Dr. Phil's Home
Updated: 04 May 2010 Tuesday.
Your Pre-Final Grades can be seen here.
IMPORTANT: If you looked at your current course grades before 1:30pm on Monday, there was a formula error in the Quiz grade calculation -- it has now been fixed. Apologies all around. -- Dr. Phil
Monday 4/26: Office Hours.
Tuesday 4/27: Office Hours. FINAL EXAM 2:45-4:45pm (2 hours) [PHYS-2070(15) 2pm]
Wednesday 4/28: Office Hours.
Thursday 4/29: Office Hours. FINAL EXAM 12:30-2:30pm (2 hours) [PHYS-2070(14) Noon]
Friday 4/30: Office Hours.
Monday 5/3: Office Hours.
Tuesday 5/4: Grades due at Noon.
Monday 1/11: Class begins. Introduction to Dr. Phil. Distribute Syllabus.
Tuesday 1/12: 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. The Greeks. Moby Dick by Herman Melville -- gold coin, lightning and the reversal of the ship's compass needle. Static electricity. The Two-Fluid Model of Electricity. Franklin's One-Fluid Model of Electricity. Occaam's Razor. The simple hydrogen atom -- whatever charge is, the charge on the electron (-e) and the proton (+e) exactly cancel. The Electric Force between two point charges, Coulomb's Law looks like Newton's Law of Universal Gravity.
Wednesday 1/13: 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 nine-billion Newtons acting on each other. "Action at a distance" -- Gravity and the Electric Force are not contact forces. The Electric Force between two point charges, Coulomb's Law looks like Newton's Law of Universal Gravity. Four Fundamental Forces in Nature: Gravity, E & M, Weak Nuclear Force, Strong Nuclear Force. The Hydrogen Atom: Gravity loses to Electric Force by a factor of 200 million dectillion (!!!). The Helium Atom: Putting more than one proton in the nucleus produces enormous forces on the tiny protons -- Need the Neutron and the Strong Nuclear Force (!!!).
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 1/14: Demo these Class Web Pages, discuss Formulas Cards and Four pages of Topic 1 assignment handed out. (Webpage here -- Full 28-page Handout as PDF File -- Searchable HTML Page ). 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.
Friday 1/15: Review of vectors and vector forces. Review of vector notation for components and Standard Form. Right Triangles and Adding and subtracting vectors: Analytical method. (Check to make sure your calculator is set for Degrees mode. Try cos 45° = sin 45° = 0.7071) Why arctangent is a stupid function on your calculator. Solving a vector Electric Force problem when there isn't symmetry to render the problem zero. The importance of "Showing All Work". Using checks to confirm you're on the right track as you solve a problem. Q1 was an in-class Attendance and Check-In form given on Friday 15 January 2010. If you missed class, then you should download quiz Q1A here, fill it out and turn it in to Dr. Phil for 8000 points. (You won't get the attendance points for the class you missed.) Q2 is a Take-Home quiz, due on Tuesday 19 January 2010, in class or by 5pm.
Monday 1/18: Dr. Martin Luther King, Jr. Memorial Observance -- No Classes.
Tuesday 1/19: Non-point charge systems -- Putting a charge on an extended object requires us to know something about the material, in addition to the dimensions and geometry of the extended object. Conductors (metals) versus non-conductors (insulators). Conduction electrons in metals -- free to move around. Insulator electrons very hard to move around. Semi-Conductors sit in the middle. Sometimes they conduct and sometimes they don't. This means they act like a switch or valve, and this is the basis for the entire electronics semi-conductor industry. Charging a conductor by induction. How does q1 know that q2 is there? -- "Action at a Distance" -- Gravity and the Electric Force are not contact forces. The mathematical construct of the Electric Field. E is not an observable quantity. (Side example: Methods of measuring speed v, do not directly measure speed v.) Electric Field is a vector. FE = q E. For a point charge, E = k q1 /r2. Q3 is a Take-Home quiz handed out on Tuesday 19 January 2010, and due on Friday 22 January 2010, in class or by 5pm.
