Updated: 16 December 2014 Tuesday.
FINAL COURSE GRADES AND BREAKDOWN BY CATEGORY FOR PHYS-1150 Fall 2014
Reminder that ICES Student Course Evaluations are available now online via GoWMU .
Monday 12/8: FINAL EXAM (2:45pm-4:45pm)
Tuesday 12/9: Office Hours.
Wednesday 12/10: Not on cammpus today. See Office Hours.
Thursday 12/11: Office Hours.
Friday 12/12: LAST DAY TO MAKE UP EXAMS.
Monday 12/15: Office Hours.
Tuesday 12/16: Grades due at Noon.
Monday 9/1: Labor Day <No Classes>
Tuesday 9/2: Class begins. Introduction to Dr. Phil. E&M is a triumph of 19th century Physics. But Physics is cumulative -- you won't believe how much from PHYS-1130 we will use this week!
Wednesday 9/3: 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. Real Electric Charges. Two charges: like charges repel, unlike (opposite) charges attract. 1 Coulomb of charge is an enormous amount of charge. "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.
Thursday 9/4: Real Electric Charges. Two charges: like charges repel, unlike (opposite) charges attract. 1 Coulomb of charge is an enormous amount of charge. 1 Coulomb of charge is an enormous amount of charge. Two 1.00 C charges separated by 1.00 meters have a force of one-billion Newtons acting on each other. The Hydrogen Atom: Gravity loses to Electric Force by a factor of 200 million dectillion (!!!). Q1 (Attendance -- Q1A coming if you weren't in class.)
Friday 9/5: 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 (!!!) Distribute syllabus. Q1½ in-class.
Monday 9/8: Finding the net vector electric force F_{E} for a system of point charges. Remember: In PHYS-1150, 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. Review of Vector Force problems (don't blink or you'll miss it). How does q_{1} know that q_{2} 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.)
From PHYS-1130 notes on vectors: Now we need to start dealing with two-dimensional problems in general. Two kinds of numbers: Scalars (magnitude and units) and Vectors (magnitude, units and direction). Adding and subtracting vectors: Graphical method. To generate an analytical method, we first need to look at some Trigonometry. Right Triangles: Sum of the interior angles of any triangle is 180°, Pythagorean Theorem (a² + b² = c²). Standard Angle (start at positive x-axis and go counterclockwise). Standard Form: 5.00m @ 30°. Practical Trigonometry. SOHCAHTOA. 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 calculatur.
Tuesday 9/9: Electric Fields: For point charges, F_{E} = kq_{1}q_{2}/r^{2}. E_{1}= kq_{1}/r^{2}. So F_{1on2} = q_{2}E_{1}. But it's more than that. FE = q_{E} is true for any net electric field E. Example: Four charges with q = 2.00 x 10-6 C are arranged in a square with the sides d = 10.0 cm = 0.100 m. Find the vector force on the charge at the lower right. The forces on the x- and y-axes have a magnitude of 3.595 N. The force on the diagonal is half that. The final force is down and to the right with a magnitude of (0.500 + sqrt(2)) (3.595 N) or (0.500 + 2cos45°) (3.595 N).
Wednesday 9/10: Return Q1½. Topic 1 assigned. (Updated Searchable booklist available online here .) Q2 in-class.
Thursday 9/11: 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. Note that far from the charges, these systems look like just a single point charge with a net charge of (1) 0, (2) +2q and (3) +q. Finite objects: Near distances, Far distances (looks like a point charge) and inbetween (requires PHYS-207 and calculus). Conductors (metals) versus non-conductors (insulators). Conduction electrons in metals -- free 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.
Friday 9/12: Charge distributions -- λ (lambda) (linear charge density, C/m), σ (sigma) (surface charge density, C/m²), ρ (rho) (volume charge density, C/m³). Two terms with unfortunately close sounding names: Electric Potential Energy (in Joules) and Electric Potential (in Volts).Electric Potential versus Electric Potential Energy. P.E. is minus the Work. Potential V is similar, but comes from the E-field not Force. More importantly the Potential V (really ΔV) is an observable quantity. SI units: Volts. Simplified equation V = E d. Example: Lighting. Strength of E-field is important, in part because in air there is a limit to how big E can get. At breakdown, the E-field is large enough to start ripping electrons from gas molecules and the positive ions go one way, the electrons go another, and then we get a "spark" and no longer are static. E_{max} in air is 3,000,000 N/C. Example: Lighting. Strength of E-field is important, in part because in air there is a limit to how big E can get. At breakdown, the E-field is large enough to start ripping electrons from gas molecules and the positive ions go one way, the electrons go another, and then we get a "spark" and no longer are static. E_{max} in air is 3,000,000 N/C = 3,000,000 Volts/m.
Monday 9/15: "Fishing" for lightning on a mountain top with rockets and a voltmeter. 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. Reminder: F_{E} and E are vectors, E.P.E. and V are scalars. E-Fields and Conductors (in Electrostatic Equilibrium). Charges accumulate on outside of conductors. (1) E-Field lines terminate normal (perpendicular) to the conductor's surface. (2) E = 0 inside the conductor. (3) Charges tend to accumulate more on pointy tips of things, which means more E-field lines and stronger E-fields. Easy to show for a uniformly charged sphere that (2) is true at center of the sphere, harder to show (2) is true everywhere inside the sphere. Conductor in equilibrium is an equipotential throughout. Equipotential lines, where V is constant, are always perpendicular to E-field lines. 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, σ = q / Area , is much higher.
