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Lectures in PHYS-2070 (17)

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CURRENT MID-TERM GRADES FOR PHYS-2070 CAN BE FOUND HERE

Updated: 13 November 2015 Friday.

Week of 9-13 November 2015.

Monday 11/9: 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.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.

Tuesday 11/10: 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. 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. (Step up or step down.) 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).

Wednesday 11/11: Homework: Repeat the power loss and efficiency work we did Tuesday, but since we're using AC, we can change the voltage at a transformer just outside to a higher voltage. This changes the current I, the power loss due to Joule heating in the powerlines (P = I²R) and the efficiency (Useful Power / Total Power). Remember that the Useful power we are planning to deliver is 50,000 W, or 466.7A at 120 Vrms. Change the voltage to 1200 volts, 12,000 volts, 120,000 volts and 1,200,000 volts. Q10 in-class.

Thursday 11/12: and Friday 11/13: Dr. Phil sick -- substitutes. Available material includes:

Why A.C. power? (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). For resistive only circuits, can still use Ohm's Law, V = I R. Current and Voltage are both sine waves. 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. 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. Phase angle, phi, for resultant Vmax vector relative to the Imax vector, is phi = tan-1((XL-XC)/R). 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 .

Quiz 11 Take-Home quiz, handed out on Friday 13 November 2015 and due on Tuesday 17 November 2015, on AC circuit variables. (Click here for a copy. fixed 11-15-15)

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Week of 7-11 September 2015.

Monday 9/7: Labor Day <No Classes>

Tuesday 9/8: Class begins. Intro to Dr Phil. Static electricity. Amber = elektros in Greek. 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. 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. Intro Handout. (Click here for a copy if you didn't make it to the first class.)

Wednesday 9/9: Real Electric Charges. Two charges: like charges repel, unlike (opposite) charges attract. A Nickel coin has a mass of 5 grams, so about 1/10th of a mole. "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. 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. 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 (The repulsive force between protons in Helium is about 231N !!!).

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

Thursday 9/10: Distribute Syllabus. (This is the corrected version.) 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. We need the Strong Nuclear Force to overcome the Coulombic replusion between protons in the nucleus. Protons repel protons, neutrons don't really stick to neutrons, but protons do stick to neutrons. Z = # of protons, determines the element. N = # of neutrons. A = Z + N. Isotopes: Nuclei with same Z, but different N. Hydrogen-1, chemical symbol H (Z=1, N=0, A=1). Hydrogen -2, Deuterium, chemical symbol D (Z=1, N=1, A=2). Hydrogen-3, Tritium, chemical symbol T, radioactive (Z=1, N=2, A=3). Helium-3, chemical symbol He, stable (Z=2, N=1, A=3). Helium-4, chemical symbol He, extremely stable (Z=2, N=2, A=4). There are about 118 elements (Z), but tens of thousands of nuclides (Z,N). Quiz 1 was an in-class Attendance and Check-In form. Q1A coming if you weren't in class.

Friday 9/11: Topic 1 assigned. (Updated Searchable booklist available online NOT UP YET -- use last years for now .) Q1½ In-Class worksheet -- "No Stress Quiz".

Week of 14-18 September 2015.

Monday 9/14: 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. Example: vector-Fx = -3.16 N i-hat, vector-Fy = +4.60 N j-hat. vector-F = 5.581 N @ 124.5°. Example: q = q1 = q2 = q3 = 2.56 × 10-6 C. d = 2.00 cm = 0.0200 m. vector-F2 = vector-F1on2 + vector-F3on2 = 0. But vector-F3 does not equal zero. And vector-F2 does not equal zero if q3 = -q.

Tuesday 9/15: Find the electric force acting on q2 for a system of three charges. 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)"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 (a) consider large charged systems which are not just a few sums of points and (b) examine the environment without needing another charge.

Wednesday 9/16: More examples of symmetric 2-D problems for FE and E-fields. 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.

Thursday 9/17: 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). Q2 in-class.

Friday 9/18: 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. 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).

Week of 21-25 September 2015.

Monday 9/21: 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. Note: Coulomb constant k is not the fundamental constant for electricity -- epsilon-naught is. Coulomb's constant k versus Permitivity of Free Space (epsilon-naught).

Tuesday 9/22: 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.

Wednesday 9/23: Return Q2. Using Gauss' Law for Insulating Sphere. The solutions for r>R (outside) and r<R (inside), must match at r=R. And they do. What happens if the volume charge density rho is not constant? If rho only depends on r, not theta or phi, then when one finds the qinside, one has to integrate rho dV, where rho=rho(r). Q3 in-class.

