Physics 2112 Unit 19

Transcription

1 Physics 11 Unit 19 Today s oncepts: A) L circuits and Oscillation Frequency B) Energy ) RL circuits and Damping Electricity & Magnetism Lecture 19, Slide 1

2 Your omments 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 111 so that pre-lecture just was rugged. The pre-lecture said without a battery an L 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. an 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! an we relate flux to snow? Electricity & Magnetism Lecture 19, Slide

3 L ircuit - I + di V L L dt L Q V Q + - I dq dt Q di ircuit Equation: + L 0 dt d Q dt - Q L d Q dt - Q where 1 L Solution to this DE: Q( t) Q cos( t + ) max Electricity & Magnetism Lecture 19, Slide 3

4 Just like in 111 Q dt m L d - Q 1 L L d x dt - x k m k a m F = -kx x Same thing if we notice that k 1 and m L Electricity & Magnetism Lecture 19, Slide 4

5 Also just like in mv 1 kx k a m F = -kx Total Energy constant (when no friction present) x 1 LI 1 Q Total Energy constant (when no resistor present) L Electricity & Magnetism Lecture 19, Slide 5

6 Prelecture Question The switch is initially open and the circuit is oscillating with a frequency ω =1/sqrt(L). What happens to the frequency ω of the oscillations when the switch is closed? A. Increases (faster oscillations) B. Remains the same. Decreases (slower oscillations) Unit 19, Slide 6

7 Q and I Time Dependence L I harge and urrent 90 o out of phase Electricity & Magnetism Lecture 19, Slide 7

8 heckpoint 1(A) At time t = 0 the capacitor is fully charged with Q max and the current through the circuit is 0. L What is the potential difference across the inductor at t = 0? A) V L = 0 B) V L = Q max / ) V L = Q max / since V L V The voltage across the capacitor is Q max / Kirchhoff's Voltage Rule implies that must also be equal to the voltage across the inductor Electricity & Magnetism Lecture 19, Slide 8

9 heckpoint 1(B) At time t = 0 the capacitor is fully charged with Q max and the current through the circuit is 0. L What is the potential difference across the inductor at when the current is maximum? A) V L = 0 B) V L = Q max / ) V L = Q max / di/dt is zero when current is maximum Electricity & Magnetism Lecture 19, Slide 9

10 heckpoint 1() At time t = 0 the capacitor is fully charged with Q max and the current through the circuit is 0. L How much energy is stored in the capacitor when the current is a maximum? A) U Q max /() B) U Q max /(4) ) U 0 Total Energy is constant! U Lmax ½ LI max U max Q max / I max when Q 0 Electricity & Magnetism Lecture 19, Slide 10

11 Prelecture Question The switch is closed for a long time, resulting in a steady current V b /R through the inductor. At time t = 0, the switch is opened, leaving a simple L circuit. Which formula best describes the charge on the capacitor as a function of time? A. Q(t) = Q max cos(ωt) B. Q(t) = Q max cos(ωt + π/4). Q(t) = Q max cos(ωt + π/) D. Q(t) = Q max cos(ωt + π) Electricity & Magnetism Lecture 19, Slide 11

12 Example 19.1 (harged L circuit) After being left in position 1 for a long time, at t=0, the switch is flipped to position. All circuit elements are ideal. V=1V 1 L=0.8H R=10W =0.01F 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? Unit 19, Slide 1

13 Example 19.1 (harged L circuit) 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: Q( t) Qmax cos( t + ) How will I figure out Q max? 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. ) Find the voltage drop across the inductor when the switch is in position 1. The inductor and the capacitor will be in parallel. Unit 19, Slide 13

14 Example 19.1 (harged L circuit) After being left in position 1 for a long time, at t=0, the switch is flipped to position. All circuit elements are ideal. V=1V 1 L=0.8H R=10W =0.01F 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? Unit 19, Slide 14