Tuesday 9/16: Coulomb's constant k versus Permitivity of Free Space (ε_{0} epsilon-naught). Introduce the permeability of free space for magnetism (μ_{0} mu-naught), and show how these two fundamental constants are connected to the speed of light, c. Working toward Gauss's Law for Electricity. Analogy of a bag around a light bulb. All the light rays are contained within the bag, no matter the shape or distance or size. Electric Flux: Electric field times Area, modified by the angle. Simplest examples are E-field perpendicular to surface, max flux and parallel to surface, no flux. Gauss's Law for Electricity. Derive for point charge, turns out to be general equation. Φ_{E} = EA = q_{inside} / ε_{0} .Take advantage of geometry and symmetry. For conducting charged metal sphere of radius R, all the charge is on the outside. Two cases: (1) r < R, no q-inside Gaussian suface, no E-field and (2) r > R, q-inside = Q, so E-field same as a point charge. For insulated charged sphere of radius R, allow the charge to be distributed evenly throughout. Two cases: (1) r > R, same as metal sphere, same as point charge, and (2) r < R, E is zero at center and builds up linearly as a function of r. Note that at r = R, both cases give same vaŒue.
Wednesday 9/17: Q3 in-class.
Thursday 9/18: Gauss's Law for Electricity. Derive for point charge, turns out to be general equation. Take advantage of geometry and symmetry. For conducting charged metal sphere of radius R, all the charge is on the outside. Two cases: (1) r < R, no q-inside Gaussian suface, no E-field and (2) r > R, q-inside = Q, so E-field same as a point charge. For insulated charged sphere of radius R, allow the charge to be distributed evenly throughout. Two cases: (1) r > R, same as metal sphere, same as point charge, and (2) r < R, E is zero at center and builds up linearly as a function of r. Note that at r = R, both cases give same value. Two more Gauss's Law cases: A line of charge, λ = Q / L. E-field falls off as 1/r, not 1/r². A sheet of charge, σ = Q / A. E-field is constant. Infinite line or sheet of charge versus finite line or sheet (need to stay close to charge and near center, away from edges.) Moving from Field Theory to Applications leading to Devices. A battery is a device which stores energy in a chemical reaction. When connected to a circuit, postive charges flow from + terminal to - terminal (or negative charges flow from - terminal to + terminal). There is a potential difference or delta-V between the terminals of a battery. A real battery will run down if used too long or if one tries to work it too hard. We will start with "perfect" batteries which never run down.
Friday 9/19: A sheet of charge, σ = Q / A. E-field is constant. Infinite line or sheet of charge versus finite line or sheet (need to stay close to charge and near center, away from edges.) Moving from Field Theory to Applications leading to Devices. A battery is a device which stores energy in a chemical reaction. When connected to a circuit, postive charges flow from + terminal to - terminal (or negative charges flow from - terminal to + terminal). There is a potential difference or delta-V between the terminals of a battery. A real battery will run down if used too long or if one tries to work it too hard. We will start with "perfect" batteries which never run down. 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. Stories: Dr. Phil & the camera flash. US Navy seaman vs. the tank capacitor (Cap-2, Seaman-0). Use Gauss' Law for Electricity to find the E-field between the plates. Turns out to be twice the E-field from a single sheet of charge. (Important to know why.) 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). Circuit diagrams represent the elements of a circuit. So far: battery, wires, capacitor. Parallel Plate Capacitor. Example: Consider a parallel plate capacitor with plates 1.00 m x 1.00 m, separated by a gap d = 1.00 mm = 0.00100 m. The Parallel plate equation tells us C = 8805 pF = 8.85 x 10^{-9} C. The Capacitor equation allows us to the voltage, if 1.00 billion electrons are on the -Q plate, and a like positive charge on the +Q plate, giving a voltage of 0.01810 volts. This isn't very high. Q4 Take-Home quiz on Gauss' Law, now due Tuesday 23 September 2014.
Monday 9/22: 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). Circuit diagrams represent the elements of a circuit. So far: battery, wires, capacitor. Parallel Plate Capacitor. Two devices connected together in a circuit can only be connected two ways: series or parallel. In Parallel, same voltage, share charge. Equivalent capacitor is always larger. In Series, same charge, share voltage. Equivalent capacitor is always smaller. NOTE: Remember to take the last reciprocal!
Tuesday 9/23: Exam 1 Review. Two devices connected together in a circuit can only be connected two ways: series or parallel. In Parallel, same voltage, share charge. Equivalent capacitor is always larger. In Series, same charge, share voltage. Equivalent capacitor is always smaller. NOTE: Remember to take the last reciprocal!
Wednesday 9/24: Exam 1.
Thursday 9/25: Capacitor Network Reduction problem. Carefully analyze the network, reducing series or parallel capacitors to equivalent capacitors, redrawing the circuit each time. Use a table with rows for all four real capacitors and the equiavalent capacitor the battery sees, and three columns for the capacitor equation Q = C V. By going back through the intermediate diagrams, it is possible to know every value of every capacitor in the network. Add a new column for the energy in each capacitor: The Work to assemble charges on a capacitor = Energy stored in the capacitor. U_{c} = ½CV², so add a fourth column to our table 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!
Friday 9/26: Energy stored in the capacitor. U_{c} = 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! Making a real capacitor. What if not filled with air? Dielectrics -- an insulator where the +/- charge pairs are free to rotate, even if they do not move. Polarization. Dielectric constant (κ kappa) and Dielectric strength (E_{max}). Dielectrics -- an insulator where the +/- charge pairs are free to rotate, even if they do not move. Polarization. Dielectric constant (kappa) and Dielectric strength (E_{max}). Dielectic constant increases capacitance over air gap. Dielectric strength usually bigger than E_{max} 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. Using dielectric to change capacitance for detection: Studfinder (for carpentry), some computer keyboard keys, biometrics for security systems.
Monday 9/29: Electrostatics versus Electrodynamics. The difference between stationary charges in equilibrium, and charges in motion. Resistors and Resistance. Current defined: i = Δ Q / Δ t (SI units, Ampere = A). Microscopic view of what is going on inside the wire. Positive current with positive charge carriers is the same as negative charge carriers (like electrons) going the other way. Inside the wire, electrons move randomly anyway. There are so many, they don't need to go fast to create a current -- the "drift velocity" of electrons is very slow, << 1 m/s. (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. 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 δV. We can have an E-field and δV inside a conductor in a circuit because this is no longer an electrostatic equilibrium problem. The Simplest Circuit: Battery, wires, load (resistor). Ohm's Law: V=IR form. (Ohm's "3 Laws"). Joule Heating, Power Law: P = IV (also 3 forms).
