Dr. Phil's Home

## Lectures in PHYS-2070 (16) / PHYS-2150

Updated: 23 August 2010 Monday.

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### Week of August 16-20, 2010.

Monday 8/16: Working with 3rd and 4th of Maxwell's Equations to generate partial differential equations of E(x,t) and B(x,t). (see pp. 958-959 in Serway) Looking at the solution to Traveling E-M Wave, with v in x-direction, E in y-direction and B in z-direction. Angular frequency omega, wave number k. c = Emax / Bmax. Derivation of c = E / B. Similar to the relationship between the E-field and the B-field in the velocity selector, where v = E / B. Poynting Vector, S. Rate of energy flow over an area -- SI units (W/m²). Traveling E-M Wave, Poynting Vector. Poynting Vector, S. Traveling E-M Wave, Poynting Vector and Intensity. Energy stored equally in E- and B-fields of the E-M wave. Momentum and Pressure of light waves absorbed or reflected on contact. (Complete absorption like totally inelastic collision; complete reflection like totally elastic collision). Discussion of solar energy -- Serway calculates 160,000 W available on the roof of a house, but we only need about 10,000 W. Even accounting for angles, clouds and night, we don't need 100% capture to significantly reduce power from external sources. Light pressure and momentum transfer, despite the fact that light as mass = zero. NASA used solar panels as solar sails on Mariner 10 near the planet Mercury. Light as a particle. The energy of a single photon ("particle" of light) is E = h f, where h = 6.626 × 10 -34 J·s is Planck's constant, a fundamental constant involved in Modern Physics. (If there was only Classical Physics, then h = 0.) Pure colors (EM waves, ROYGBIV for visible light) vs. Perceived colors (RGB, CMY, pinks, browns, "white" light). "Normal" human vision, some types of color blindness. The Electromagnetic Spectrum. Visible light (ROYGBIV=red orange yellow green blue indigo violet). Visible light is 400nm to 750nm (4000 angstroms to 7500 angstroms). Cannot "see" atoms with visable light, because the atom is about 1 angstrom across (1.00E-10 meters). The visible light wave is too large to see something that small. Frequencies LOWER and wavelengths LONGER than visible light (IR infrared, Microwave, Radio waves, ELF extremely low frequency). Frequencies HIGHER and wavelengths SHORTER than visible light (UV ultraviolet, X-rays, Gamma rays). IR perceived as heat. UVA and UVB, sunglasses. Quiz 18/19 DOUBLE Take-Home on E-M Waves and Photons, due Tuesday 17 August 2010. Because this is TWO quizzes, there are 30,000 points here -- even if you leave it blank, please turn in a copy with your name on it, at the very least, for attendance purposes.

• FYI: Handout: SI Prefixes and Dr. Phil's Simplified Significant Figures.
• Note: Dr. Phil has been using capital P for Power and capital P for Pressure. Serway uses script-P for Power. If you are confused in this section, trying spelling out Power and Pressure, rather than abbreviating it to "P". And, of course, momentum is little p.
• Interesting: In looking up the Asahi Pentax Quartz Takumar 85mm lens online, I found a reference to the slightly later and more advanced Ultra Achromatic Takumar 85mm f4.5 lens which had quartz and fluorite lenses (NO glass) and could be used from 200 nm (UV) to 1000 nm (IR).
• And yes, the Final Exam is cummulative.

Tuesday 8/17: Modern Physics -- goes to size/time/length scales far outside our normal experience. Classical Relativity (two observers, two frames of reference), Special Relativity (speed constant), General Relativity (accelerations or gravity). Einstein's postulates: (1) All observers see the same Physics laws. (2) All observers measure the speed of light in vacuum as c. Beta, gamma, Length Contraction and Time Dilation. Alpha Centauri is 4.20 LY from Earth (proper length). Those on a starship see a different distance and experience a different time than the observer left on the Earth. But both think the other observer is moving at v < c. No preferred observer in Special Relativity. Two observers cannot agree on what they see, distance or time. They can only agree that the speed of light in vacuum is c. One sees the proper length: a length measurement where both ends are measured at the same time. One sees the improper length: a length measurement made at two different times. Neither observer is preferred -- that is one is not "more right" than the other. They are both right. These differences in time and length measurements have been confirmed by experiment. Experimental confirmation of Special Relativity: put atomic clocks on aircraft, spacecraft. Two observers cannot agree on the order of events, either. The concept of "simultaneity" is gone. Another confirmation of Special Relativity: Muons (a form of heavy electron) are created in the upper atmosphere -- they're unstable and will decay. Muons measured at mountaintop -- by sea level, nearly all should have already decayed. But you detect almost as many at sea level as on the mountaintop, because the muon lifetime is measured in the muon's rest frame not while we are watching it moving. The Correspondence Principle -- at some point our Classical Physics results need to match the Modern Physics results. So when do we need Special Relativity? For eyeball measurements, we have trouble distiguishing the size of things that are only off by 10%. That would correspond to a gamma = 1.10, and a beta = 0.417 c. Difference in time with identical clocks left on the ground. Relativistic momentum, prel = gamma mv, and relativistic Kinetic Energy, KErel = (gamma - 1) mc². Total Energy, Etotal = gamma mc². The Einstein Relation, E = m c², and conversion between energy and matter (mass). Pair creation of an electron-positron pair from two high energy gamma rays. Once we used to talk of a "relativistic mass", to try to explain why an object of matter cannot be accelerated up to the speed of light in vacuum, c. You can just as easily use the Work-Energy theorem to show that it would take infinite work to get a matter object up to c. (British SF writer Charlie Stross talks about the difficulties of space travel, including traveling to another star here.) Faster than the speed of light? The tachyon -- a hypothetical particle, whose time properties are very confusing. Second set of Sample Final Exams handed out. (Click here and here for a copy.) Quiz 20/21 DOUBLE Take-Home on Special Relativity, due Thursday 19 August 2010. Because this is TWO quizzes, there are 30,000 points here -- even if you leave it blank, please turn in a copy with your name on it, at the very least, for attendance purposes.

• NOTE: In Q20, all calculations are based on when the arrow is moving (from the archer's point of view). As usual, we neglect the time it takes to accelerate and decelerate.

Wednesday 8/18: No class, but Dr. Phil will have special Office Hours from Noon to 4pm.

• We now have Pre-Final Exam Grades here.