Thursday 9/24: Gauss' Law for Thick-Walled Insulated Sphere of Charge. A thick-walled sphere is the volume of the sphere minus the hole in the middle. Similar for thick-walled cylinders. 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.

Friday 9/25: Return Q3. Note: It is annoying that electric potential, V, and its SI unit, V=volts, are both "V". Be careful between units and variable -- Dr. Phil sometimes spells out the unit as volts. (The textbooks can get away with this by putting one of the two in italics, but it is hard to write italics in your notes or on the board.) Simplified equation V = E d. (But remember that it's really delta-V = - E d .) Example: (1) Can we get sparks from the two slots of an electrical outlet at 125V? (2) Lightning. 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. Q4 is a Take-Home on Gauss's Law on a Cylinder, due on Monday 28 September 2015 at the beginning of class (or by 4pm). (Click here for a copy.)

Week of 28 September-2 October 2015.

Monday 9/28: What is an electron volt (eV)? (1) It is literally the charge e times 1 volt -- for things with a charge q=±e, then qV=(e)(1 volt)=1.602 × 10-19 J. (2) Given the relationship between UE and V, UE = qV. The Work Energy Theorem says W = delta-K and delta-K = - delta-UE , so W = delta-K = -qV. So electron volts are an easy way to figure out the accelerating potential V if we express the change in kinetic energy in electron volts and not Joules. Note: can only use K = ½mv² up to about v = c/2, before we need relativistic equations. Problem: A spherical insulator with a non-constant volume charge density, rho = C r². Have to integrate to find total Q of the sphere for r > R. And integrate to find the qinside for r < R. More about charge accumulating 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.

Tuesday 9/29: Last thoughts on the Electric Potential V -- it really is a scalar, not a vector. Problems with point charges may be zero for E and not for V and vice versa. Also, can integrate easily over part of the perimeter of a circle, since it just adds as a scalar. And we can recover the components of E: Ex = - partial derivative with respect to x of V. Next Unit: Moving from Field Theory to Applications leading to Devices. Start of Capacitors and Capacitance. The Capacitor stores charge +Q on one plate and -Q on second plate, stores energy in the E-field between the plates. This is different from a battery, which has energy stored in its chemical reaction. Stories: Dr. Phil & the camera flash. US Navy seaman vs. the tank capacitor (Cap-2, Seaman-0). Capacitor Equation. SI unit for Capacitance is the Farad. 1F is a large capacitor. Usually deal with µF (microfarad = 1/1,000,000th of a Farad) and pF (picofarad = 1/1,000,000,000,000th of a Farad).

Wednesday 9/30: Exam 1.

Thursday 10/1: Capacitors and Capacitance. The Capacitor stores charge +Q on one plate and -Q on second plate, stores energy in the E-field between the plates. This is different from a battery, which has energy stored in its chemical reaction. Stories: Dr. Phil & the camera flash. US Navy seaman vs. the tank capacitor (Cap-2, Seaman-0). Capacitor Equation. SI unit for Capacitance is the Farad. 1F is a large capacitor. Usually deal with µF (microfarad = 1/1,000,000th of a Farad) and pF (picofarad = 1/1,000,000,000,000th of a Farad). Apply Gauss' Law for Electricity to the constant E-field of the Parallel Plate Capacitor. We now have an "operational equation", true for all capacitors, and a "by geometry" equation for the special case of the parallel plate capacitor. Example: A parallel plate capacitor has plates 1.00 m x 1.00 m and a gap of 10.0 cm. C = 88.5 pF. Such a big capacitor with such a tiny capacitance. Need either more plate (increase A) or narrow gap (decrease d) to increase C. Work to assemble charges on a capacitor = Energy stored in the capacitor = U = ½CV² . Basic circuit diagrams -- elements include wires, battery, capacitor, and possibly a switch.

Friday 10/2: 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, UE=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!

Week of 5-9 October 2015.

Monday 10/5: 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. 791 ***) 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.

Tuesday 10/6: Examples of the uses of capacitors and dielectrics. Computer keyboards with switches which have "no moving parts". Capacitative stylus for tablets. Biometric security devices. 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.

Wednesday 10/7: Q5 in-class.

Thursday 10/8: 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. 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.

Friday 10/9: Return X1. 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 R1 sees the largest current and dissipates the largest amount of energy per second (Power in Watts). This means it is also the most vulnerable. (Story of radio "repair" call from 4,000,000,000 miles.)

Week of 12-16 October 2015.