Wednesday 1/20: Electric Field is a vector. FE = q E. For a point charge, E = k q1 /r2. SI units for E-field: (N/C). 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. Emax in air is 3,000,000 N/C. E-field lines radiate away from a positive point charge; converge towards a negative point charge. If the universe is charge neutral, can have all E-field lines from + charges terminating on - charges.Why use E-fields, when you need the force F = q E anyway? Because it allows us to examine the environment without needing another charge. E-field lines allow us to qualitatively sketch what happens when two charges are near to each other. (1) +q and -q, (2) +q and +q. Very close to each point charge, the E-field lines are radial outward, evenly spaced. In the system, the E-field lines interact with each other -- but E-field lines can never cross. Long range, the system of point charges looks like a single net charge. The density of E-field lines in an area gives you an indication of the strength of the E-field line. The numbers of E-field lines attached to a point charge is proportional to the charge. E-field lines radiate away from positive charges and terminate on negative charges. (3) +2q and -q. First Sample Exam 1 handed out. (Click here and here and here for a copy.)
Thursday 1/21: So far we've looked at Electric Forces and Fields from discrete charges. Now we will look at extended continuous and uniform charges. Direct integration of Electric Force and Electric Field are similar, so we'll just go over direct integration of the E-field. Charge distributions -- lamda (linear charge density, C/m), sigma (surface charge density, C/m²), rho (volume charge density, C/m³). Note the similarity to mass distributions from PHYS-2050. Examples: Rod in-line with line from point P (1-dimensional integration). Rod perpendicular to line from point P. Note that in all these cases, we can predict the long range behavior (E-field behaves as a single point net charge), and anticipate the close-in short range behavior. Check Serway's examples (that's your textbook) -- watch out that his notation may be different.
Friday 1/22: 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). Q4 is a Take-Home quiz handed out on Friday 22 January 2010, and due on Tuesday 26 January 2010, in class or by 5pm.
Monday 1/25: 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.
Tuesday 1/26: 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. Second Sample Exam 1 handed out. (Click here for a copy.)
Wednesday 1/27: Gauss' Law for Electricity.
Using Gauss' Law for Point Charge, Conducting
Sphere, Insulating Sphere, Infinite Line of Charge.
Gauss' Law for Infinite Sheet of Charge. Q5 is
a Take-Home quiz handed out Wednesday 27 January 2010, and
Friday 29 January 2010 now due on Monday 1
February 2010, in class or by 5pm.
Thursday 1/28: 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.
Friday 1/29: Handy chart of the four quantities: FE (vector, 2 charges), E (vector, 1 charge), UE (scalar, 2 charges), V (scalar, 1 charge) .Simplified equation V = E d. (But remember that it's really delta-V = - E d .) Example: Lightning. Equipotential surfaces -- lines of constant Electric Potential (voltage). Analogy: Topographic maps, the equipotential lines are like the altitude contour lines. A skier's line of maximum descent down a mountain corresponds to the E-field lines. Conductor in equilibrium is an equipotential throughout. In electrostatic equilibrium, E = 0 inside a charged conductor, but V = constant, not V = 0 automatically. Why charge accumulates on the tips of "pointy things". Model a conducting blob with a blunt end and a pointy end, sort of like a piece of candy corn, by a large conducting sphere and a smaller conducting sphere, connected together by a wire so they are all equipotentials, i.e. V = constant. For a charged sphere, same as a point charge: V = kq/r. While the charge on the tip is less than the charge on the rest, the surface charge density, sigma = q / Area, is much higher. NOTE: The book is effectively closed for Exam 1 topics now, except for finding V by direct integration, which you might want to look at in the textbook before Monday's class. Remember, V really is a scalar quantity and not a vector.
Monday 2/1: Finding V by direct integration. Direct integration of V for a circular line of charge -- V really is a scalar, not a vector. In electrostatic equilibrium, E = 0 inside a charged conductor, but V = constant, not V = 0 automatically. Exam 1 Review.
Tuesday 2/2: Exam 1.
Wednesday 2/3: Two things about the electric potential, V: (1) Sketch of equipotential lines and perpendicular E-field lines for the irregular pointy conductor with charge Q. Conductor in equilibrium is an equipotential throughout. Equipotential lines, where V is constant, are always perpendicular to E-field lines. (2) Direct integration of V for a whole and a half of a circular line of charge -- V really is a scalar, not a vector. New 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). Circuit diagrams represent the elements of a circuit. So far: battery, wires, capacitor.