Thursday 8/19: Return X3. (Click here for a solution.) Maxwell's Equations and Hertz's radio wave LC oscillator -- the spark gap radio. The Marconi wireless telegraph of the RMS Titanic, and the modern cellphone. AM (amplitude modulation) radio versus FM (frequency modulation) and digital radio. For traveling waves in general, a wave is function of both space and time, x and t. The repeat time is T, the Period. Light as a wave. The frequency is f = 1/T. The repeat length is lamda, the wavelength. For all traveling waves, v = frequency × wavelength. For light v = c = 2.998 × 108 m/s (in vacuum). Classic Farmer vs. Pole Vaulter problem. Each believes that they win the bet, both are entitled to their opinion. Cannot realistically do it though -- cannot build a mechanism that would work as the farmer wants it to work. FYI Only Topics: The deBroglie wavelength -- Wave/Particle Duality for Matter. Planck's constant -- a very small number, but it is NOT zero ( h = 0 in Classical Physics). So the deBroglie wavelength only matters for very small objects, not Buicks. A slowly moving electron is more wavelike, while an electron moving at 99.99% of c is very particle-like. The Heissenberg Uncertainty Principle, means that there are limits to how well we can know (measure) pairs of certain quantities at the same time. delta-p and delta-x, also delta-E and delta-t -- where delta means the uncertainty or error in measuring the quantity. Review. Quiz 22 was an In-Class quiz on the Speed of Light and Attendance. Third set of Sample Final Exams handed out. (Click here for a copy.) There is also another Sample Final put online only.

• ICES Student Course Evaluations available online via GoWMU 8/20 through 8/23.
• For Your Amusement: Werner Heisenberg was pulled over by a traffic cop. The cop asks, "Do you know how fast you were going?" Heisenberg responds, "No, but I know exactly where I am!" The cop says, "I clocked you going exactly 92.67 miles per hour!" and Heisenberg says, "Oh, *thanks*, now I'm lost!"

Friday 8/20: FINAL EXAM: Noon to 2pm, 1110 Rood Hall.

Monday 8/23: Office Hours -- Noon to 3pm. Last chance to make up any missed Exams.

Tuesday 8/24: Grades due at Noon.

### Week of June 28-July 2, 2010.

Wednesday 6/30: Office Hours.

Thursday 7/1: Class begins. Introduction to Dr. Phil. Distribute Syllabus. Electricity & Magentism are related -- one of the great triumphs of 19th century Physics was the realization that Electricity and Magnetism were two sides of the same coin. Moby Dick by Herman Melville -- gold coin, lightning and the reversal of the ship's compass needle.

Friday 7/2: Demo these Class Web Pages and Two pages of Topic 1 assignment handed out. (Webpage here -- Full 28-page Handout as PDF File -- Searchable HTML Page ). Static electricity. The Two-Fluid Model of Electricity. Franklin's One-Fluid Model of Electricity. Occaam's Razor. The simple hydrogen atom -- whatever charge is, the charge on the electron (-e) and the proton (+e) exactly cancel. The Electric Force between two point charges, Coulomb's Law looks like Newton's Law of Universal Gravity. Real Electric Charges. Two charges: like charges repel, unlike (opposite) charges attract. 1 Coulomb of charge is an enormous amount of charge. Two 1.00 C charges separated by 1.00 meters have a force of nine-billion Newtons acting on each other. Quiz 1 was an in-class Attendance and Check-In form. If you were not here, you can still get some of the points by printing out Quiz 1A and giving it to Dr. Phil. (Click here for a copy.)

FYI: Handout: SI Prefixes and Dr. Phil's Simplified Significant Figures. List of Topics covered in PHYS-2050 when Dr. Phil taught it last.

• NOTE: Monday 5 July 2010 is a WMU Holiday, so no classes.
• HOWEVER: Thursday 1 July, Friday 2 July and Tuesday 6 July 2010 are NOT holidays and class DOES meet.
• REMEMBER: If you wish to join the Facebook group, follow the directions here. Do not try to "friend" me. (grin)
• NOTE: If you are using the same computer to access the class web page or these lecture notes, you might want to hit the Refresh button if you don't see any updates. Some browsers tend to use a cached copy of the webpage rather than checking to see if there is an updated page.

### Week of July 5-9, 2010.

Monday 7/5: Independence Day/Fourth of July (Observed), No Classes.

Tuesday 7/6: "Action at a distance" -- Gravity and the Electric Force are not contact forces. The Electric Force between two point charges, Coulomb's Law looks like Newton's Law of Universal Gravity. Four Fundamental Forces in Nature: Gravity, E & M, Weak Nuclear Force, Strong Nuclear Force. The Hydrogen Atom: Gravity loses to Electric Force by a factor of 200 million dectillion (!!!). The Helium Atom: Putting more than one proton in the nucleus produces enormous forces on the tiny protons -- Need the Neutron and the Strong Nuclear Force (!!!). Finding the net vector electric force FE for a system of point charges. Remember: In PHYS-2070, Looking at Symmetry and Zeroes (problems where the answer is zero) as a way of solving problems. Solving a vector Electric Force problem when there isn't symmetry to render the problem zero. The importance of "Showing All Work". Using checks to confirm you're on the right track as you solve a problem. Leaving you to solve for net vector electric force on q1 for problem in class -- q1 = +1.00 × 10-6 C = +1.00 µC,   q2 = +2.00 × 10-6 C ,   q3 = +3.00 × 10-6 C ,   d = 10.0 cm = 0.100 m. Give final vector in Standard Form (Magnitude w/ Units @ Standard Angle), expecting Standard Angle to be just greater than 90°. (see below) Quiz 2 Take-Home on Coulomb's Law, due Thursday 8 July 2010, in class or by 5pm.

• Review of vectors and vector forces. Review of vector notation for components and Standard Form. Right Triangles and Adding and subtracting vectors: Analytical method. (Check to make sure your calculator is set for Degrees mode. Try cos 45° = sin 45° = 0.7071) Why arctangent is a stupid function on your calculator.

Wednesday 7/7: No class on Wednesdays.