Tuesday 9/30: The Simplest Circuit: Battery, wires, load (resistor). Ohm's Law: V=IR form. (Ohm's "3 Laws"). Joule Heating, Power Law: P = IV (also 3 forms). Electrocution -- it's not the voltage, it's the current and the resistance affects the current. 11mA = 0.011A is the danger value across the heart. Resistivity is a property of the material used in a resistor. Conductance and conductivity are the reciprocal of resistance and resistivity. Resistance by geometry. R = ρ (L / A), where ρ = rho = resistivity of the material (omega·m), L = length and A = cross-sectional area. 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 ohm meter is useless. Kammerleigh Onnes 1916 work on extending the R vs. T curve toward T = 0 Kelvin. Discovered Superconductivity, where R=0 identically. We usually treat the wires in a circuit as having R=0, but they usually are not superconductors. "High temperature" superconductors (liquid nitrogen temperature, not liquid helium).
Wednedsday 10/1: Quiz 5 in-class on Wednesday 1 October 2014, on
Capacitor Circuit Reduction and Dielectrics in Capacitors now a
Take-Home on Capacitor Circuit Reduction, due Friday 3 October 2014.
Thursday 10/2: Review capacitor series-parallel reduction for Q5... Kammerleigh Onnes 1916 work on extending the R vs. T curve toward T = 0 Kelvin. Discovered Superconductivity, where R=0 identically. We usually treat the wires in a circuit as having R=0, but they usually are not superconductors. "High temperature" superconductors (liquid nitrogen temperature, not liquid helium). The "Woodstock of Physics" in 1987. 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.
Friday 10/3: 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.) Discussion of types of real batteries, especially the rechargeables (Nicad, NIMH and Li-ion, plus lead-acid) and why things have changed. Real batteries consist of a "perfect" battery (Electromotive force = emf, also shown as script-E) in series with a small internal resistance, r. As chemical reaction in battery runs down, the internal resistance increases. The actual voltage from the battery is the emf voltage minus the loss from the internal resistance, V = emf - i r . 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.
Monday 10/6: Return X1. Real batteries consist of a "perfect" battery (Electromotive force = emf, also shown as script-E) in series with a small internal resistance, r. As chemical reaction in battery runs down, the internal resistance increases. The actual voltage from the battery is the emf voltage minus the loss from the internal resistance, V = emf - i r . 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. How to jump a car battery correctly and safely. (Improper jump can result in hydrogen explosions, boiling sulfuric acid, etc.)
Tuesday 10/7: 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, then (2) will give you (M-1) 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 . (As suspected, doing the work fast on the board, I probably made a mistake -- use the linked PDF solution.) Note that our assumption that i_{2} goes to the left was wrong, because the solution gives i_{2} as slightly negative. Minus signs mean "other direction". In this case, there is a small current going backwards through the smaller battery, V_{2}. Perhaps this is a charging circuit for a rechargeable battery?
Wednesday 10/8: Yesterday's 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 i_{2} goes to the left was wrong, because the solution gives i_{2} as slightly negative. Minus signs mean "other direction". In this case, there is a small current going backwards through the smaller battery, V_{2}. Perhaps this is a charging circuit for a rechargeable battery? A look at other Kirchhoff's Law problems -- symmetry sometimes means that one critical resistor sees zero current in some cicumstances. . It doesn't exist. Without it, the circuit may not be reducible by series and parallel means. Discuss 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. Measuring things has the potential to change that which we are measuring. In PHYS-1160 Lab, you are likely using digital multimeters. Ammeters measure current by connecting in series to the circuit. Voltmeters measure potential difference by connecting in parallel to the circuit. Because they are attached to the circuit, they change the circuit -- we must take this into account when designing an ammeter or a voltmeter. (The third most common meter is the ohmeter, which measures resistance using a 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. We will not talk further about designing ohmeters.)
Thursday 10/9: Q6 in-class. Q7 Take-Home on Kirchhoff's Laws, due Monday 20 October 2014.
Friday 10/10: Measuring things has the potential to change that which we are measuring. In PHYS-1160 Lab, you are likely using digital multimeters. 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 and the needle moves in response to a current through a tiny coil. Ammeter: a galvanometer and a very small shunt resistor in parallel, together connected in series with the circuit. Voltmeter: a galvanometer amd a very large resistor in series, together connected in parallel with the circuit. 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. Finally, we checked to see if, in putting a real ammeter and voltmeter in a circuit, whether the very act of measuring V and I changes their values. (If you have the numbers from class, you can show that for a resistor expected 5.00 A and 5.00 V that the changes in the readings with both meters present won't change the values in our example by more than 0.01%.) Next up -- the RC circuit.
Monday 10/13: RC series circuit. Circuit diagram for charging capacitor (ignore the calculus part). 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. The current, for both charging and discharging capacitor in an RC circuit, is maximum at time = 0 and goes to zero over time. The natural number e = 2.7182818284590452353602874713527... provides a way for us to calculate exponential curves using the e^{x} function of your calculators. To find t for any given charge q or current i level, you can use the natural log function, ln x^{}, to "cancel" the e^{x} function. The Magnetism part of Electricity & Magentism. "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: isolated North or South poles).
e^{0} | 1 | 1 - e^{0} | 0 |
e^{-1} | 0.3679 | 1 - e^{-1} | 0.6321 |
e^{-2} | 0.1353 | 1 - e^{-2} | 0.8647 |
e^{-3} | 0.0498 | 1 - e^{-3} | 0.9502 |
Tuesday 10/14: The Magnetism part of Electricity & Magentism. "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: isolated North or South poles). Rules similar to Electric Charges: Unlike poles attract, like poles repel. 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. Broken cow magnets show a radial crystalline structure due to the alignment of small magnetic domains -- what makes the magnetic field strong also makes the magnet physically weak in some directions, hence the breakage across the radially aligned grains. Dropping a permanent magnet can result in a reduction of its magnetic field as the shock allows some iron atoms to flip 180° and therefore cancel instead of add to an adjacent iron atom's magnetic field. 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. Example of the 4T NMR magnet at Michigan Tech and the 10-foot radius line on the floor and erasing ATM cards within that circle.