Thursday 2/4: 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. Work to assemble charges on a capacitor = Energy stored in the capacitor = U = ½CV² . Energy density, uE = U/vol. = ½(epsilon-naught)E² . While this was derived for the parallel plate case, it turns out to be true in general. Two devices connected together in a circuit can only be connected two ways: series or parallel. In Series, same charge, share voltage. Equivalent capacitor is always smaller. NOTE: Remember to take the last reciprocal! In Parallel, same voltage, share charge. Equivalent capacitor is always larger. Q6 is a Take-Home quiz due on Monday 8 February 2010, in class or by 5pm. Q7 is a Take-Home quiz due on Tuesday 9 February 2010, in class or by 5pm.
Friday 2/5: Capacitor Network Reduction problem. Carefully analyze the network, reducing series or parallel capacitors to equivalent capacitors, redrawing the circuit each time. Use table with columns for Q = C V. By going back through the intermediate diagrams, it is possible to know every value of every capacitor in the network. Extend the example in class with a fourth column, U=½CV², and find the energy stored in the equivalent capacitor and the sum of the energy stored in all four of the real capacitors -- if they agree, then our analysis and calculations are correct -- the battery cannot tell the difference!
Monday 2/8: 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). 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.
Tuesday 2/9: Examples of the uses of capacitors and dielectrics. Capacitive studfinder, uses edge effects of E-field from a capacitor to "see" the dielectric material behind the wall. Computer keyboards with switches which have "no moving parts". Electrostatics (equilibrium) to Electrodynamics (moving charges). Current defined: i = delta-Q/delta-t = dq/dt. 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.
Wednesday 2/10: Dr. Phil's PHYS-2070 classes at Noon and 2pm are canceled today.
Thursday 2/11: Joule Heating, Power Law: P = IV (also 3 forms). Resistance by geometry. R = rho (L / A), where rho = resistivity of the material, L = length and A = cross-sectional area. Continuing with Simple Circuits... Series and Parallel Resistors: Two devices connected together in a circuit can only be connected two ways: series or parallel. In Series, same current, share voltage. Equivalent resistance is always larger. In Parallel, same voltage, share current. Equivalent resistance is always smaller. Resistor Network Reduction. (Similar rules to Capacitor Network Reduction except "opposite".) For example given in class, Resistor 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.) Q8 is a Take-Home quiz handed out Thursday 11 February 2010, and due Tuesday 16 February 2010, in class or by 5pm.
Friday 2/12: Go over Q7 solution. "Catch-up Lecture". (1) 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. (2) 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. (3) What if R = 0? Finish discussion of high temperature superconductors. "High temperature" superconductors (liquid nitrogen temperature, not liquid helium). The "Woodstock of Physics" in 1987. (4) Discuss power cords -- flexible but hot cords for hair dryers, why power cords get recalled. (5) Real batteries consist of a "perfect" battery (Electromotive force = emf) in series with a small internal resistance, r. As chemical reaction in battery runs down, the internal resistance increases. 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. First Sample Exam 2. (Click here and here for a copy.)
Monday 2/15: PRESIDENT'S DAY (Not a WMU holiday). Return X1. Real batteries consist of a "perfect" battery (Electromotive force = emf) in series with a small internal resistance, r. As chemical reaction in battery runs down, the internal resistance increases. 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).
Tuesday 2/16: Not all circuits can be reduced by serial and parallel network analysis. Kirchhoff's Laws: (1) The sum of all currents in and out of any junction must be zero. (2) The sum of all voltage gains and voltage drops about any closed loop is zero. Practically speaking, if there are N junctions, then (1) will give you (N-1) unique equations, and if there are M loops that can be made in the circuit by going around the perimeter of each "puzzle piece", then (2) will give you sufficient unique equations. You will get the same number of equations as you unknown currents through the resistors. NOTE: EE students and those who have had ECT-2100 (?) may know a "better" way to solve Kirchhoff's problems. But the brute force algebra approach has the advantage of being based on the Physics, so has instructional value. Example in class had 3 equations in 3 unknowns. (Solution by brute force algebra here). Q9 in-class quiz on real batteries and internal resistance.