Thursday 7/8: Solution for Problem given in class on Tuesday. Non-point charge systems -- Putting a charge on an extended object requires us to know something about the material, in addition to the dimensions and geometry of the extended object. Conductors (metals) versus non-conductors (insulators). Conduction electrons in metals -- free to move around. Insulator electrons very hard to move around. Semi-Conductors sit in the middle. Sometimes they conduct and sometimes they don't. This means they act like a switch or valve, and this is the basis for the entire electronics semi-conductor industry. Charging a conductor by induction. How does q1 know that q2 is there? -- "Action at a Distance" -- Gravity and the Electric Force are not contact forces. The mathematical construct of the Electric Field. E is not an observable quantity. (Side example: Methods of measuring speed v, do not directly measure speed v.) Electric Field is a vector. FE = q E. For a point charge, E = k q1 /r2. SI units for E-field: (N/C). E-field lines radiate away from a positive point charge; converge towards a negative point charge. If the universe is charge neutral, can have all E-field lines from + charges terminating on - charges.Why use E-fields, when you need the force F = q E anyway? Because it allows us to examine the environment without needing another charge. E-field lines allow us to qualitatively sketch what happens when two charges are near to each other. (1) +q and -q, (2) +q and +q. Very close to each point charge, the E-field lines are radial outward, evenly spaced. In the system, the E-field lines interact with each other -- but E-field lines can never cross. Long range, the system of point charges looks like a single net charge. The density of E-field lines in an area gives you an indication of the strength of the E-field line. The numbers of E-field lines attached to a point charge is proportional to the charge. E-field lines radiate away from positive charges and terminate on negative charges. (3) +2q and -q. Quiz 3 Take-Home on Vector Electric Field and Electric Forces, due Friday 9 July 2010, in class or by 5pm.

Friday 7/9: Review of 2-D and 3-D Integration. Rectangular (area, volume), Polar (circumference, area), Cylindrical (volume, surface area). Spherical Co-ordinates (volume, surface area, hollow volume). So far we've looked at Electric Forces and Fields from discrete charges. Now we will look at extended continuous and uniform charges. Direct integration of Electric Force and Electric Field are similar, so we'll just go over direct integration of the E-field. Charge distributions -- lamda (linear charge density, C/m), sigma (surface charge density, C/m²), rho (volume charge density, C/m³). Note the similarity to mass distributions from PHYS-2050. Examples: Rod in-line with line from point P (1-dimensional integration). Rod perpendicular to line from point P. Note that in all these cases, we can predict the long range behavior (E-field behaves as a single point net charge), and anticipate the close-in short range behavior. Check Serway's examples (that's your textbook) -- watch out that his notation may be different. First Sample Exam 1 handed out. (Click here and here and here for a copy.) Quiz 4 Take-Home on Linear Charge Distributions, due Monday 12 July 2010, in class or by 5pm.

• Note that Sample Exams are for your use -- they are NOT to be turned in.
• Q4 Hints: (a) You can use the results from today's lecture. (b) You will need to use Calculus to get a solution.

### Week of July 12-16, 2010.

Monday 7/12: Direct integration of Electric Field continued. Thin ring of charge to center point P. (Symmetry!) Thin ring of charge perpendicular to line from point P. Note that in all these cases, we can predict the long range behavior (E-field behaves as a single point net charge), and anticipate the close-in short range behavior. Check Serway's examples (that's your textbook) -- watch out that his notation may be different. Disk of charge to center point P. Harder to see 1/r² dependence at long range, but it is clear that E goes to zero. Electric Flux: Electric field times Area. Analogy of a bag or box around a light, captures all the light rays no matter the size or shape. Use known E-field of a point charge to evaluate what the Electric Flux must be equal to. Review of Dot Product. Gauss' Law for Electricity. Using Gauss' Law for Point Charge, Conducting Sphere (case 1: r < R). Note that E-field is zero inside a spherical conducting sphere (solid or hollow). If the Earth were hollow, there'd be no gravity inside the Earth either, besides being zero-gee at center of core. Using Gauss' Law for Point Charge, Conducting Sphere, Insulating Sphere, Infinite Line of Charge. Second Sample Exam 1 handed out. (Click here for a copy.) Quiz 5 Take-Home on Gauss' Law, due Tuesday 13 July 2010, in class or by 5pm.

Tuesday 7/13: Gauss' Law for Infinite Sheet of Charge. P.E. is minus the Work. Potential V is similar, but the integral is done on E-field not Force. More importantly the Potential V is an observable quantity. Find components of E by negative of the partial derivative of Electric Potential function V. It will turn out that charge accumulates on the tips of long pointy things -- applies in why some things seem to always get hit by lightning (golfers, people standing in an open field, church steeples). Emax = 3,000,000 N/C = 3,000,000 V/m, in dry air. Ben Franklin and lightning rods. Why your hair stands up warning you that you are getting charged. Handy chart of the four quantities: FE (vector, 2 charges), E (vector, 1 charge), UE (scalar, 2 charges), V (scalar, 1 charge) .Simplified equation V = E d. (But remember that it's really delta-V = - E d .) Example: Lightning. Equipotential surfaces -- lines of constant Electric Potential (voltage). Analogy: Topographic maps, the equipotential lines are like the altitude contour lines. A skier's line of maximum descent down a mountain corresponds to the E-field lines. Conductor in equilibrium is an equipotential throughout. In electrostatic equilibrium, E = 0 inside a charged conductor, but V = constant, not V = 0 automatically (it's delta V = 0 inside the conductor). Why charge accumulates on the tips of "pointy things". Model a conducting blob with a blunt end and a pointy end, sort of like a piece of candy corn, by a large conducting sphere and a smaller conducting sphere, connected together by a wire so they are all equipotentials, i.e. V = constant. For a charged sphere, same as a point charge: V = kq/r. While the charge on the tip is less than the charge on the rest, the surface charge density, sigma = q / Area, is much higher. NOTE: The book is effectively closed for Exam 1 topics now, except for finding V by direct integration, which you might want to look at in the textbook before Thursday's class. Remember, V really is a scalar quantity and not a vector. Quiz 6 Take-Home on Electric Potential (V), due Thursday 15 July 2010, in class or by 5pm.

Wednesday 7/14: No class on Wednesdays.

Friday 7/16: Exam 1.

### Week of July 19-23, 2010.

Monday 7/19: Return Q4, Q5, Q6. Work to assemble charges on a capacitor = Energy stored in the capacitor = U = ½CV² . Energy density, uE = U/vol. = ½(epsilon-naught)E² . While this was derived for the parallel plate case, it turns out to be true in general. Two devices connected together in a circuit can only be connected two ways: series or parallel. In Series, same charge, share voltage. Equivalent capacitor is always smaller. NOTE: Remember to take the last reciprocal! In Parallel, same voltage, share charge. Equivalent capacitor is always larger. Capacitor Network Reduction problem. Carefully analyze the network, reducing series or parallel capacitors to equivalent capacitors, redrawing the circuit each time. Use table with columns for Q = C V. By going back through the intermediate diagrams, it is possible to know every value of every capacitor in the network. Extend the example in class with a fourth column, U=½CV², and find the energy stored in the equivalent capacitor and the sum of the energy stored in all four of the real capacitors -- if they agree, then our analysis and calculations are correct -- the battery cannot tell the difference! Quiz 8 is a Take-Home on Series-Parallel Reduction of a Capacitor Network, due Thursday 22 July 2010.