Wednesday 10/15: Q8 in-class.
Thursday 10/16: Although we know that there is a magnetic force, F_{B}, between two magnets, we're not really interested in that. Instead, we're going straight to magnetic force on a moving electric charge. Magnetic Force on a Moving Electric Charge - Right-Hand Rule & Uniform Circular Motion. 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 -- they form what is called a 3-tuple. 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... The Velocity Selector works with both positive and negative charges, but there must be a charge. A mass spectrometer -- a device that sorts particles by mass. 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 chemical means). Mass Spectrometer as Calutron -- detecting or separating isotopes, something that cannot be done by chemical means.
Friday 10/17: 1104 Rood Hall -- Demo Day. Light bulb boxes for PARALLEL RESISTORS (unscrew a bulb, remainder stay lit at same brightness), SERIES RESISTORS (unscrew a bulb and they all go out; put a 100W and a 15W bulb in series and the smaller bulb lights up okay, but little/no glow from bigger bulb -- remember that light bulb filaments heat up and are non-ohmic resistors, so the resistance changes over time), and RC CIRCUITS (start with three caps in series, connected in series with a light bulb. large (bright) initial current, fades over time as δV on plates builds up. 1 cap or 3 caps in parallel, takes longer -- RC constant is larger.). Broken cow magnets show a radial crystalline structure due to the alignment of small magnetic domains -- what makes the magnetic field strong also makes the magnet physically weak in some directions, hence the breakage across the radially aligned grains. Dropping a permanent magnet can result in a reduction of its magnetic field as the shock allows some iron atoms to flip 180° and therefore cancel instead of add to an adjacent iron atom's magnetic field. For the future -- Lenz Law race between cow magnets dropped through (a) plastic pipe and (b) non-magnetic aluminum pipe. And second race between (b) non-magnetic aluminum pipe and (c) non-magnetic copper 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. It is Lenz's Law that gives us the minus sign in Faraday's Law of Induction. Right-Hand Rule & Uniform Circular Motion. The origin of Auroras (aurora borealis = northern lights, aurora australis = southern lights): charged particles from the Sun end up following the Earth's B-field lines in helical (screwlike) paths towards the poles -- when these fast moving particles hit the upper atmosphere, they cause a glow.
Monday 10/20: 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 chemical means). Mass Spectrometer as Calutron -- detecting or separating isotopes, something that cannot be done by chemical means. In the Mass Spectrometer the distance d between two masses after going through a semi-circular path is equal to the difference in the diameters, not the difference in the radii. D = 2r , so... d = D_{2} - D_{1} = 2(r_{2} - r_{1}). Magnetic Force on a Current Carrying Wire. An electric current is a series of moving electric charges, after all.
Tuesday 10/21: Exam 2 Review.
Wednesday 10/22: Exam 2.
Thursday 10/23: : Magnetic Force on a Current Carrying Wire. An electric current is a series of moving electric charges, after all. Magnetic Torque on a Current Carrying Wire Loop. Torque = IAB or IABsin(θ), where A = area enclosed by the coil. Though derived for a rectangular coil with sides a and b, it turns out this is a general result. Left as is, this system is an osciallator -- the torque goes to zero after 90° and then points the other way and reaches a maximum at 180°. 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 electric motor. The magnetic constant -- the Permeability of Free Space, µ_{0}, is unusual in that we know the exact mathematical representation, which is why is given as 4π × 10^{-7} T·m/A. As we saw earlier, if we calculate 1/sqrt(ε_{0} × µ_{0}), we get the c = speed of light in vacuum -- again showing the fundamental connection between electricity and magnetism. Magnetic Field loops from a Current Carrying Wire. B = µ_{0} I / 2 π r , where r is the radius of the B-field loops. 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.
Friday 10/24: Return Q8. Magnetic Field loops from a Current Carrying Wire. B = µ_{0} I / 2 π r , where r is the radius of the B-field loops. 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. Magnetic field from a single loop of wire. B = (μ_{0})I / 2 R. Approximating a circular current by using a square of four straight wires. For N loops in the same space, B = N(μ_{0}) I / 2 R. 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 winding real coils with thin, varnish insulation. Gauss' Law for Magnetism is 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 along a B-field and the current(s) contained inside -- Ampere's Law.
Monday 10/27: Return X2. Gauss' Law for Magnetism is 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 along a B-field and the current(s) contained inside -- Ampere's Law: B L = (μ_{0}) I_{enclosed}. 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.
Tuesday 10/28: X2 slution. Gauss' Law for Magnetism is 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 along a B-field and the current(s) contained inside -- Ampere's Law: B L = (μ_{0}) I_{enclosed}. 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. In a Solenoid, the N loops are spread out over a length L, so n = N / L. (NOTE: Italics added to emphasize that the N-loop coil and the N-loop solenoid are different.) B-field of a Solenoid: constant and uniform B-field inside (away from edges) and zero outside the coil. Use Ampere's Law to find B-field inside a Solenoid: B = (μ_{0}) N I / L = (μ_{0}) n I. (NOTE: The BL Amperean paths 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 path are perpendicular, so the term is zero as well.) The Solenoid is the magnetic analog of the parallel plate capacitor for E-fields. 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 winding real coils with thin, varnish insulation.