Wednesday 2/17: Other examples of systems which require Kirchhoff Laws. Sometimes a resistor has zero current, in which case it does not contribute to the circuit. Proper procedure for jump starting a car. (And why doing it wrong ranges from dangerous to deadly. Improper jump can result in hydrogen explosions, boiling sulfuric acid, etc.) 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. Q10 Take-Home quiz, due Tuesday 23 February 2010, in class or by 5pm.
Thursday 2/18: Note that you can use Kirchhoff's law even when you CAN reduce a circuit by series and parallel network reduction. For example, our Q8 problem has three unknown currents (1 junction equation, 2 loop equations). 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. Second set of Sample Exam 2's handed out: (Click here and here for copies.)
Friday 2/19: 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. Q11 is a Take-Home quiz handed out Friday 19 February 2010, and due Tuesday 23 February 2010, in class or by 5pm. NOTE: I'd do Q11 first before having all that algebra fun of Q10, to get Q11 out of the way.
Monday 2/22: "Magnetism is just like Electricity, only different." Most people are familiar with (1) magnets sticking to some metals, not others such as stainless steel and (2) if you have two magnets, they may attract or repel. North and south are analogous to plus and minus charges. Real Magnets are dipoles (North and South ends, linked). Break a magnet in half, and you either get two new magnets -- or nothing. So far, there is no evidence that there are Magnetic Monopoles (magnetic charges: qM , isolated North or South poles). Rules similar to Electric Charges: Unlike poles attract, like poles repel. The horizontal compass needle rotates until its North end points North (or rather to the North Magnetic Pole, which is of course a South pole of the Earth's magnetic core); the vertical compass rotates so that it lines up with the B-field along the surface of the Earth at the point. At the Equator, the vertical magnetic should be parallel to the ground, at the magnetic poles, it should be perpendicular to the ground. Is the Earth's magnetic field going to flip some day? And what about Mars? Magnetic Force on a Moving Electric Charge - The Cross Product and Right-Hand Rule (R.H.R.). The Cross Product (or Vector Product) is the exact opposite of the Dot Product (or Scalar Product). Multiplying two vectors together by a cross product gives us another vector (instead of a scalar). And the cross product is not commutative, vector-A × vector-B = - (vector-B × vector-A), so the order is paramount. Using Right Hand Rule to assign directions to x,y,z coordinates. Constant speed, perpendicular constant magnetic force --> Uniform Circular Motion. Cyclotron frequency -- no dependence on the radius (constant angular velocity).
Tuesday 2/23: 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... 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.
Wednesday 2/24: Hand back Q2, Q3, Q4, Q6. Review for Exam 2.
Thursday 2/25: Exam 2.
Friday 2/26: Spirit Day - No Classes
WMU SPRING BREAK
Monday 3/8: A current carrying wire consists of moving electric charges, and so therefore would see a magnetic force from a magnetic field. Discussion of microscopic theory of charges in a conductor. Drift velocity is the very slow net movement of the electrons moving randomly in the wire. Magnetic Force on a Current Carrying Wire. Demo -- hey it works and even in the right direction! Technically current is not a vector, despite the fact we talk of direction of current. J = current density = current/cross-sectional area is the vector related to current. NOTE: J-vector = sigma × E-vector (current density = conductivity × E-field) is the vector version of Ohm's Law. This was NOT on Exam 2. So we use the displacement vector L for the direction. For a Closed Loop, the net Magnetic Force from a constant B-field is zero. Magnetic Torque on a Current Carrying Wire. We use the enclosed area vector A, whose direction is defined by using the Mode 2 R.H.R. (fingers curled around the direction of the current loop, thumb is the area vector A perpendicular to the plane of the loop). Left as is, this system is an oscillator -- the torque goes to zero after 90° and then points the other way. But if we can reverse the direction of the current after the torque goes to zero, then the rotation can continue -- and we have a primitive DC electric motor.
Tuesday 3/9: Return X2. 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).