• NOTE: Filling in the table is a case where you might want to use 3 + 2 sig. figs., in order to get the ratios to come out really close in the end.
• NOTE: Regarding Q8: Especially important that students who know circuits and/or are taking or have taken an ECT circuits course, should not try to solve this with your circuits course methods.
• Note that the energy density, uE , can be important in the real world, because there may be limits. For example, if a parallel plate capacitor is in air, then E < Emax . This business of limits to energy density is something that is frequently wrong in science fiction movies and TV, where you have near infinite power supplies and batteries on devices, especially powerful weapons, which seem to run on and on forever, like the Eveready Energizer Bunny. No -- can't happen like that.

Tuesday 7/20: Making a real capacitor. What if not filled with air? Filling with conductor, must have at least one gap, otherwise will short outthe plates. A conducting slab inside a parallel plate capacitor makes two capacitors in series. Charge neutral slab stays charge neutral, but +Q of top plate attracts -Q on top of slab, and -Q of bottom plate attracts +Q on bottom of slab. Dielectrics -- an insulator where the +/- charge pairs are free to rotate, even if they do not move. Dielectric constant (kappa) and Dielectric strength (E-max). (See Table 26-1, p. 736) Dielectic constant increases capacitance over air gap. Dielectric strength usually bigger than Emax in air. Both allow you to (a) make bigger capacitors (or smaller for the same values) and (b) make non-hollow, self-supporting components. Electrolytic capacitors -- must be connected into the circuit with correct + and - polarity. Examples of the uses of capacitors and dielectrics. Capacitive studfinder, uses edge effects of E-field from a capacitor to "see" the dielectric material behind the wall. Computer keyboards with switches which have "no moving parts". Electrostatics (equilibrium) to Electrodynamics (moving charges). Current defined: i = delta-Q/delta-t = dq/dt. (1) The net charge of a current carrying wire remains zero, so have 1.00 A = 1.00 C/sec isn't the same as having 1.00 C of bare charge lying around. (2) Positive charges moving in same direction as a positive current is the same as negative charges moving the other way. (3) When people say electricity moves "at/near the speed of light", it does not mean the electrons in the wire are moving at the speed of light. It is the E-field which is moving at the speed of light in the material. (4) Discussion of microscopic theory of charges in a conductor. Drift velocity is the very slow net movement of the electrons moving randomly in the wire. See pp. 753-755. Drift velocity of electrons in copper wire is about 2.23×10-4 m/s. This microscopic theory becomes more important as we go to smaller and smaller circuit elements in our microchips. Moore's Law. (5) The charges are moving in response to an E-Field, because we have a non-zero delta-V. We can have an E-field and delta-V inside a conductor in a circuit because this is no longer an electrostatic equilibrium problem. The Simplest Circuit: Battery, wires, load (resistor). Resistance vs. Conductance. Ohm's Law: V=IR form. (Ohm's "3 Laws") We usually treat the wires in a circuit as having R=0, but they usually are not superconductors. Resistance is a function of temperature. Kammerleigh Onnes 1916 work on extending the R vs. T curve toward T = 0 Kelvin. Discovered Superconductivity, where R=0 identically.

• Week 4 Checklist. (Updated 7-20-10 Tu)
• Quiz 9 will be handed out Thursday 22 July 2010. No new quiz on Tuesday 20 July 2010.
• Second video: Helpful motivation for studying. (grin)
• Exam 1 should be returned on Thursday. CONTACT DR. PHIL IF YOU NEED TO MAKE UP EXAM 1.

Wednesday 7/21: No class on Wednesdays.

Thursday 7/22: Return X1. (Click here for a solution.) Joule Heating, Power Law: P = IV (also 3 forms). Resistance by geometry. R = rho (L / A), where rho = resistivity of the material, L = length and A = cross-sectional area. Continuing with Simple Circuits... Series and Parallel Resistors: Two devices connected together in a circuit can only be connected two ways: series or parallel. In Series, same current, share voltage. Equivalent resistance is always larger. In Parallel, same voltage, share current. Equivalent resistance is always smaller. Resistor Network Reduction. (Similar rules to Capacitor Network Reduction except "opposite".) For example given in class, Resistor R3 sees the largest current and dissipates the largest amount of energy per second (Power in Watts). This means it is also the most vulnerable. R1 is second largest. Had we used identical resistors in this example as we did with the capacitor network, then R1 would generate the largest amount of heat by a long shot, and be the most vulnerable resistor. (Story of radio "repair" call from 4,000,000,000 miles.) First Sample Exam 2. (Click here and here for a copy.) Quiz 9 Take-Home on Series-Parallel Reduction of a Resistor Network, due Monday 26 July 2010.

• NOTE: You can turn in Q8 in class on Friday.
• NOTE: I wouldn't even look at Q9 until you're done with Q8, to avoid confusion between capacitors and resistors. (grin)
• Also, I had a mistake in the posted solution to Q7 -- I had cut-and-pasted E into the last equation instead of V -- it's corrected now.

Friday 7/23: Real batteries consist of a "perfect" battery (Electromotive force = emf, or script E) in series with a small internal resistance, r. As chemical reaction in battery runs down, the internal resistance increases. V = (emf - i r) = i R. Don't cut open batteries. Comments on different types of disposable (carbon-zinc, alkaline, lithium) and rechargable (Rayovac Renewal alkaline, NiCad, NiMH, Li-ion) batteries. Multi-cell batteries (6V lattern battery, 9V transistor/smoke alarm, 510V dry cell). Tip for weak car battery on cold day: Run headlights for 30 to 90 seconds. High internal resistance will warm the battery and make it more efficient. Not all circuits can be reduced by serial and parallel network analysis. Kirchhoff's Laws: (1) The sum of all currents in and out of any junction must be zero. (2) The sum of all voltage gains and voltage drops about any closed loop is zero. Practically speaking, if there are N junctions, then (1) will give you (N-1) unique equations, and if there are M loops that can be made in the circuit by going around the perimeter of each "puzzle piece", then (2) will give you sufficient unique equations. You will get the same number of equations as you unknown currents through the resistors. NOTE: EE students and those who have had ECT-2100 (?) may know a "better" way to solve Kirchhoff's problems. But the brute force algebra approach has the advantage of being based on the Physics, so has instructional value. Example in class had 3 equations in 3 unknowns -- (1) i3 is opposite direction than in class, (2) resistors are divided by 10 from class.. (Solution by brute force algebra here). Quiz 10 Take-Home on Real Batteries and Internal Resistance, due Monday 26 July 2010.

