1. LC Oscillations L C I
++++
Kirchoff’s loop rule
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2. LC Oscillations Q V C I V t L dI dt
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3. Example 1 (a) Vab < 0 (b) Vab = 0 (c) Vab > 0 L C t=0 L C t=t1+ - Q Qo = L C
t=t1 Q =
At t=0, the capacitor in the LC circuit shown has a total charge Q0. At t = t1, the capacitor is uncharged.
What is the value of Vab, the voltage across the inductor at time t1?
(a) Vab < 0
(b) Vab = 0
(c) Vab > 0
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4. Example 2 (a) I2 = I0 (b) I2 = 2I0 (c) I2 = 4I0 t=0+ - Q Qo =
At t=0 the capacitor has charge Q0; the resulting oscillations have frequency 0. The maximum current in the circuit during these oscillations has value I0.
What is the relation between I0 and I2 , the maximum current in the circuit when the initial charge = 2Q0?
(a) I2 = I0
(b) I2 = 2I0
(c) I2 = 4I0
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5. Example 3 (a) 2 = 1/2 0 (b) 2 = 0 (c) 2 = 20 t=0+ - Q Qo =
At t=0 the capacitor has charge Q0; the resulting oscillations have frequency 0. The maximum current in the circuit during these oscillations has value I0.
What is the relation between 0 and 2, the frequency of oscillations when the initial charge = 2Q0?
(a) 2 = 1/2 0
(b) 2 = 0
(c) 2 = 20
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6. LC Oscillations: Energy Check
Oscillation frequency has been found from the loop equation.
The other unknowns ( Q0,  ) are found from the initial conditions. e.g. in our original example we took as given, initial values for the charge (Qi) and current (0). For these values: Q0 = Qi,  = 0.
Question: Does this solution conserve energy?
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7. Energy Check Energy in Capacitor Energy in Inductor t t U U E Bt U B
Therefore,
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8. UB versus UE
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9. Example 4 (a) UL1 < UC1 (b) UL1 = UC1 (c) UL1 > UC1 L C t=0 L C+ - Q Qo = L C
t=t1 Q =
At t=0, the capacitor in the LC circuit shown has a total charge Q0. At t = t1, the capacitor is uncharged.
What is the relation between UL1, the energy stored in the inductor at t = t1, and UC1 , the energy stored in the capacitor at t = t1?
(a) UL1 < UC1
(b) UL1 = UC1
(c) UL1 > UC1
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10. LC Oscillations with Finite R
If L has finite R, then energy will be dissipated in R and the oscillations will become damped.
R = 0 Q t
R  0
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11. Driven Oscillations R C L An LC circuit is a natural oscillator.+ + C L - -
In a real LC circuit, we must account for the resistance of the inductor. This resistance will damp out the oscillations. Q t
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12. AC Circuits: Series LCR
Statement of problem:
Given  = msint , find I(t).
Everything else will follow.
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13. Phasors: LCR
Given:
Assume: Þ Þ   I m R  X C L
From these equations, we can draw the phasor diagram at the right.
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14. Phasors: LCR  I m R  X C L I m R   (X L -X C ) ß
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15. Phasors: LCR
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16. QUIZ lecture 20 L C
At time t = 0 the capacitor is fully charged with Qmax and the current through the circuit is 0.
How much energy is stored in the capacitor when the current is a maximum?
U=0 stored in capacitor when current is maximum
Total Energy is constant!
ULmax = ½ LImax2
UCmax = Qmax2/2C
I = max when Q = 0
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17. QUIZ lecture 20 L C
At time t = 0 the capacitor is fully charged with Qmax and the current through the circuit is 0.
What is the potential difference across the inductor at when the current is maximum? A)  VL = 0
B)  VL = Qmax/2C
C)  VL = Qmax/sqrt(2)C
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18. Resonance
For fixed R,C,L the current Im will be a maximum at the resonant frequency 0 which makes the impedance Z purely resistive.
ie:
reaches a maximum when: X C = L
the frequency at which this condition is obtained is given from:
Note that the resonant frequency is identical to the natural frequency of the LC circuit by itself!
At this frequency, the current and the driving voltage are in phase!
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19. Power in LCR Circuit
The power supplied by the emf in a series LCR circuit depends on the frequency . The maximum power is supplied at the resonant frequency 0.
The instantaneous power (for some frequency, ) delivered at time t is given by:
The most useful quantity to consider here is not the instantaneous power but rather the average power delivered in a cycle.
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20. Power in LCR Circuit
This result is often rewritten in terms of rms values:
Power delivered depends on the phase, , the “power factor”
Phase depends on the values of L, C, R, and w and therefore...
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21. Maxwell Equations: Electromagnetic Waves
Maxwell’s Equations contain the wave equation
The velocity of electromagnetic waves:
c = x 108 m/s
The relationship between E and B in an EM wave
Energy in EM waves: the Poynting vector x z y
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22. The equations so far..... Gauss’ Law for E Fields
Gauss’ Law for B Fields
Faraday’s Law
Ampere’s Law
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23. A problem with Ampere’s Law
Consider a wire and a capacitor. C is a loop.
Time dependent situation: current flows in the wire as the capacitor charges up or down.
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24. Maxwell’s Displacement Current, Id
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25. Calculating Displacement Current
Consider a parallel plate
capacitor with circular plates of radius R. If charge is flowing onto one plate and off the other plate at a rate I = dQ/dt what is Id ?
The displacement current is not a current. It represents magnetic fields generated by time varying electric fields.
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26. Calculating the B field
Example
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27. Maxwell’s Equations (1865)
in Systeme International (SI or mks) units
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