Wednesday 3/10: Gauss' Law for Magnetism. Not as useful as Gauss' Law for Electricity, because it is always zero (no magnetic monopoles). The Biot-Savart Law. B-field from a infinitely long straight current carrying wire by direct integration. (Serway has a similar example, but rather than do the integral in x, he does this theta substitution which Dr. Phil does not think is straight forward.) Circular loop of current carrying wire by integration for P at the center of the loop. (Serway's example allows for P to be on a line perpendicular to the loop.) B-field for a circular current carrying wire at the center -- or any part of a circle. Note that a wire coming in along the r-hat direction makes no contribution to the B-field. Q12 Take-Home, due Friday 12 March 2010, in class or by 5pm. (Click here for a copy.)
Thursday 3/11: Returned Formula Cards. Magnetic Field loops from a Current Carrying Wire. RHR has "two modes". Mode 1 uses three mutually perpendicular directions for when you have three vectors (A × B = C is 1-2-3, x-y-z). Mode 2 uses the curling of the fingers to represent the circulation of a field or the motion of a current, etc., with the thumb representing the relevent vector or direction. Magnetic Force between Two Current Carrying Wires. Combining problems, we find that for two parallel current carrying wires, with the currents in the same direction, the magnetic field from wire 1 creates an attractive magnetic force on wire 2. And the magnetic field from wire 2 creates an attractive magnetic force on wire 1. (Two forces, equal and opposite, acting on each other -- this is exactly as it should be with Newton's 3rd Law.) Anti-parallel currents (wires parallel, but currents in opposite directions) repel. Crossed currents (wires perpendicular to each other) see no magnetic force on each other. Operational defnition of the ampere and the coulomb: If the Force per length for two wires with a current I separated by 1 meter is F/L = 2 × 10-7 N/m, then I = 1 A exactly. Then in 1 second, 1 C of charge is moved by this 1 A current. Gauss' Law for Magnetism. Not as useful as Gauss' Law for Electricity, because it is always zero (no magnetic monopoles). However, there is something we can use in a similar way which involves involving a path integral along a B-field and the current(s) contained inside -- Ampere's Law. Use in a way similar to the way we used Gauss' Law for Electricity. Use symmetry and geometry to select your Amperean Loop to your advantage.
Friday 3/12: 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.) Coils can have right-hand or left-hand turns -- but it is the direction that the current wraps around the coil which determines which way the B-field points. 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! Q13 Take-Home, due Tuesday 16 March 2010, in class or by 5pm. (Click here for a copy.) NOTE: Part (c) may take you a little bit of time, so don't put this off to the last minute!
Monday 3/15: Edge effects: E-field of parallel plate capacitor vs. B-field of solenoid. More Comments about making a real velocity selector -- trying to stuff a capacitor for the E-field and a solenoid for the B-field in the same space! Comments about making real coils. Insulating varnish, heat damage. Yields affect time and money. B-field of an infinite sheet of current. Note that Ampere's Law has a flaw, which we will correct at a later date. If a current carrying wire can create a magnetic field, can a magnetic field passing through a coil create an electrical current? Demos: Cow magnets -- powerful cylindrical, rounded end magnets which get dropped into a cow's first stomach, to collect nails, bits of barbed wire, etc. from continuing on to the cow's other stomachs. Demo: Lenz Law race between cow magnets dropped through (a) plastic pipe, (b) non-magnetic aluminum pipe and (c) non-magnetic copper pipe. Something is going on, such that the magnets travel much slower through the metal pipes -- and the thicker copper pipe was much slower than the thinner aluminum pipe.
Tuesday 3/16: Demo: Magnet moving into a coil, causing current to flow through galvanometer. Faraday's Law of Induction. A changing magnetic flux induces a current, induces an e.m.f., in the circuit, substituting for the battery as the power source. Lenz's Law "of maintaining the status quo." The coil acts as if it opposes any change of the magnetic flux inside, by inducing a magnetic field to cancel and increasing flux or maintain a decreasing flux. To create this induced magnetic field, one needs an induced current, which is powered by an induced emf. It is Lenz's Law that gives us the minus sign in Faraday's Law of Induction. Recall yesterday's Lenz Law race between cow magnets dropped through (a) plastic pipe, (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. Turn a coil in a magnetic field and the flux changes, thereby inducing a B-field, emf and current. Has same 180° problem that a DC motor has. Hand-crank generators. Electric generators and electric motors differ in which way the arrow points toward or away from mechanical energy. Regenerative braking (also called dynamic braking) -- diesel-electric and electric trains, also hybrid cars like the Toyota Prius -- turn electric motors into generators. Energy either wasted as heat through a resistor, or recharge battery.