15 heckpoint (A) The capacitor is charged such that the top plate has a charge +Q 0 and the bottom plate -Q 0. At time t 0, the switch is closed and the circuit oscillates with frequency 500 radians/s. L L = 4 x 10-3 H = 500 rad/s What is the value of the capacitor? A) = 1 x 10-3 F B) = x 10-3 F ) = 4 x 10-3 F 1 L L (5 10 )(4 10 ) Electricity & Magnetism Lecture 19, Slide 15

16 heckpoint (B) closed at t = 0 L +Q 0 -Q 0 Which plot best represents the energy in the inductor as a function of time starting just after the switch is closed? U L 1 LI Energy proportional to I cannot be negative urrent is changing U L is not constant Initial current is zero Electricity & Magnetism Lecture 19, Slide 16

17 heckpoint () 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? closed at t = 0 L +Q 0 -Q 0 A) +Q 0 B) +Q 0 / ) 0 D) -Q 0 / E) -Q 0 Q is maximum when current goes to zero dq I dt urrent goes to zero twice during one cycle Electricity & Magnetism Lecture 19, Slide 17

18 Add Resistance 1 Switch is flipped to position. R Use characteristic equation d Q dq 1 - L - R - Q dt dt at + bt + c 0 0 Unit 19, Slide 18

19 Damped Harmonic Motin Q Q Max e - t cos( ' t + ) Damping factor R L Natural oscillation frequency o 1 L Damped oscillation frequency ' o - Unit 19, Slide 19

20 Remember from 111? Overdamped ritically damped Under damped Which one do you want for your shocks on your car? Mechanics Lecture 1, Slide 0

21 Example 19. (harged L circuit) After being left in position 1 for a long time, at t=0, the switch is flipped to position. The inductor now nonideal has some internal resistance. V=1V 1 R L =7W L=0.8H R=10W =0.01F What is the equation for the charge on the capacitor at any given time t? How long does it take the lower plate of the capacitor to fully discharge and recharge? Unit 19, Slide 1

22 Example 19. (harged L circuit) 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 0.57sec B. less than 0.57sec. the discharge time is not effected by the resistance Unit 19, Slide

23 V(t) across each elements The elements of a circuit are very simple: di V L L dt V V + V + V L R Q V dq I dt V R IR 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 3

24 How would we actually do this? Start with some initial V, I, Q, V L Now take a tiny time step dt (1 ms) VL d I dt L dq Idt V Q V R IR Repeat V V -V -V L R What would this look like? Electricity & Magnetism Lecture 19, Slide 4

25 V L V R V I Unit 19, Slide 5

26 Question In the circuit to the right R 1 = 100 Ω, L 1 = 300 mh, L = 180 mh, = 10 μf and V = 1 V. (The positive terminal of the battery is indicated with a + sign. All elements are considered ideal.) The switch has been closed for a long time. What is the energy stored on the capacitor? A. 0 J B J J D J E. 1.J Electricity & Magnetism Lecture 19, Slide 6

27 Question In the circuit to the right R 1 = 100 Ω, L 1 = 300 mh, L = 180 mh, = 10 μf and V = 1 V. (The positive terminal of the battery is indicated with a + sign. All elements are considered ideal.) The switch has been closed for a long time. What is the energy stored in the inductors? A. 0 J B J J D J E. 1.J Electricity & Magnetism Lecture 19, Slide 7

28 Example 19.3 In the circuit to the right R 1 = 100 Ω, L 1 = 300 mh, L = 180 mh, = 10 μf and V = 1 V. 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.8 seconds after switch is open? onceptual 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 Electricity & Magnetism Lecture 19, Slide 8

29 Example 19.4 In the circuit to the right R 1 = 100 Ω, L 1 = 300 mh, L = 180 mh, = 10 μf and V = 1 V. 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.8 seconds after switch is opened? onceptual Analysis Use conservation of energy Strategic Analysis Find energy stored on capacitor using information from Ex 19.3 Subtract of the total energy calculated in the clicker question Electricity & Magnetism Lecture 19, Slide 9