1. LC Oscillators PH 203 Professor Lee Carkner Lecture 22

2. LC Circuit   The capacitor discharges as a current through the inductor   This plate then discharges backwards through the inductor  This process will cycle over and over  Like a mass on a swing

3. LC Oscillations Figure

4. Circuit Properties  Energy   Sum must be constant  Charge   Electrons switch plates  Current  Current in the circuit will vary sinusoidally from max one way to zero to max the other way

5. Oscillation Frequency   Like all sinusoidal patterns, we can define a angular frequency  = 1/(LC) ½   There are 2  radians in a complete cycle   The value of  tells us how rapidly the properties of the circuit cycle

6. Current and Charge   Similarly, q is the charge at a given time and Q is the maximum charge  q = Q cos (  t +  ) i = -I sin (  t +  )  Where  is the phase constant   Note that I and Q are related I =  Q

7. Energy  U E = q 2 /2C  U B = Li 2 /2  We can substitute our expressions for i and q U E = Q 2 /2C cos 2 (  t+  ) U B = Q 2 /2C sin 2 (  t+  )

8. Energy Variations  Unlike q and i, U is always positive   Both energies have the same maximum = Q 2 /2C   The total amount of energy in the system   When one is a maximum the other is zero

9. Simple Harmonic Motion   Velocity and position vary sinusoidally   Parameterized by an angular frequency that depends on two key properties (spring constant and mass)

10. Damping   It will go on forever with total energy never changing   Energy, current and charge decrease with time  Just like a damped mechanical oscillator

11. Damping Factors   Frequency  The frequency of a damped oscillator is less than that on an undamped one  ’ = (  2 – (R/2L) 2 ) ½   The amplitudes are lower by an exponential factor e (-Rt/L)  Note that the higher the resistance the more damping 

12. Next Time  Read 31.6-31.8  Problems: Ch 31, P: 13, 17, 18, 28, 29

13. A switch is closed, starting a clockwise current in a circuit. What direction is the magnetic field through the middle of the loop? What direction is the current induced by this magnetic field? A)Up, clockwise B)Down, clockwise C)Up, counterclockwise D)Down, counterclockwise E)No magnetic field is produced

14. The switch is now opened, stopping the clockwise current flow. Is there a self- induced current in the loop now? A)No, since the magnetic field goes to zero B)No, self induction only works with constant currents C)Yes, the decreasing B field produces a clockwise current D)Yes, the decreasing B field produces a counterclockwise current E)Yes, it runs first clockwise then counterclockwise

15. Consider an inductor connected in series to a battery and a resistor. If the value of the resistor is doubled what happens to the maximum current and the time it takes to reach the maximum current? A)Both increase B)Both decrease C)Max current increases, time decreases D)Max current decreases, time increases E)Neither will change