### Week of July 26-30, 2010.

• Quiz 11 will be a Take-Home on Kirchhoff's Laws, to be handed out Monday 26 July 2010, and due Thursday 29 July 2010.
• Week 5 Checklist.

Monday 7/26: Note that you can use Kirchhoff's law even when you CAN reduce a circuit by series and parallel network reduction. For example, our Q9 problem has three unknown currents (1 junction equation, 2 loop equations). RC series circuit. Use Kirchhoff's 2nd Law to get a loop equation for voltage gains and drops around charging capacitor. q=q(t) and i=i(t)=dq/dt means that we can use calculus to find the current through the resistor and the charge on the capacitor. Calculus derivation of q(t) for charging capacitor and discharing circuits. RC current i(t) will be the same in both cases. Who knew that (ohms) × (farads) = (seconds)? By time t=3RC, a charging capacitor will reach 95% of its top charge, or a discharging capacitor will be down to 5% of its original charge. Either way the current will be down to 5% of its maximum value. Measurement: Building an ammeter or voltmeter -- non-digital version with a needle. The Galvanometer is a generic meter. It has a resistance and the needle moves in response to a current through a tiny coil. Since meters must be connected to the circuit, technically they change the circuit. However, we will show that the design of an ammeter and a voltmeter minimizes these changes. Ammeters measure current by connecting in series to the circuit. Voltmeters measure potential difference by connecting in parallel to the circuit. The Galvanometer is a generic meter. It has a resistance RG and the needle moves in response to a current through a tiny coil. The full-scale deflection current, iFS , is the current needed to move the needle to the maximum value on the scale -- it is very small. Quiz 11 Take-Home on Kirchhoff's Laws, due Thursday 29 July 2010.

• NOTE: You can still turn Q9 in on Tuesday 27 July 2010 at the beginning of class.

Tuesday 7/27: Return Q8. Ammeter: a galvanometer and a very small shunt resistor in parallel, together connected in series with the circuit. We can make this small resistance by using a short length of "high resistance" wire. Voltmeter: a galvanometer amd a very large resistor in series, together connected in parallel with the circuit. Using a decade box. In both cases, the role of the second resistor is to limit the current to the galvanometer, no matter what the design criteria of the meter in question. Does putting a real ammeter and voltmeter in a circuit, whether the very act of measuring V and I changes their value? It can't be by much, because the full-scale deflection current and the voltage drop across the galvanometer are so small, compared to the values we are measuring. "Magnetism is just like Electricity, only different." Most people are familiar with (1) magnets sticking to some metals, not others such as stainless steel and (2) if you have two magnets, they may attract or repel. North and south are analogous to plus and minus charges. Real Magnets are dipoles (North and South ends, linked). Break a magnet in half, and you either get two new magnets -- or nothing. So far, there is no evidence that there are Magnetic Monopoles (magnetic charges: qM , isolated North or South poles). Rules similar to Electric Charges: Unlike poles attract, like poles repel. The horizontal compass needle rotates until its North end points North (or rather to the North Magnetic Pole, which is of course a South pole of the Earth's magnetic core); the vertical compass rotates so that it lines up with the B-field along the surface of the Earth at the point. At the Equator, the vertical magnetic should be parallel to the ground, at the magnetic poles, it should be perpendicular to the ground. Is the Earth's magnetic field going to flip some day? And what about Mars? Magnetic Force on a Moving Electric Charge - The Cross Product and Right-Hand Rule (R.H.R.). The Cross Product (or Vector Product) is the exact opposite of the Dot Product (or Scalar Product). Multiplying two vectors together by a cross product gives us another vector (instead of a scalar). And the cross product is not commutative, vector-A × vector-B = - (vector-B × vector-A), so the order is paramount. Using Right Hand Rule to assign directions to x,y,z coordinates. Constant speed, perpendicular constant magnetic force --> Uniform Circular Motion. Cyclotron frequency -- no dependence on the radius (constant angular velocity). Velocity Selector - the Magnetic Force is speed dependent, the Electric Force is not. So we can use an E-field to create an Electric Force to cancel the Magnetic Force on a moving charged particle, such that at the speed v = E / B, the particle travels exactly straight with no net force -- any other speed and the particle is deflected into a barrier. Hence a velocity selector "selects" velocities... Second set of Sample Exam 2's handed out: (Click here and here for copies.) Quiz 12 was an In-Class quiz on Attendance, the By-Geometry Resistor equation, q(t) for a Charging Capacitor and a Reality Check on whether your Formula Card has been kept up to date.

• NOTE: The numbers we found for the 5.00 A ammeter and 5.00 volt voltmeter resistors were: rs = 0.001262 ohms and Rv = 49,940 ohms. The galvanometer had a resistance RG = 63.1 ohms and a full-scale deflection current iFS = 1.00 ×10-4 A. The high resistance wire we used for the shunt resistor in the ammeter had an R/L = 0.000147 ohm/cm. For I = 5.00 A and V = 5.00 volts, the load resistor would be R = 1.00 ohms.
• One last topic for Exam 2 to cover at the start of Thursday -- The Mass Spectrometer.

Wednesday 7/28: No class on Wednesdays.

Thursday 7/29: Return Q10. Velocity Selector - the Magnetic Force is speed dependent, the Electric Force is not. So we can use an E-field to create an Electric Force to cancel the Magnetic Force on a moving charged particle, such that at the speed v = E / B, the particle travels exactly straight with no net force -- any other speed and the particle is deflected into a barrier. Hence a velocity selector "selects" velocities... Velocity Selector. Mass Spectrometer - different semi-circular paths for ions of different mass but same velocity. Can determine chemicals, molecules, and separate isotopes (same element, different number of neutrons in nucleus, so different mass -- cannot be separated by ordinary chemical means). Mass Spectrometer as Calutron -- detecting or separating isotopes, something that cannot be done by ordinary chemical means. NOTE: The book is effectively closed for Exam 2 topics now. UPCOMING TOPICS: Since there is a magnetic force on a moving electric charge in a magnetic field, and a current carrying wire is a collection of moving electric charges, there must be a magnetic force on a current carrying wire in a perpendicular B-field. Later we will see that a current carrying wire creates a B-field, so there ends up being a magnetic force between two current carrying wires. Review for X2.

