Your Comments
I am not feeling great about this midterm...some of this stuff is really confusing still and I don't know if I can shove everything into my brain in time, especially after spring break. Can you go over which topics will be on the exam Wednesday?
This was a very confusing prelecture. Do you think you could go over thoroughly how the LC circuits work qualitatively?
I may have missed something simple, but in question 1 during the prelecture why does the charge on the capacitor have to be 0 at t=0? I feel like that bit of knowledge will help me with the test wednesday
I remember you mentioning several weeks ago that there was one equation you were going to add to the 2013 equation sheet... which formula was that? Thanks!
Electricity & Magnetism Lecture 19, Slide 1 Some Exam Stuff
Exam Wed. night (March 27th) at 7:00
Covers material in Lectures 9 – 18
Bring your ID: Rooms determined by discussion section (see link)
Conflict exam at 5:15 Don’t forget:
• Worked examples in homeworks (the optional questions)
• Other old exams For most people, taking old exams is most beneficial
Final Exam dates are now posted The Big Ideas L9-18
Kirchoff’s Rules
Sum of voltages around a loop is zero
Sum of currents into a node is zero
Kirchoff’s rules with capacitors and inductors
• In RC and RL circuits: charge and current involve exponential functions with time constant: “charging and discharging” Q dQ Q • E.g. IR R V C dt C Capacitors and inductors store energy Magnetic fields
Generated by electric currents (no magnetic charges)
Magnetic forces only on charges in motion Fmag qvxB
Easiest to calculate with Ampere’s Law B d I 0 enclosed Changing magnetic fields can generate electric fields! FARADAY’S LAW d d E d EMF V B dA mag dt dt Physics 212 Lecture 19, Slide 3 Physics 212 Lecture 19
Today’s Concepts: A) Oscillation Frequency B) Energy C) Damping
Electricity & Magnetism Lecture 19, Slide 4 LC Circuit
I
dI L C Q Q V L VC L dt C
Q dI Circuit Equation: L 0 C dt
dQ d 2Q Q d 2Q I 2Q dt dt2 LC dt2 where 1 LC
Electricity & Magnetism Lecture 19, Slide 5 CheckPoint 1a
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 t = 0?
A) VL = 0
B) VL = Qmax/C since VL VC C) VL = Qmax/2C
The voltage across the capacitor is Qmax/C Kirchhoff's Voltage Rule implies that must also be equal to the voltage across the inductor Pendulum.
Electricity & Magnetism Lecture 19, Slide 6 LC Circuits analogous to mass on spring
d 2Q 1 2Q L C dt2 LC
F = -kx k a d 2 x k 2 x m dt2 m
x 1 Same thing if we notice that k and m L C
Electricity & Magnetism Lecture 19, Slide 7 Time Dependence
I L C
Electricity & Magnetism Lecture 19, Slide 8 CheckPoint 1b
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
dI/dt is zero when current is maximum
Electricity & Magnetism Lecture 19, Slide 9 CheckPoint 1c
At time t = 0 the capacitor is fully charged with Qmax and the current through the circuit is 0. L C
How much energy is stored in the capacitor when the current is a maximum ?
2 A) U Qmax /(2C) 2 B) U Qmax /(4C) C) U 0
Total Energy is constant! 2 ULmax ½ LImax 2 UCmax Qmax /2C I max when Q 0
Electricity & Magnetism Lecture 19, Slide 10 CheckPoint 2a
The capacitor is charged such that the L = 4 x 10-3 H top plate has a charge Q0 and the bottom plate Q . At time t 0, the = 500 rad/s 0 switch is closed and the circuit L C 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
1 1 1 3 C 2 4 3 10 LC L (2510 )(410 )
Electricity & Magnetism Lecture 19, Slide 11 CheckPoint 2b
closed at t = 0
Q0 L C Q0
Which plot best represents the energy in the inductor as a function of time starting just after the switch is closed? 1 U LI 2 L 2 Energy proportional to I2 C cannot be negative
Current is changing UL is not constant Initial current is zero
Electricity & Magnetism Lecture 19, Slide 12 CheckPoint 2c
When the energy stored in the closed at t = 0 capacitor reaches its maximum Q0 again for the first time after t 0, L C how much charge is stored on the Q0 top plate of the capacitor?