Wednsday 10/29: Faraday's Law of Induction. A changing magnetic flux induces a current in a loop of wire. Think of the coil as wanting to maintain "the status quo". (The crankly old man who doesn't want anything to change. "You're adding flux! I don't want more flux!" "You've added flux, it's steady, I can live with that." "Hey! You're taking my flux away! Give me back my flux!") So if the magnetic flux is increasing, the induced current creates an induced magnetic field to try to cancel the flux. If the magnetic flux is decreasing, the induced current creates an induced magnetic field to try to bolster the flux. 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. It is Lenz's Law that gives us the minus sign in Faraday's Law of Induction. Demo: (We did the racing cow magnets the other week.) Lenz Law race between cow magnets dropped through (a) plastic pipe and (b) non-magnetic aluminum pipe. And second race between (b) non-magnetic aluminum pipe and (c) non-magnetic copper 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. It is Lenz's Law that gives us the minus sign in Faraday's Law of Induction. Demo: (If we had a real Physics lecture hall...)Magnet moving into a coil, causing current to flow through galvanometer. Turning coil in a constant magnetic field -- creates a generator (A.C. or D.C.). See notes for 10/10 on a primitive electric motor. Essentially electric motors and generators are the same devices. Dynamic or Regenerative Braking turns electric motors into generators -- it takes energy (work) to turn the generator and this energy is robbed from the vehicle's Kinetic Energy. Hybrid gas-electric cars use this as do most railroad trains (either electric or diesel-electric). Induction Cook Surfaces: 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.
Thursday 10/30: 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. It is Lenz's Law that gives us the minus sign in Faraday's Law of Induction. Induction Cook Surfaces: 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. Or we can prevent heating from Eddy Currents in a piece of metal by cutting slots (like a comb) or turning a slab of metal into a stack of thin sheets separated by insulation, so that we cannot get large circles of induced currents from changing magnetic flux. Mutual Inductance between 2 coils (1 & 2, or Primary & Secondary). With an AC power source, there is always a changing flux in the secondary coil from the changing current in the primary coil. The Inductor (L). (SI units = Henry = H) Self-Inductance. Back emf, back current. Opposing the status quo -- even of its own B-field! Why induction is a big deal in electronics, industrial motors and electrical power distribution. More practical uses for induction: (First, how regular fuses and circuit breakers work -- and why that isn't fast enough to prevent some types of accidents.) ... Q9 Take-Home on Magnetic Coils and Ampere's Law, due Monday 3 November 2014.
Friday 10/31: More practical uses for induction: (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. Hall Effect -- a device with no moving electrical parts -- proves that charge carriers in a current carrying wire are negative, not positive, by looking at a Hall current perpendicular to the wire when there is an applied B-field passing through the wire. "The 200 Year Hall Effect Keyboards", will last "forever", but made obsolete in two years when Windows 95 added three keys. Why A.C. power? Because we can make a transformer (mutual inductance) and easily raise or lower the voltage by using two coils connected magnetically by an iron core. V_{2} = V_{1}N_{2}/N_{1}. D.C. power lines have huge power losses due to Joule heating, very low efficiency. Efficiency = Power Used ÷ Total Power Generated. Power lines run at higher voltages to minimize power losses due to Joule heating in the powerlines.
Monday 11/3: Why A.C. circuits are different than D.C. 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 V_{RMS} = 0.7071 V_{Max}. Similar for RMS Current. Real AC circuits: For resistive only A.C. 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. For A.C. circuits with a Resistor only: I and V stay in phase with each other. RC Circuits: I and V out of phase by -90°. (We say the voltage lags the current.) RL Circuits: I and V out of phase by +90°. (We say the voltage leads the current.) Inductive Reactance, Capacitive Reactance.
Tuesday 11/4: Inductive Reactance, 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). Treat Resistive Voltage as one vector and Inductive Voltage minus Capacitive Voltage as a perpendicular vector. Whole structure rotates at an angular frequency ω = 2π f and we take the y-components at the angle θ = ω t to get the sine-function of the voltage. V_{max} and Phase angle (φ) just like finding the magnitude and standard angle from an x- and y-components of a vector. 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 or even 57Hz.Phase Angle φ (phi): Phasor diagrams (see textbook for diagrams). Treat Resistive Voltage as one vector and Inductive Voltage minus Capacitive Voltage as a perpendicular vector. Whole structure rotates and we take the y-components to get the sine-function of the voltage. V-max and Phase angle φ (phi), or impedance Z, just like finding the magnitude and standard angle from an x- and y-components of a vector. φ = tan^{-1} ( (V_{L}-V_{C}) / V_{R} ) = tan^{-1} ( (X_{L}-X_{C}) / R ).
Wednesday 11/5: Q10 In-Class/Open Quiz.
Thursday 11/6: (We won't be doing problems with today's material. It closes the loop, so to speak, informationwise.) The Series RL Circuit. Very much like the Series RC Circuit, but everything the opposite. RC: Charges stored on capacitor plates. The current stops when fully charged. Energy is stored in the electric field between the plates (PE = ½ C V²). RL: Current (moving charges) moving through coil. The current is at a maximum when fully energized. Energy is stored in the magnetic field inside the coil (PE = ½ L I²). Who knew that (Henrys) ÷ (ohms) = (seconds)? NOTE: Technically it is very hard to have a purely inductive circuit, that is just a battery and an inductor L, because the long thin wire used in the coil's windings generally has a resistance. Therefore real L circuits are really RL circuits. This is analagous to the capacitor, where there is a resistance in the wire connected to the plates of the capacitor, and therefore real C circuits are really RC circuits. In both cases this means it takes a finite, non-zero time for the device (L or C) to store energy in its respective field. LC oscillator. Comments ONLY about the RLC Damped Harmonic Oscillator. Mechanical analogue is the mass-on-a-spring with shock absorbers. Three cases: Underdamped, overdamped, critically damped. Need a tuned suspension with shock absorbers to drive a car safely on the road. The Driven Damped Harmonic Oscillator -- apply an AC signal and if close to the resonant frequency of the oscillator, becomes an amplifier. Changing from AC to DC or DC to AC: Old way using motor-generator sets. A diode is a device which only allows current to flow in one direction. A diode in AC only takes one side of the AC, get choppy DC (pulse power). A diode bridge of four diodes gets DC from both above and below. Brief discussion of semi-conductors. n-type semi-conductors are electron donors -- extra electrons free to move. p-type semi-conductors have "holes" which move around -- effectively positive charges. A current going from p to n at a p-n junction will go through, opposite current will not. The transistor -- a switch, an amplifiier -- with three types of material, p-n-p or n-p-n.
