Physics 121: Electricity & Magnetism – Lecture 13 E-M Oscillations and AC Current

1. Physics 121: Electricity & Magnetism – Lecture 13E-M Oscillations and AC Current Dale E. Gary Wenda Cao NJIT Physics Department

2. Electromagnetic Oscillations

3. Oscillating Quantities • We will write oscillating quantities with a lower-case symbol, and the corresponding amplitude of the oscillation with upper case. • Examples:

4. Derivation of Oscillation Frequency • We have shown qualitatively that LC circuits act like an oscillator. • We can discover the frequency of oscillation by looking at the equations governing the total energy. • Since the total energy is constant, the time derivative should be zero: • But and , so making these substitutions: • This is a second-order, homogeneous differential equation, whose solution is • i.e. the charge varies according to a cosine wave with amplitude Q and frequency w. Check by taking two time derivatives of charge: • Plug into original equation:

5. Which Current is Greatest? • The expressions below give the charge on a capacitor in an LC circuit. Choose the one that will have the greatest maximum current? • q = 2 cos 4t • q = 2 cos(4t+p/2) • q = 2 sin t • q = 4 cos 4t • q = 2 sin 5t

6. Time to Discharge Capacitor • The three circuits below have identical inductors and capacitors. Rank the circuits according to the time taken to fully discharge the capacitor during an oscillation, greatest first. • I, II, III. • II, I, III. • III, I, II. • III, II, I. • II, III, I. I. II. III.

7. Charge, Current & Energy Oscillations • The solution to the equation is , which gives the charge oscillation. • From this, we can determine the corresponding oscillation of current: • And energy • But recall that , so . • That is why our graph for the energy oscillation had the same amplitude for both UE and UB. • Note that Constant

8. Power lost due to resistive heating • As before, substituting and gives the differential equation for q Solution: Damped Oscillations • Recall that all circuits have at least a little bit of resistance. • In this general case, we really have an RLC circuit, where the oscillations get smaller with time. They are said to be “damped oscillations.” • Then the power equation becomes Damped Oscillations

9. Resonant Frequency • How does the resonant frequency w for an ideal LC circuit (no resistance) compare with w’ for a non-ideal one where resistance cannot be ignored? • The resonant frequency for the non-ideal circuit is higher than for the ideal one (w’ > w). • The resonant frequency for the non-ideal circuit is lower than for the ideal one (w’ < w). • The resistance in the circuit does not affect the resonant frequency—they are the same (w’ = w).

10. Alternating Current • The electric power out of a home or office power socket is in the form of alternating current (AC), as opposed to the direct current (DC) of a battery. • Alternating current is used because it is easier to transport, and easier to “transform” from one voltage to another using a transformer. • In the U.S., the frequency of oscillation of AC is 60 Hz. In most other countries it is 50 Hz. • The figure at right shows one way to make an alternating current by rotating a coil of wire in a magnetic field. The slip rings and brushes allow the coil to rotate without twisting the connecting wires. Such a device is called a generator. • It takes power to rotate the coil, but that power can come from moving water (a water turbine), or air (windmill), or a gasoline motor (as in your car), or steam (as in a nuclear power plant).

11. RLC Circuits with AC Power • When an RLC circuit is driven with an AC power source, the “driving” frequency is the frequency of the power source, while the circuit can have a different “resonant” frequency . • Let’s look at three different circuits driven by an AC EMF. The device connected to the EMF is called the “load.” • What we are interested in is how the voltage oscillations across the load relate to the current oscillations. • We will find that the “phase” relationships change, depending on the type of load (resistive, capacitive, or inductive).

12. A Resistive Load • Phasor Diagram: shows the instantaneous phase of either voltage or current. • For a resistor, the current follows the voltage, so the voltage and current are in phase (f = 0). • If • Then f

13. Power in a Resistive Circuit • The plot below shows the current and voltage oscillations in a purely resistive circuit. Below that are four curves. Which color curve best represents the power dissipated in the resistor? • The green curve (straight line). • The blue curve. • The black curve. • The red curve. • None are correct. PR t

14. A Capacitive Load • For a capacitive load, the voltage across the capacitor is proportional to the charge • But the current is the time derivative of the charge • In analogy to the resistance, which is the proportionality constant between current and voltage, we define the “capacitive reactance” as • So that . • The phase relationship is that f = -90º, and current leads voltage.

15. An Inductive Load • For an inductive load, the voltage across the inductor is proportional to the time derivative of the current • But the current is the time derivative of the charge • Again in analogy to the resistance, which is the proportionality constant between current and voltage, we define the “inductive reactance” as • So that . • The phase relationship is that f = +90º, and current lags voltage.

16. Units of Reactance • We just learned that capacitive reactance is and inductive reactance is . What are the units of reactance? • Seconds per coulomb. • Henry-seconds. • Ohms. • Volts per Amp. • The two reactances have different units.

17. Summary Table

18. Summary Energy in magnetic field • Energy in inductor: • LC circuits: total electric + magnetic energy is conserved • LC circuit: • LRC circuit: • Resistive, capacitive, inductive Charge equation Current equation Oscillation frequency Charge equation Oscillation frequency Reactances:

19. Thoughts on Clickers • How did you like using the clickers in this class? • Great! • It had its moments. • I could take it or leave it. • I would rather leave it. • It was the worst!

20. Thoughts on Clickers • Which answer describes the most important way that the clicker aided you in learning the material? • It helped me to think about the material presented just before each question. • It broke up the lecture and kept me awake. • It tested my understanding. • It provided immediate feedback. • It showed me what others were thinking.

21. Thoughts on Clickers • Which answer describes the second most important way that the clicker aided you in learning the material? • It helped me to think about the material presented just before each question. • It broke up the lecture and kept me awake. • It tested my understanding. • It provided immediate feedback. • It showed me what others were thinking.

22. Thoughts on Clickers • How would you react to clickers being used in other classes at NJIT? • I think it would be excellent. • I think it is a good idea. • I wouldn’t mind. • I would rather not. • I definitely hope not.

23. Thoughts on Clickers • What problems did you have with your clicker? • I had no problems with my clicker. • It was too big or bulky, a pain to carry around. • I had trouble remembering to bring it to class. • My clicker had mechanical problems. • I lost or misplaced it (for all or part of the semester).

24. Thoughts on Clickers • If you had the choice between using a clicker versus having a lecture quiz where you had to fill in a scantron, which would you prefer? • I would prefer the clicker. • I would prefer the scantron quiz.

25. Have a Nice Day • Please click any button on your clicker as you turn your clicker in. This will register your name as having turned in your clicker.