• NOTE: J-vector = sigma × E-vector (current density = conductivity × E-field) is the vector version of Ohm's Law, where J-vector is the Current Density (although we assign a direction, technically current is a scalar and current density (I / Area) is the vector quantity). There are at least two Sample Exam 2 problems with J-vector. This will NOT be on Exam 2.
• Q11 due date now Monday 2 August 2010. You can safely set it aside until after Exam 2.
• Week 5 Checklist. UPDATED 7-29-2010 Th.
• NOTE: T1 Book Reports due starting Thursday 5 August 2010 through Monday 9 August 2010.
• Last day to turn in a Draft Book Report if you wish to, is Monday 2 August 2010. NOT Required, unless you had a book approved that was not on the booklist.

Friday 7/30: Exam 2.

### Week of August 2-6, 2010.

• Week 6 Checklist.
• Last day to turn in a Draft Book Report if you wish to, is Monday 2 August 2010. NOT Required, unless you had a book approved that was not on the booklist.
• NOTE: T1 Book Reports due starting Thursday 5 August 2010 through Monday 9 August 2010.

Monday 8/2: Returned Formula Cards. A current carrying wire consists of moving electric charges, and so therefore would see a magnetic force from a magnetic field. Discussion of microscopic theory of charges in a conductor. Drift velocity is the very slow net movement of the electrons moving randomly in the wire. Magnetic Force on a Current Carrying Wire. Demo -- hey it works and even in the right direction! Technically current is not a vector, despite the fact we talk of direction of current. J = current density = current/cross-sectional area is the vector related to current. NOTE: J-vector = sigma × E-vector (current density = conductivity × E-field) is the vector version of Ohm's Law. This was NOT on Exam 2. So we use the displacement vector L for the direction. For a Closed Loop, the net Magnetic Force from a constant B-field is zero. Magnetic Torque on a Current Carrying Wire. We use the enclosed area vector A, whose direction is defined by using the Mode 2 R.H.R. (fingers curled around the direction of the current loop, thumb is the area vector A perpendicular to the plane of the loop). Left as is, this system is an oscillator -- the torque goes to zero after 90° and then points the other way. But if we can reverse the direction of the current after the torque goes to zero, then the rotation can continue -- and we have a primitive DC electric motor. Hall Effect -- a device with no moving electrical parts -- proves that charge carriers in a current carrying wire are negative, not positive. "The 200 Year Hall Effect Keyboards", will last "forever", but made obsolete in two years when Windows 95 added three keys. Gauss' Law for Magnetism. Not as useful as Gauss' Law for Electricity, because it is always zero (no magnetic monopoles). It turns out that a moving electric charge or a current carrying wire creates a magnetic field. This makes sense that a moving charge or current would have a magnetic force from an external magnetic field, if its own magnetic field is interacting with the external field, just as two charges "see" each other via their electric fields. The permeability of free space, µ0, is the fundamental constant of magnetism. It is unusual in that we know the exact mathematical representation, which is why is given as 4pi × 10-7 T·m/A. If we calculate 1/sqrt(epsilon0 × µ0), we get the c = speed of light in vacuum -- again showing the fundamental connection between electricity and magnetism. To find a magnetic field from a current carrying wire we use the Biot-Savart Law. Note that we use ds-vector × r-hat, the unit vector pointing towards the point P, so that the Biot-Savart Law looks like a 1/r² law. If we used ds-vector × r-vector, it would look like a 1/r³ law. Circular loop of current carrying wire by integration for P at the center of the loop. Quiz 13 Take-Home on the Magnetic Force and Torque on a Current Carrying Wire/Loop, due Tuesday 3 August 2010.

Tuesday 8/3: Gauss' Law for Magnetism. Not as useful as Gauss' Law for Electricity, because it is always zero (no magnetic monopoles). The Biot-Savart Law. B-field from a infinitely long straight current carrying wire by direct integration. (Serway has a similar example, but rather than do the integral in x, he does this theta substitution which Dr. Phil does not think is straight forward.) Circular loop of current carrying wire by integration for P at the center of the loop. (Serway's example allows for P to be on a line perpendicular to the loop.) B-field for a circular current carrying wire at the center -- or any part of a circle. Note that a wire coming in along the r-hat direction makes no contribution to the B-field. Magnetic Field loops from a Current Carrying Wire. RHR has "two modes". Mode 1 uses three mutually perpendicular directions for when you have three vectors (A × B = C is 1-2-3, x-y-z). Mode 2 uses the curling of the fingers to represent the circulation of a field or the motion of a current, etc., with the thumb representing the relevent vector or direction. Magnetic Force between Two Current Carrying Wires. Combining problems, we find that for two parallel current carrying wires, with the currents in the same direction, the magnetic field from wire 1 creates an attractive magnetic force on wire 2. And the magnetic field from wire 2 creates an attractive magnetic force on wire 1. (Two forces, equal and opposite, acting on each other -- this is exactly as it should be with Newton's 3rd Law.) Anti-parallel currents (wires parallel, but currents in opposite directions) repel. Crossed currents (wires perpendicular to each other) see no magnetic force on each other. Operational defnition of the ampere and the coulomb: If the Force per length for two wires with a current I separated by 1 meter is F/L = 2 × 10-7 N/m, then I = 1 A exactly. Then in 1 second, 1 C of charge is moved by this 1 A current. Gauss' Law for Magnetism. Not as useful as Gauss' Law for Electricity, because it is always zero (no magnetic monopoles). However, there is something we can use in a similar way which involves involving a path integral along a B-field and the current(s) contained inside -- Ampere's Law. Use in a way similar to the way we used Gauss' Law for Electricity. Use symmetry and geometry to select your Amperean Loop to your advantage. 3-D directions and R.H.R. You can make a list of axes directions or unit vectors (x y z x y z) and (i j k i j k) and find the 3rd direction of the cross product by going to the right (+) or to the left (-) in the list. Example: i-hat × j-hat = +k-hat, since order is "i j k", but j-hat × i-hat = -k-hat, since "j i k" goes to the left. Ampere's Law. Use in a way similar to the way we used Gauss' Law for Electricity. Use symmetry and geometry to select your Amperean Loop to your advantage. B-field of a Torroid (torroidal coil; a torus is like a donut). B-field of a Solenoid. (NOTE: The integrals for the L and R sides of the Amperean Loop for Ampere's Law are zero because: (1) the B-field is zero outside the solenoid and (2) for that part of the path which is inside the solenoid, the B-field and the ds-vector are perpendicular, so the dot product is zero as well.) Quiz 14 Take-Home on Creating Magnetic Fields, due Thursday 5 August 2010.