Friday 11/7: Maxwell's Equations and Hertz's radio wave LC oscillator -- the spark gap radio. AM (amplitude modulation) radio versus FM (frequency modulation) and digital radio. The Marconi wireless telegraph of the RMS Titanic, and the modern cellphone. The Electromagnetic Wave travels at the speed of light. c = 300,000,000 m/s = 186,000 miles/sec, in vacuum. The Great 19th Century Debate: Is Light a Particle or a Wave? (Wave-Particle Duality did not seem obvious at the time.) Something that is both a wave and a particle is a "wavicle". 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 HIGHER and wavelengths SHORTER than visible light (UV ultraviolet, X-rays, Gamma rays). UVA and UVB, sunglasses.
Monday 11/10: 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 HIGHER and wavelengths SHORTER than visible light (UV ultraviolet, X-rays, Gamma rays). UVA and UVB, sunglasses. Discussion of why Superman's X-ray vision cannot work: Normal human vision -- light either reflected off of objects or directly from light source. Few X-rays get reflected, so what is the source of the X-rays for Superman to see? (If he's projecting the X-rays, he's killing anyone he looks at.) Gamma rays and food irradiation. 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.) Frequencies LOWER and wavelengths LONGER than visible light (IR infrared, Microwave, Radio waves, ELF extremely low frequency). IR perceived as heat. Microwave ovens have metal screens in their windows -- the centimeter-range sized EM waves cannot see the "small" holes in the screen, so they bounce off the window as if it were just like the metal in the other five walls. Discussion of how microwave ovens "cook" food.
Tuesday 11/11: Optics: Geometric Optics (empirical) and Physical Optics (more wave and fieldlike). Ray Tracing: Rays from a spherical source become essentially parallel rays when you are far away. When a light ray reaches a new material, it can undergo (1) reflection, (2) absorption, (3) transmission. The Law of Reflection. Measure all angles from the normal line perpendicular to the surface. Rough surfaces -- scattering. The Optical Lever -- move a mirror by 10° and the reflected ray moves by 20°. (Dr. Phil's theory on the origin of "seven years of bad luck for breaking a mirror".) People tend to not like photographs of themselves, because they are used to seeing their mirror image -- their normal image, which the rest of us sees, looks "wrong". Corner and Corner Cube reflectors. 2 or 3 perpendicular mirros cause incoming rays to be reflected out in the same directio, except with an offset. The Law of Refraction - Snell's Law. Light bent at the interface between two media, because the speed of light changes in the media. (Analogy: If you are driving along the road and your right tires go off onto the soft shoulder, they can't go as fast and the car turns towards the shoulder until all four wheels are driving off the road.) Whether refaction bends light towards normal or away, depends on whether going from low to high index of refraction or high to low index of refraction.
Wednesday 11/12: Q11 in-class / Open Quiz. Topic 2 Worksheet (1 of 4) handed out, due Thursday 4 December 2012. (Click here for a copy.)
Thursday 11/13: The Law of Refraction - Snell's Law. Light bent at the interface between two media, because the speed of light changes in the media. (Analogy: If you are driving along the road and your right tires go off onto the soft shoulder, they can't go as fast and the car turns towards the shoulder until all four wheels are driving off the road.) Whether refaction bends light towards normal or away, depends on whether going from low to high index of refraction or high to low index of refraction. If going from an high index of refraction media to a lower index media, have a chance for Total Internal Reflection (T.I.R.). This is a "perfect" reflection, better than a mirror. Used in high-end optical systems instead of mirrors. Also useful in fiber optics cables. Dispersion -- in vacuum all speeds of light are the same, but in a medium, there are slighltly different n's for each wavelength. As one goes from a high index of refraction to a low index, increasing the angle of incidence, white light will start breaking into the rainbox spectrum of colors. Light going through a parallel-plano sheet of glass. Two interfaces: air-to-glass and glass-back-to-air. Coming down the normal, 0°, no deviation. At any other angle from the normal in air, the ray is refracted towards the normal in glass and then back out at the original angle when back in air. However, the light ray is offset from the line the ray would've followed had the glass not been there. Discussion of trying to see forward (large angle from normal) through the windows of a bus, train or plane. Very hard and very expensive to make true parallel-plano surfaces.
Friday 11/14: Return Q9. Ray tracing gets same results as doing Snell's Law on mulitple curved lens surfaces. Handy not to have to do all that refraction calculations! Real image formed by passing three rays through a positive thin lens (biconvex, converging lens). f = focal length, p = object distance, q = image distance, h = object height, h' = image height. Three cases: object distance p > 2f (real, inverted, reduced image), 2f < p < f (real, inverted, magnified image), p < f (virtual, upright, magnified image = magnifying glass) -- latter two not shown here. Zone of Confusion: No solution for p = f. Analytic formula for object and image. Magnification: M = h' / h = -q / p. Diopters are the reciprocal of focal length in meters. Note that some eyepieces to optical devices have some diopter built in, so that if you add an eyepiece to the eyepiece, the second unit may be marked for the total diopter, not the diopter of that particular lens. In multiple lenses the effective diopter is just the sum of the diopters. A flat piece of glass is 0 diopters.
