x 10-3 H

= 500 rad/s+ +

--

CheckPoint 2(A)

Electricity & Magnetism Lecture 19, Slide 15

LC

1

3

34210

)104)(1025(

11 -

-

LC

Which plot best represents the energy in the inductor as a function of time starting just after the switch is closed?

Current is changing UL is not constant

Initial current is zero

L C

closed at t = 0

+Q0

-Q0

CheckPoint 2(B)

Electricity & Magnetism Lecture 19, Slide 16

Energy proportional to I2 C cannot be negative

2

2

1LIUL

When the energy stored in the capacitor reaches its maximum again for the first time after t 0, how much charge is stored on the top plate of the capacitor?

A) +Q0

B) +Q0 /2

C) 0

D) -Q0/2

E) -Q0

CheckPoint 2(C)

Electricity & Magnetism Lecture 19, Slide 17

L C

closed at t = 0

+Q0

-Q0

Current goes to zero twice during one cycle

Q is maximum when current goes to zero

dt

dQI

Add Resistance

Unit 19, Slide 18

1 2

Switch is flipped to position 2.

R

01

2

2

--- QCdt

dQR

dt

QdL

02 ++ cbtatUse “characteristic

equation”

Damped Harmonic Motin

Unit 19, Slide 19

)'cos( + - teQQ t

Max

L

R

2

222' - o

Damping

factor

Damped

oscillation

frequency

LCo

1

Natural

oscillation

frequency

Remember from 2111?

Mechanics Lecture 21, Slide 20

Overdamped

Critically damped

Under damped

Which one do you want for your shocks

on your car?

L=0.82H

Example 19.2 (Charged LC circuit)

Unit 19, Slide 21

1 2After being left in position 1 for a long time, at t=0, the switch is flipped to position 2.

The inductor now non-ideal has some internal resistance.

What is the equation for the charge on the capacitor at any given time t?

C=0.01FV=12V

RL=7W

R=10W

How long does it take the lower plate of the capacitor to

fully discharge and recharge?

Example 19.2 (Charged LC circuit)

Unit 19, Slide 22

When we did this problem before without any resistance, we found that it takes 0.57 sec for the lower plate of the capacitor to fully discharge and recharge.

How long will it take now that I am accounting for the resistance of the inductor?

A. more than 0.57secB. less than 0.57secC. the discharge time is not effected by the

resistance

The elements of a circuit are very simple:

This is all we need to know to solve for anything.

But these all depend on each other and vary with time!!

Electricity & Magnetism Lecture 19, Slide 23

dt

dILVL

C

QVC

dt

dQI

IRVR

RCL VVVV ++

V(t) across each elements

Start with some initial V, I, Q, VL

Now take a tiny time step dt (1 ms)

Repeat…

How would we actually do this?

Electricity & Magnetism Lecture 19, Slide 24

C

QVC

IRVR

dtL

VId L

IdtdQ

CRL VVVV --

What would this look like?

Unit 19, Slide 25

VL

VR

VC

I

In the circuit to the right • R1 = 100 Ω,• L1 = 300 mH, • L2 = 180 mH,• C = 120 μF and • V = 12 V.

(The positive terminal of the battery is indicated with a + sign. All elements are considered ideal.)

Question

Electricity & Magnetism Lecture 19, Slide 26

The switch has been closed for a long time. What is the energy stored on the capacitor?

A. 0 JB. 0.0015JC. 0.0034JD. 0.0069JE. 1.2J

In the circuit to the right • R1 = 100 Ω,• L1 = 300 mH, • L2 = 180 mH,• C = 120 μF and • V = 12 V.

(The positive terminal of the battery is indicated with a + sign. All elements are considered ideal.)

Question

Electricity & Magnetism Lecture 19, Slide 27

The switch has been closed for a long time. What is the energy stored in the inductors?

A. 0 J

B. 0.0015J

C. 0.0034J

D. 0.0069JE. 1.2J

In the circuit to the right • R1 = 100 Ω,• L1 = 300 mH, • L2 = 180 mH,• C = 120 μF and • V = 12 V.

Example 19.3

Electricity & Magnetism Lecture 19, Slide 28

The positive terminal of the battery is indicated with a + sign. All elements are considered ideal. Q(t) is defined to be positive if V(a) – V(b) is positive. The switch has been closed for a long time.

What is the charge on the bottom plate of the capacitor 2.82 seconds after switch is open? Conceptual Analysis

Find the equation of Q as a function of time Strategic Analysis

Determine initial current

Determine oscillation frequency 0

Find maximum charge on capacitor

Determine phase angle

In the circuit to the right • R1 = 100 Ω,• L1 = 300 mH, • L2 = 180 mH,• C = 120 μF and • V = 12 V.

Example 19.4

Electricity & Magnetism Lecture 19, Slide 29

The positive terminal of the battery is indicated with a + sign. All elements are considered ideal. Q(t) is defined to be positive if V(a) – V(b) is positive.

What is the energy stored in the inductors 2.82 seconds after switch is opened?

Conceptual AnalysisUse conservation of energy

Strategic AnalysisFind energy stored on capacitor using information from Ex 19.3

Subtract of the total energy calculated in the clicker question