Wednesday 8/4: No class on Wednesdays.

Thursday 8/5: Edge effects: E-field of parallel plate capacitor vs. B-field of solenoid. More Comments about making a real velocity selector -- trying to stuff a capacitor for the E-field and a solenoid for the B-field in the same space! Comments about making real coils. Insulating varnish, heat damage. Yields affect time and money. B-field of an infinite sheet of current. Note that Ampere's Law has a flaw, which we will correct at a later date. If a current carrying wire can create a magnetic field, can a magnetic field passing through a coil create an electrical current? Demo: Magnet moving into a coil, causing current to flow through galvanometer. Faraday's Law of Induction. A changing magnetic flux induces a current, induces an e.m.f., in the circuit, substituting for the battery as the power source. Lenz's Law "of maintaining the status quo." The coil acts as if it opposes any change of the magnetic flux inside, by inducing a magnetic field to cancel and increasing flux or maintain a decreasing flux. To create this induced magnetic field, one needs an induced current, which is powered by an induced emf. It is Lenz's Law that gives us the minus sign in Faraday's Law of Induction. Demos: Cow magnets -- powerful cylindrical, rounded end magnets which get dropped into a cow's first stomach, to collect nails, bits of barbed wire, etc. from continuing on to the cow's other stomachs. Demo: Lenz Law race between cow magnets dropped through (a) plastic pipe, (b) non-magnetic aluminum pipe and (c) non-magnetic copper pipe. Something is going on, such that the magnets travel much slower through the metal pipes -- and the thicker copper pipe was much slower than the thinner aluminum pipe. Induced B-fields due to changing B-fields of falling magnets are created by induced currents and induced emf -- as the magnet enters and leaves a circular region of metal pipe, it is slowed by magnetic forces between its magnetic field and the induced B-field. Turn a coil in a magnetic field and the flux changes, thereby inducing a B-field, emf and current. Has same 180° problem that a DC motor has. Hand-crank generators. Electric generators and electric motors differ in which way the arrow points toward or away from mechanical energy. Regenerative braking (also called dynamic braking) -- diesel-electric and electric trains, also hybrid cars like the Toyota Prius -- turn electric motors into generators. Energy either wasted as heat through a resistor, or recharge battery. Discussion of electrical issues in two scenes of Steven Spielberg's movie Jurassic Park. FIRST DAY to turn in Topic 1 Book Reports.

• Dr. Phil likes to think of the analogy of the "cranky old guy" who complains whenever you try to change this.

Friday 8/6: Return X2, and Q9/11/12/13. Demo: "Jumping Rings", making the bulb light, by Eddy Currents and Induction. Note that the metal rings get HOT, because there is a large induced current and metal has a low resistance. Adding metal increases the mass, but provides more current loops and therefore more induced B-fields repelling the solenoid's B-field. A split metal ring (a) does not get hot and (b) does not jump, because there is no circuit enclosing the changing magnetic flux. Ford test electric vehicle with inductive charger -- no exposed metal contacts, everything covered in smooth plastic. Dr. Phil has been advocating for 54 semesters that small charging bricks for all our electronic devices be replaced with universal charging-by-induction systems. Beginning to see first practical systems such as Powermat. Practical uses for induction: (1) Heating: The Good -- Heating the bottom of a metal cooking pot by induction: New type of cooking range uses sealed induction heating elements instead of exposed hot resistors or open gas fed flames -- usable for metal pans only. The Bad -- A slab of metal used to conduct a B-field can waste energy as heat if the B-field is changing, such as in an AC circuit. By making the slab out of thin plates insulated from each other, the B-field still can go around the metal, but the perpendicular loops of Eddy Currents can only have a diameter equal to the thickness of the metal. Small eddy currents cannot generate much heat because induced emf is too small. (2) Safety: (First, how regular fuses and circuit breakers work -- and why that isn't fast enough to prevent some types of accidents.) Ground Fault Interupt -- if the current doesn't return via the return wire, because it has found another conductive path, then the 2 wires (hot and return) total a net-enclosed-current for Ampere's Law, generating a B-field in a metal ring, detected by an induction coil wrapped around the ring and this sets off the relay which breaks the circuit. An Inductor is a coil in a circuit. Why an Inductor has Self-Inductance -- running a current through a coil creates a magnetic field and therefore changes the magnetic flux in the coil. The inductor has to respond to that change. Inductance can be a big deal. Even our Simplest Circuit (a resistor hooked up to a battery) forms a loop, and the loop must respond to the circuit being turned on. SECOND DAY to turn in Topic 1 Book Reports. First Set Sample Exan 3 (Click here for a copy.) Quiz 15 is a Take-Home on Ampere's Law and Faraday's Law of Induction, handed out Friday 6 August 2010, and due Monday 9 August 2010. NOTE: This is something of a "catch-up" quiz, covering several problems from Ampere's law, toroidal coil, solenoid and Faraday's Law of Induction, so make sure you give it enough time to do.

• Powermat Wireless Charging System.
• REMINDERS: Monday 9 August 2010 is the last day to turn in a Topic 1 paper (unless you had a draft evaluated by Dr. Phil). And Friday 13 August 2010 is Exam 3.
• DON'T TRY THIS AT HOME: I have never heard of Coin Shrinking before, but it seems to me that they are getting extremely large induced currents flowing in these coins, something in the 750,000 A range, which probably heats them to be soft enough that the attraction between parallel current paths might cause the coins to contract. Quarters shrunk down to the size of dimes? I suspect that the reason it didn't work so well with a nickel, is that nickel is a pretty hard metal and has a higher melting point. So they probably didn't get a big enough current, so instead of shrinking the nickel, they just sort of wrinkled it. (grin).
• If you know your 5-digit PID number from Q1, then your Mid-Term Grades can be seen here.