Monday 11/17: Showing how two lenses can be combined, where a Real Image from Lens 1 becomes a Virtual Object for Lens 2 to create a Real Image from Lens 2. (You will NOT be asked to do this drawing.) Physical Optics. Based on wave properties of light. Constructive and Destructive Interference. Anti-Reflection coating for lenses. Step 1 - A thin coating of thickness t applied to a glass surface, means that light rays coming in from the air make two reflections (air to coating, and coating to glass). If the roundtrip distance of the 2nd reflection is out of phase with the 1st reflection (off by ½ wavelength) then the two reflections can cancel each other. Thickness = half the round-trip distance, yields a "quarter-wave coating". Step 2 - But the roundtrip distance is done in the coating, so need to find the wavelength in the coating, not the air or glass. Step 3 - When a light ray goes from a low index of refraction to a higher index of refraction, the reflection gains a ½ wavelength phase shift. If both reflections are the same kind (low to high, or high to low), then we still get the same quarter-wave anti-reflection solution. If both reflections are different, we get half-wave anti-reflection coatings. Back to Anti-Reflection and Max-Reflection coatings with 0, 1 or 2 half-wavelength shifts upon reflections. This closes the material for Exam 3. (Special relativity problems on some Sample Exam 3's will be saved for Final Exam.)
Tuesday 11/18: Classes canceled by winter storm.
Wednesday 11/19: Exam 3 Review. Anti-Reflection coating for lenses. Step 1 - A thin coating of thickness t applied to a glass surface, means that light rays coming in from the air make two reflections (air to coating, and coating to glass). If the roundtrip distance of the 2nd reflection is out of phase with the 1st reflection (off by ½ wavelength) then the two reflections can cancel each other. Thickness = half the round-trip distance, yields a "quarter-wave coating". Step 2 - But the roundtrip distance is done in the coating, so need to find the wavelength in the coating, not the air or glass. Step 3 - When a light ray goes from a low index of refraction to a higher index of refraction, the reflection gains a ½ wavelength phase shift. If both reflections are the same kind (low to high, or high to low), then we still get the same quarterwave anti-reflection solution. If both reflections are different, we get halfwave anti-reflection coatings. Back to Anti-Reflection and Max-Reflection coatings with 0, 1 or 2 half-wavelength shifts upon reflections. Example: Maximum Reflection coating for 650 nm light. Example: For what wavelength in air, is a 125nm coating anti-reflection or max reflection, depending on having 1 or 2 half-wavelength shifts?
Thursday 11/20: Exam 3.
Friday 11/21: Young's Double-Slit Interference. Light from two slit sources -- classically the light should travel in a straight line, get two "light shadows" or bright spots on the target wall. In wave theory, however, light from two coherent (in-phase) point sources will reach a point along the centerline, both traveling the same distance so get constructive interference and a bright spot. At some place offset from centerline, the two distances will vary by half a wavelength and get destructive interference and a dark spot. Alternating bright and dark spots as the two distances change. Single-Slit Diffraction. Sikmilarly, we can divide a single slit into an upper and lower half, and see interference between them. Double-Slit plus Single Slit. (Because those two slits separated by a distance d also have a width a.) Get the double-slit's alternating light and dark bands, attenuated by the wider envelope of the single slit pattern. Diffraction Limited. Resolution limitations due to the circular aperture diffraction pattern of the "hole" in a lens (pupil in the eye). A smaller opening makes picture look "sharper" for a while, due to increased "depth of field". Then diffraction limiting takes over -- far away cannot tell if a bright light is one source or two.
Monday 11/24: X3 Returned. 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). Start with: Special Relativity (speed constant). Einstein's postulates: (1) All observers see the same Physics laws. (2) All observers measure the speed of light in vacuum as c. Two equations: Beta, gamma β = v / c ; γ = 1 / sqrt (1 - β²) . β has no units and must be < 1. γ also has no units and must be 1 or greater. Problem: Find beta for a car traveling at 31.3 m/s (70 mph). If you use your calculator, you probably can't get gamma to be anything other than 1 exactly. Topic 2 Worksheets 2-4 handed out, due Thursday 4 December 2014. (Click here for a copy.) .
Tuesday 11/25: 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. One sees the proper length: a length measurement where both ends are measured at the same time. One sees the proper time: a time measurement where beginning and end are measured at the same place. Two observers cannot agree on the order of events, either. The concept of "simultaneity" is gone. Doppler Shifts with Light: If light source is moving toward an observer, they see shorter wavelengths and high frequencies -- blue shifted. If light source is moving away from an observer, they see longer wavelengths and lower frequencies -- red shifted. Distant objects in the universe all appear to be moving away from us, the most distant ones a considerable fraction of the speed of light. 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.
Wednesday 11/26: WMU Classes end at Noon -- Class does not meet.
Thursday 11/27: Thanksgiving Day. No classes.
Friday 11/28: No classes.
Monday 12/1: 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 γ = 1.10 -- and a beta = 0.417 c. Relativistic momentum, p_{rel} = γ mv, and relativistic Kinetic Energy, KE_{rel} = (γ - 1) mc². Total Energy, E_{total} = γ 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 = γm", 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.) The deBroglie wavelength -- λ = h / p -- Wave/Particle Duality for Matter. Planck's constant -- h = 6.626 × 10^{-34} J·s -- 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. Q12 take-home, on Special Relativity, due Wednesday 3 December 2014.