A) Q0 B) Q0 /2 C) 0
D) Q0/2 E) Q0
Q is maximum when current goes to zero dQ I dt Current goes to zero twice during one cycle Electricity & Magnetism Lecture 19, Slide 13 Add R: Damping
Just like LC circuit but energy but the oscillations get smaller because of R
Concept makes sense… …but answer looks kind of complicated
Electricity & Magnetism Lecture 19, Slide 14 Physics Truth #1:
Even though the answer sometimes looks complicated…
Q(t) Qo cos(t )
the physics under the hood is still very simple!
d 2Q 2Q dt2
Electricity & Magnetism Lecture 19, Slide 15 The elements of a circuit are very simple: dI V L L dt
Q V C C V VL VC VR dQ I dt
VR IR
This is all we need to know to solve for anything!
Electricity & Magnetism Lecture 19, Slide 16 Calculation The switch in the circuit shown has been closed for a long time. At t 0, the switch is opened. V L C
What is QMAX, the maximum charge on R the capacitor?
Conceptual Analysis Once switch is opened, we have an LC circuit
Current will oscillate with natural frequency 0
Strategic Analysis Determine initial current
Determine oscillation frequency 0 Find maximum charge on capacitor
Electricity & Magnetism Lecture 19, Slide 18 Calculation
The switch in the circuit shown has
been closed for a long time. At t = 0, IL the switch is opened. V L C R
What is IL, the current in the inductor, immediately after the switch is opened? Take positive direction as shown.
A) IL < 0 B) IL 0 C) IL > 0
Current through inductor immediately after switch is opened is the same as the current through inductor immediately before switch is opened
before switch is opened: all current goes through inductor in direction shown
Electricity & Magnetism Lecture 19, Slide 19 Calculation
The switch in the circuit shown has I been closed for a long time. At t = 0, L V the switch is opened. L VC 0 C R
IL (t 0 ) > 0 The energy stored in the capacitor immediately after the switch is opened is zero. A) TRUE B) FALSE
before switch is opened: after switch is opened:
dIL/dt ~ 0 VL 0 VC cannot change abruptly
BUT: VL VC VC 0 since they are in parallel U ½ CV 2 0 ! VC 0 C C
IMPORTANT: NOTE DIFFERENT CONSTRAINTS AFTER SWITCH OPENED CURRENT through INDUCTOR cannot change abruptly VOLTAGE across CAPACITOR cannot change abruptly
Electricity & Magnetism Lecture 19, Slide 20 Calculation
The switch in the circuit shown has IL been closed for a long time. At t = 0, V the switch is opened. L C R
IL (t 0 ) > 0 VC (t 0 ) 0
What is the direction of the current immediately after the switch is opened?
A) clockwise B) counterclockwise
Current through inductor immediately after switch is opened is the same as the current through inductor immediately before switch is opened
Before switch is opened: Current moves down through L After switch is opened: Current continues to move down through L
Electricity & Magnetism Lecture 19, Slide 21 Calculation
The switch in the circuit shown has been closed for a long time. At t = 0, V the switch is opened. L C R
IL (t 0 ) > 0 VC (t 0 ) 0 What is the magnitude of the current right after the switch is opened? C V L V V A) I V B) I o C) I o D) Io o L R2 C R 2R
Current through inductor immediately after switch is opened is the same as the current through inductor immediately before switch is opened
I L IL Before switch is opened: V VL 0 L VL 0 C
IL R V ILR
Electricity & Magnetism Lecture 19, Slide 22 Calculation
The switch in the circuit shown has IL been closed for a long time. At t = 0, V C the switch is opened. L R Hint: Energy is conserved
IL (t 0 ) V/R VC (t 0 ) 0
What is Qmax, the maximum charge on the capacitor during the oscillations?
V 1 V A) Q LC B) Q max CV C) Q max CV D) Qmax max R 2 R LC
Imax Q max 1 1 Q2 L C L C LI 2 max 2 2 C V When I is max When Q is max Qmax Imax LC LC R (and Q is 0) (and I is 0) 2 1 1 Qmax U LI 2 U 2 2 C Electricity & Magnetism Lecture 19, Slide 23 Follow-Up
The switch in the circuit shown has
been closed for a long time. At t = 0, IL the switch is opened. V L C
Is it possible for the maximum voltage on the R capacitor to be greater than V ? V I V/R QLC A) YES B) NO max max R
V V L L Qmax LC V V can be greater than V IF: > R R max R C max C
We can rewrite this condition in terms of the resonant frequency: 1 0 L > R OR > R 0C
We will see these forms again when we study AC circuits!
Electricity & Magnetism Lecture 19, Slide 24