Wednesday 3/17: Demo: "Jumping Rings", making the bulb light, by Eddy Currents and Induction. Note that the metal rings get HOT, because there is a large induced current (plus eddy currents which we'll talk about tomorrow) and metal has a low resistance. Adding metal increases the mass, but provides more current loops and therefore more induced B-fields repelling the solenoid's B-field. A split metal ring (a) does not get hot and (b) does not jump, because there is no circuit enclosing the changing magnetic flux. Ford test electric vehicle with inductive charger -- no exposed metal contacts, everything covered in smooth plastic. Dr. Phil has been advocating for 54 semesters that small charging bricks for all our electronic devices be replaced with universal charging-by-induction systems. Beginning to see first practical systems such as Powermat. Demo: Place bundle of iron rods in an AC coil and light bulb dims -- analogous to lights dimming when large electric motors driving vacuum cleaners and compressors for refrigerators, air conditioning and dehumidifiers. The "back emf" from very large coils and large industrial motors can cause problems starting and stopping -- in particularly you might be able to spark across the gap of an on/off switch if the back emf is too high when you turn off an industrial motor, thus completing the circuit again so the motor continues running. Discussion of electrical issues in two scenes of Steven Spielberg's movie Jurassic Park. Q14 is a Take-Home quiz, handed out Wednesday 17 March 2010 and due Friday 19 March 2010, in class or by 5pm. (Click here for a copy.) NOTE: This is something of a "catch-up" quiz, covering several problems from Ampere's law, toroidal coil, solenoid and Faraday's Law of Induction, so make sure you give it enough time to do.
Thursday 3/18: Practical uses for induction: (1) Heating: The Good -- Heating the bottom of a metal cooking pot by induction: New type of cooking range uses sealed induction heating elements instead of exposed hot resistors or open gas fed flames -- usable for metal pans only. The Bad -- A slab of metal used to conduct a B-field can waste energy as heat if the B-field is changing, such as in an AC circuit. By making the slab out of thin plates insulated from each other, the B-field still can go around the metal, but the perpendicular loops of Eddy Currents can only have a diameter equal to the thickness of the metal. Small eddy currents cannot generate much heat because induced emf is too small. (2) Safety: (First, how regular fuses and circuit breakers work -- and why that isn't fast enough to prevent some types of accidents.) Ground Fault Interupt -- if the current doesn't return via the return wire, because it has found another conductive path, then the 2 wires (hot and return) total a net-enclosed-current for Ampere's Law, generating a B-field in a metal ring, detected by an induction coil wrapped around the ring and this sets off the relay which breaks the circuit. An Inductor is a coil in a circuit. Why an Inductor has Self-Inductance -- running a current through a coil creates a magnetic field and therefore changes the magnetic flux in the coil. The inductor has to respond to that change. Inductance can be a big deal. Even our Simplest Circuit (a resistor hooked up to a battery) forms a loop, and the loop must respond to the circuit being turned on. The Inductor (L). (SI units = Henry = H) Self-Inductance. Back emf, back current. Opposing the status quo. Equations for Inductance. Comments about large scale inductance in the power grid, and small scale inductance problems inside cell phones, etc. First Set Sample Exan 3 (Click here for a copy.)