### Week of August 9-13, 2010.

Monday 8/9: Practically speaking, you cannot have a purely inductive circuit with just a battery and L, you really have some resistance as well. Series RL Circuit, similar to Series RC Circuit, except that energy is stored in the magnetic field at the maximum current. UL = ½ L I ². RL Circuit for energizing the coil. Equation for current i(t) is similar in appearance to the equation for q(t) for a charging capacitor. Now we will de-energize the coil. NOTE: In class I used my usual brute force approach, instead of Serway's. The Kirchhoff Loop for the de-energining coil is -iR -L(di/dt) = 0. Since the current is decreasing, di/dt is negative and the induced emf from the coil becomes a voltage gain. Solution for i(t) the same form as the current i(t) for charging/discharging capacitor. Solution for the magnitude of the induced emf, script-EL = -L di/dt, is the same for both energizing and de-energizing circuit, much like the current i(t) = dq/qt gives the same solution for both charging and discharging RC circuit. RL Circuit, similar to RC Circuit, except that energy is stored in the magnetic field at the maximum current. UL = ½ L I ². Mutual Inductance between two inductors. 2nd coil responds only to changes in magnetic flux coming from 1st coil, which is based on the changes in the current i1 in the 1st coil. And vice versa. LC Oscillator circuit. Same 2nd order differential equation as the Simple Harmonic Oscillator (PHYS-2050), such as a mass on a spring. Solutions are sines and cosines. Energy is held constant for all t between the capacitor and the inductor. Can't really have a true LC oscillator, since normal wires and coils have a resistance which dissipates energy through Joule heating. LC Oscillator solution: q(t) = Q0 cos(omega t + phi), where omega = 1 / SQRT (LC) is the angular frequency and phi is a phase angle. Energy is held constant for all t between the capacitor and the inductor. U = UC + UL = q²/2C + ½Li² = Q²/2C = ½ L I ². Can't really have a true LC oscillator, since normal wires and coils have a resistance which dissipates energy through Joule heating. Comments ONLY about the RLC Damped Harmonic Oscillator. Mechanical analogue is the mass-on-a-spring with shock absorbers. LAST DAY to turn in Topic 1 Papers -- End of Grace Period (5pm). Second set of Sample Exam 3. (Click here for a copy.) Quiz 16 Take-Home on Inductors, LR and LC circuits, due Tuesday 10 August 2010.

• NOTE: What about the frequency f of an LC oscillator? (And f = 1 /T, where T is the Period.) Technically you should know this and it is in the book, but the angular frequency (rad/sec) omega = 2 pi f. And omega = 1/SQRT(LC). You need this for Q16.

Tuesday 8/10: A.C. Circuits. Voltage is a sine or cosine functions, as is the Current. Problem: Average voltage is ZERO. Need to define a new average, the Root-Mean-Square. It is the RMS Voltage and Current that are usually reported in A.C. circuits. Typical A.C. frequency in U.S. is 60 Hz. Need to specify what type of A.C. For sine wave, define RMS Voltage as 0.7071 Maximum Voltage. Similar for RMS Current. Why A.C. power? (1) Transformers allow voltage to be raised or lowered. D.C. voltage can only be lowered by the voltage drop of a resistor, or raised by adding power sources. The transformer consists of two coils connected magnetically (i.e., mutual induction) by an iron core (made of insulated plates to minimize heating from eddy currents!). V2 = V1N2/N1. (2) D.C. power lines have huge power losses due to Joule heating, very low efficiency. Actual Efficiency = Power Used ÷ Total Power Generated. Power lines run at higher voltages to minimize power losses due to Joule heating in the powerlines (P = I²R). The Equations for A.C. Circuits By Components: For resistive only circuits, can still use Ohm's Law, V = I R. Current and Voltage are both sine waves. Phasor diagrams -- taking the y-component of a rotating vector gives the sine function. Real A.C. circuits may have a Resistive nature, a Capacitive nature and an Inductive nature. For A.C. circuits with a Resistor only: I and V stay in phase with each other. For resistive only circuits, can still use Ohm's Law, V = I R. Current and Voltage are both sine waves. RL Circuits: I and V out of phase by -90°. (The current lags behind the voltage.) Inductive Reactance. Phasor diagrams -- taking the y-component of a rotating vector gives the sine function. RC Circuits: I and V out of phase by +90°. (The current leads ahead of the voltage.) Capacitive Reactance. Many A.C. circuits have features of all three components (R, L and C), so we have to deal with Impedance, Z. Phasor diagrams (see textbook for diagrams). NOTE that when we look at the RLC AC circuit, that we rotate our previous phasor diagrams, because "There can be only ONE current." Minimum impedance is when purely Resistive or when the two Reactances cancel each other -- the latter is frequency dependent. Can run into problems if expecting f=60Hz but get f=50Hz or f=25Hz. NOTE: Mentioned this as people were handing in Q16: Phase angle, phi, for resultant Vmax vector relative to the Imax vector, is phi = tan-1((XL-XC)/R). This is the same way we always find the angle for a vector given the x- and y- components. For impedance matching , where XL=XC, we get the same equation for the angular frequency omega = 1/SQRT(LC) as for the LC oscillator -- but what's important this time is that omega is BOTH the LC oscillator frequency and the AC frequency. In other words, when the impedance Z is minimized, the LC oscillator part of the circuit isn't fighting itself. This is why power companies have to worry about maintaining their frequency -- it affects the impedance of the circuits. For DC circuits, P = IV. For AC, it is a little more complicated. Paverage = Irms Vrms cos(phi) -- also Paverage = Irms² R -- because of the phase angle between V and I. (NOTE: Serway's derivation skips a couple of steps, and uses a couple of trig identies.) For a purely resistive circuit, or one which looks like a purely resistive circuit, phi = 0°, and so get Paverage = Irms Vrms . For impedance matching , where XL=XC, we get the same equation for the angular frequency omega0 = 1/SQRT(LC) as for the LC oscillator -- but what's important this time is that omega is BOTH the LC oscillator frequency and the AC frequency. In other words, when the impedance Z is minimized, the LC oscillator part of the circuit isn't fighting itself. As a result, with Z minimized, Irms is maximized. If R=0, then Irms would become infinite, but there is always some small resistance. Irms drops off as the AC frequency omega deviates plus or minus from omega0. (see Serway pp. 937-939) Quiz 17 Take-Home on AC circuits, Inductive & Capacitative Reactances, Impedance and Phasor Diagrams, due Thursday 12 August 2010.

• NOTE: Do not put off starting Q17. Though it looks short, there are a number of things to calculate. For the Phasor Diagram, draw it at a time other than t=0 as I did in class, so the angle theta isn't zero. That way you'll get non-zero y-components for all the vectors. See NOTE above about Phase Angle phi.