Tuesday 12/2: Matter and the Atom. All models of the atom are fundamentally wrong at some level. It's the nature of Quantum Mechanics, which operates at the land of the very small. But we've already used the "planetary" model of the atom back when we looked at the electron in orbit of a hydrogen atom, using U.C.M. and Coulomb's Law. Chemistry suggests a finite number of elements or starting blocks for all materials. Everything made of individual atoms, as opposed to say the Velveeta cheese model, where you can forever slice the Velveeta ever finer. Plum pudding model of the atom -- some hard bits in a matrix. Rutherford experiment -- firing particles at a very thin piece of gold foil (discussion of gold leaf), some of the particles rebounded at nearly 180°. "It was as if I had fired cannonballs at a piece of paper and some bounced back towards me." Better model of the atom: electrons on the outside (size of the atom is about 1 angstrom = 1 ×10^{-10} meters) and protons (and neutrons, not yet discovered) concentrated into a nucleus (about 1 femtometer = 1 ×10^{-15} meters). The Bohr atom is really quite a triumph of the Physics of PHYS-1130 and PHYS-1150: We showed early in the semester that the gravitational attraction between an electron and a proton doesn't matter in the hydrogen atom, so if we have an electron in a circular orbit about the proton (or better yet, the nucleus with a total proton charge Q = +Z e, so we can do elements other than hydrogen), then the Coulomb Force provides the centripetal force for Uniform Circular Motion (UCM). That allows us to find the speed v as a function of the radius r. The notion in Quantum Mechanics that values for things like electrons can only have certain values is foreign to Classical Physics. Indeed, in Classical Physics, Planck's constant h = 0. But think of our lecture hall -- you can only sit in an unoccupied seat. Each seat exists in a row with only so many seats, each row is at a particular height above the lecture floor and a particular distance (radius) from the lecture table. So in our lecture hall, each of you exists in a unique quantized state. And it takes more work and energy to climb out of the lowest level (ground state) up to the top of the lecture hall and walk out the door, freed from 1110 Rood. (grin) Which leads us to... Niels Bohr postulated that the electron could only exist in certain orbits, so he proposed that the angular momentum (mvr) can only have integer values of h-bar. This introduces the principle quantum number n. This also means that the circumference of the orbital is some integer of the electron's deBroglie wavelength -- and we have a circular standing wave only for those orbits which are allowed. The deBroglie wavelength -- λ = h / p -- Wave/Particle Duality for Matter. Planck's constant -- h = 6.626 × 10^{-34} J·s -- 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.
Niels Bohr postulated that the electron could only exist in certain orbits, so he proposed that the angular momentum (mvr) can only have integer values of h-bar. This introduces the principle quantum number n. This also means that the circumference of the orbital is some integer of the electron's deBroglie wavelength -- and we have a circular standing wave only for those orbits which are allowed. By the time we get the radius equation, there are only two variables (and both are integers!): the quantum state number n and the proton/element number Z. All the other items are fundamental constants, which when multiplied together give us "a_{0}", the radius of the n=1 innermost state of the hydrogen atom, equal to 0.528 ×10^{-10} meters. Twice that is the diameter, so the size of the hydrogen atom is about an angstrom, as advertised. Note that the radius goes as n², so the orbitals get big quite fast. Since this is UCM, knowing the radius means we know the speed v. And that allows us to calculate the classical Kinetic Energy, KE = ½mv². (It turns out that the speeds remain below 0.42 c for all n and Z's for about half the periodic table, so we don't have to deal with Relativity.) The total energy of each state, En = KE + PE, and in this problem I shall state that the PE = -2KE. So En = -KE and it ends up being Z²/n² times more constants. The ground-state or n=1 energy for hydrogen is -2.18 ×10^{-18} J or -13.6 eV. Now for an electron to move from one orbit to another, it must gain or lose energy. Going from a higher n to a lower n, the difference in the energy is release as a photon with E = hf. To go from a lower n to a higher n, the electron has to absorb a photon of E=hf. And now we have an explanation of the spectral lines which we had once described as "fingerprints for elements". Burn hydrogen and the light emitted, when run through a prism will split not into a rainbow, but individual lines of individual colors -- these are emission lines. Take white sunlight, shine it through a prism and look at the rainbow of colors under a microscope and you will see that individual lines of color are missing -- these are absoption lines caused by the hydrogen gas in the Sun's atmosphere removing those colors and moving their electrons to higher orbits or ionizing completely. Atomic spectra were known for decades before the Bohr atom, so they were trying to solve the puzzle backwards -- what they didn't know was that any transition involving the n=1 innermost orbit in hydrogen gave photons in the UV and so the scientists couldn't see them at the time. If we try to solve the helium atom (Z=2) in a similar way, we find that with one nucleus and two electrons, we have a three-body problem and we can't solve that in closed form. However, we can use our Bohr equations for hydrogenic ions (hydrogen-like) which have only one electron, so we can solve for He^{+}, Li^{+2}, Be^{+3}, B^{+4}, C^{+5}, ... , U^{+91}, etc. (Click here for the Periodic Table with derivation of the radius and energy equations on the back.)
Wednesday 12/3: Q13 - THE LAST QUIZ - In-Class/Open Quiz.
Thursday 12/4: IMPORTANT: In class today you filled out a new sheet with a PID Personal Identification Number -- this is so I can post grading info on the website. If you weren't in class, you need to email me a 5-digit PID number. Doesn't matter if you remember the one you filled out in September. A similar form is online for "Quiz 1A" here. Turn in on Friday or email me the PID number. Heisenberg Uncertainty Principle: (1) Δp_{x} Δx greater-than-or-equal h / 4 π (h-bar / 2) and (2) ΔE Δt greater-than-or-equal h / 4 π (h-bar / 2). α, β and γ. Alpha particles are ^{4}He nuclei. Heavy, but can be stopped by paper shielding. Beta particles are electrons or their anti-matter cousins, positrons. More pentetrating, but can be stopped by foil shielding. Gamma rays are very hard, particle like photons of high energy. They require lead shielding.Alpha, Beta, Gamma decays. Gamma rays are photons, which have neither a charge nor a mass, so don't change the element and isotope. Alpha particles are the nucleus of ^{4}He, so ejecting an alpha changes both Z and N. Beta decay has three processes, all of which change the element number: (1) Beta minus is an electron, (2) Beta plus is a positron -- a positive electron, the antimatter form of an electron, and (3) Electron capture (e.c.), in which an innermost electron strays into the nucleus -- equivalent to a beta plus decay. The Neutrino -- "little neutral one". Postualted by Enrico Fermi, because the trajectories of some decays seemed to violate conservation of momentum. So the neutrino has no charge, a tiny mass, travels at nearly the speed of light and carries a bit of momentum and energy. Solar neutrinos and whether the core of the Sun is still working.
Friday 12/5: REVIEW.