Monday 10/12: Ohmic versus non-ohmic materials: If R=constant over operating range, then we say the material is "ohmic". If R is not constant, it is "non-ohmic". Example: Because of the temperature dependence of R, the filament of an incandescent light bulb has a very different R when lit or dark. Therefore measuring the resistance of a light bulb with an ohmeter is useless. The ohmeter measures resistance using a small reference voltage. The ohmeter does not get hooked up to the live circuit -- if you want to find resistance in situ, measure V and I, then use Ohm's Law. This always works. Discuss incandescent (glowing filament) light bulbs and modern Christmas tree lights. Bulbs in series, but each bulb is in parallel with a small resistor, to keep whole string lit when one bulb burns out. Older sets were either parallel (full 120 volts) or series (one burned out bulb takes out the whole set). Newest sets use LED lights, much less current, much longer estimated lifetimes. 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.

Tuesday 10/13: How to jump a car battery correctly and safely. (Improper jump can result in hydrogen explosions, boiling sulfuric acid, etc.) 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 with two batteries and 3 resistors had 3 equations in 3 unknowns. Solution by brute force algebra here. Note that our assumption that i2 goes to the left was wrong, because the solution gives i2 as slightly negative. Minus signs mean "other direction". In this case, there is a small current going backwards through the smaller battery, V2. Perhaps this is a charging circuit for a rechargeable battery?

Wednesday 10/14: Q6 in-class.

Thursday 10/15: 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). Example: A case with five resistors, where some arrangements can be reduced by series/parallel reduction, others only by Kirchhoff's laws. The special case with symmetry and R1=R2 and R3=R4. Then the voltage on either side of R5 is the same and the voltage different is zero -- that means no current through R5 and it's as if R5 doesn't exist. With the troublesome resistor gone, you can reduce by series/parallel reduction. Note that getting "rid" of a resistor means using an eraser -- replace with a plain wire introduces a new circuit element and is NOT the same. 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.

Friday 10/16: 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. Solve q(t) charging or discharging equation for t, using the natural log (ln) function to get rid of the exponential. Applications for slowing down the charging/discharging of a capacitor. Backup power (short duration) -- not quite the same as a battery. Next week: Building an ammeter and a voltmeter -- does the very act of taking a measurement, change the thing you are trying to measure? Q7 Take-Home on Kirchhoff's Laws, due Monday 26 October 2015. (Click here for a copy)

Week of 19-23 October 2015.

Monday 10/19: 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. 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.

Tuesday 10/20: Return Q5, Q6. HW problem still pending... can't talk about effect of measurement on a circuit... 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. "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). B-field = magnetic field. B-field of a bar magnet resembles the E-field of an electric dipole (+q and -q separated by a distance).

          the data point

Wednesday 10/21: "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. SI Units for B-field: (1 Tesla = 1 T). "Other" unit for B-field: ( 1 Gauss = 1 G ; 1 T = 10,000 G ). Earth's B-field is about 1 Gauss at the Earth's surface. 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? Q8 in-class.

Thursday 10/22: 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...

Friday 10/23: 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. 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. Technically current is not a vector, despite the fact we talk of direction of vector-J = current density = current/cross-sectional area -- the vector related to current. Since we want to use the current I, we use the displacement vector L for the direction.

Week of 26-30 October 2015.

Monday 10/26: 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. Q8 Take-Home due.

Tuesday 10/27: 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. Review for X2.

Wednesday 10/28: Exam 2.

Thursday 10/29: 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.

Friday 10/30: Magnetic Force on a Current Carrying Wire. Demo -- hey it works and even in the right direction! 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.)

Week of 2-6 November 2015.

Monday 11/2: 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 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.) 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. The B-fields are constant and parallel to the surface and perpendicular to the current elements. (Similar to the E-field of an infinite sheet of charge.) 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. 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.

Tuesday 11/3: 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: Magnet moving into a coil, causing current to flow through galvanometer. 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. Ford test electric vehicle with inductive charger -- no exposed metal contacts, everything covered in smooth plastic. Dr. Phil has been advocating for 69 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 and some new non-Apple cellphones. (2) Safety: (First, how regular fuses and circuit breakers work -- and why that isn't fast enough to prevent some types of accidents.)

Wednesday 11/4: (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. Q9 in-class.

Thursday 11/5: Return X2. 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. 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.

Friday 11/6: 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. Practically speaking, you cannot have a purely inductive circuit with just a battery and L, you really have some resistance as well. L = self-inductance, SI units (Henry) = (H). Use Self-Inductance, Faraday's Law and the solenoid equation to generate the by-geometry equation for an air-filled solenoid. 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) for energizing the coil, 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 the de-energizing coil, i(t) is the same form as the current i(t) for 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. 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.