Levitation above a Superconductor

©2008 by W.H. Freeman and Company

READING QUIZ 1 Which of the following statements is incorrect? A| In an A. C. circuit an ideal resistor is a frequency independent component. B| At zero frequency the reactance of an ideal capacitor is infinite. C| At resonance in a driven series RLC circuit the circulating current has a minimum value. D| In an A. C. circuit the inductive reactance of an ideal inductor increases with frequency.

Resistive Load Start by considering simple circuits with one element (R, C, or L) in addition to the driving emf. Pick a resistor R first. -- + I(t) Kirchhoff’s Loop Rule: Ipeak vR(t) and I(t) in phase

©2008 by W.H. Freeman and Company

I(t) leads v(t) by 90o (1/4 cycle) Capacitive Load Loop Rule: + -- I(t) leads v(t) by 90o (1/4 cycle) Power:

©2008 by W.H. Freeman and Company

vL(t) leads I(t) by 90o (1/4 cycle) Inductive Load Kirchhoff’s Loop Rule: + -- vL(t) leads I(t) by 90o (1/4 cycle) Power:

©2008 by W.H. Freeman and Company

LC Oscillations No Resistance = No dissipation

More on LC Oscillations Charge and current: Period is half that of Q(t) With R=0 Energy stored in capacitor: t t Energy stored in inductor: where so

Series RLC Circuits The resistance R may be a separate component in the circuit, or the resistance inherent in the inductor (or other parts of the circuit) may be represented by R. Finite R Energy dissipation damped oscillation only if R is “small” For large R multiply by I

Driven Series RLC Circuit Kirchhoff’s Loop Rule: common current I must be determined

Voltage and Current in Driven Series RLC Circuit ϕ Phasors

Impedance in Driven Series RLC Circuit ϕ impedance, Z

WARM UP QUIZ 2 Consider an inductor with reactance XL and a capacitor with reactance XC that carry an AC current with frequency f. If the frequency is doubled, what happens to the inductive reactance X’L (2f) and to the capacitive reactance X’C (2f) ? XL’ = 2XL XC’ = 2XC XL’ = 1/2 XL XC’ = 2XC XL’ = 2XL XC’ = 1/2 XC XL’ = 1/2 XL XC’ = 1/2 XC

Resonance For given ε peak , R, L, and C, the current amplitude Ipeak will be at the maximum when the impedance Z is at the minimum. i.e., load purely resistive ε and I in phase This is called resonance. Resonance angular frequency:

Resonance (continued) angular frequency (radians/s): In a steady, driven RLC circuit, power dissipated = power supplied by ac source. This power is dissipated only in R. At resonance, this power is maximum. Power dissipated: frequency (Hz): Phase difference between ε and I: 

Power Delivered ϕ Power factor

DEMO Resonance 6C-12

Transformer AC voltage can be stepped up or down by using a transformer. AC current in the primary coil creates a time-varying magnetic flux through the secondary coil via the iron core. This induces EMF in the secondary circuit. Ideal transformer (no losses and magnetic flux per turn is the same on primary and secondary). (With no load) step-up step-down With resistive load R in secondary, current I2 flows in secondary by the induced EMF. This then induces opposing EMF back in the primary. The latter EMF must somehow be exactly cancelled because V1 is a defined voltage source. This occurs by another current I1 which is induced on the primary side due to I2.

Transformer with a Load With switch S closed: conservation of energy proportional to average power S Imag+I1 I2 equivalent resistance Req The generator “sees” a resistance of Req Impedance Matching: Maximum energy transfer occurs when impedance within the EMF source equals that of the load. Transformer can vary the “effective” impedance of the load.

DEMO JACOB’S LADDER 6D13

Physics 241 – 10:30 QUIZ 3 NOVEMBER 3, 2011 An LC circuit has a natural frequency of 100 MHz. If you want to decrease the natural frequency to 71 MHz, which of the following will accomplish that? a) Double C b) Double both L and C c) Halve C d) Halve both L and C e) Double L and halve C

Physics 241 -- 11:30 Quiz 3. NOVEMBER 3, 2011 An LC circuit has a natural frequency of 100 MHz. If you want to increase the natural frequency to 141 MHz, which of the following will accomplish that? a) Double L b) Double both L and C c) Halve L d) Halve both L and C e) Double L and halve C

Physics 241 -- 11:30 Quiz 3 March 29, 2011 An LC circuit has a natural frequency of 141 MHz. If you want to decrease the natural frequency to 100 MHz, which of the following will accomplish that? a) Double L b) Double both L and C c) Halve L d) Halve both L and C e) Double L and halve C