Friday 3/19: Practically speaking, you cannot have a purely inductive circuit with just a battery and L, you really have some resistance as well. Series RL Circuit, similar to Series RC Circuit, except that energy is stored in the magnetic field at the maximum current. UL = ½ L I ². RL Circuit for energizing the coil. Equation for current i(t) is similar in appearance to the equation for q(t) for a charging capacitor. Now we will de-energize the coil. NOTE: In class I used my usual brute force approach, instead of Serway's. The Kirchhoff Loop for the de-energining coil is -iR -L(di/dt) = 0. Since the current is decreasing, di/dt is negative and the induced emf from the coil becomes a voltage gain. Solution for i(t) the same form as the current i(t) for charging/discharging capacitor. Solution for the magnitude of the induced emf, script-EL = -L di/dt, is the same for both energizing and de-energizing circuit, much like the current i(t) = dq/qt gives the same solution for both charging and discharging RC circuit. Q15 is a Take-Home quiz, handed out Friday 19 March 2010 and due Tuesday 23 March 2010, in class or by 5pm. NOTE: As of Friday, we have not done the LC oscillator for the last part.
Monday 3/22: RL Circuit, similar to RC Circuit, except that energy is stored in the magnetic field at the maximum current. UL = ½ L I ². Mutual Inductance between two inductors. 2nd coil responds only to changes in magnetic flux coming from 1st coil, which is based on the changes in the current i1 in the 1st coil. And vice versa. LC Oscillator circuit. Same 2nd order differential equation as the Simple Harmonic Oscillator (PHYS-2050), such as a mass on a spring. Solutions are sines and cosines. Energy is held constant for all t between the capacitor and the inductor. Can't really have a true LC oscillator, since normal wires and coils have a resistance which dissipates energy through Joule heating. LC Oscillator solution: q(t) = Q0 cos(omega t + phi), where omega = 1 / SQRT (LC) is the angular frequency and phi is a phase angle. Energy is held constant for all t between the capacitor and the inductor. U = UC + UL = q²/2C + ½Li² = Q²/2C = ½ L I ².
Tuesday 3/23: 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. Why A.C. power? (1) Transformers allow voltage to be raised or lowered. D.C. voltage can only be lowered by the voltage drop of a resistor, or raised by adding power sources. The transformer consists of two coils connected magnetically (i.e., mutual induction) by an iron core (made of insulated plates to minimize heating from eddy currents!). V2 = V1N2/N1. (2) D.C. power lines have huge power losses due to Joule heating, very low efficiency. Actual Efficiency = Power Used ÷ Total Power Generated. Power lines run at higher voltages to minimize power losses due to Joule heating in the powerlines (P = I²R).
Wednesday 3/24: 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. Phasor diagrams -- taking the y-component of a rotating
vector gives the sine function. Real A.C. circuits may have a Resistive nature,
a Capacitive nature and an Inductive nature. For A.C. circuits with a Resistor
only: I and V stay in phase with each other. RL Circuits: I and V out of phase
by 90°. Second set of Sample Exam 3. (Click
here for a copy.) Q16 Take-Home quiz, due
Friday 26 March 2010 Tuesday 30 March
2010, in class or by 5pm.
Thursday 3/25: 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).
Friday 3/26: 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 .
Monday 3/29: For impedance matching , where XL=XC, we get the same equation for the angular frequency omega0 = 1/SQRT(LC) as for the LC oscillator -- but what's important this time is that omega is BOTH the LC oscillator frequency and the AC frequency. In other words, when the impedance Z is minimized, the LC oscillator part of the circuit isn't fighting itself. As a result, with Z minimized, Irms is maximized. If R=0, then Irms would become infinite, but there is always some small resistance. Irms drops off as the AC frequency omega deviates plus or minus from omega0. (see Serway pp. 937-939) 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 (bridge rectifier) gets DC from both above and below. In either case, you can start to smooth the pulses by putting a capacitor in parallel with the load resister R, so when the current i(t) = 0, the capacitor discharges to prop up the DC current. Serway also mentions high- and low-pass filters. The problem with Ampere's Law -- it doesn't work properly in the gap between the plates of a capacitor while it is charging. So James Clerk Maxwell fixed it with a "displacement current" term, involving the time derivitive of the Electric flux in the gap. Ampere-Maxwell Law. Maxwell's Equations in integral form. Note that Maxwell didn't invent the four equations, only half of one, but he figured out what todo with them. E & M Waves. In vacuum (free space): a traveling set of perpendicular E-fields and B-fields, as sine waves constantly changing in space and time, moving with wave speed c (the speed of light in vacuum). With the Ampere-Maxwell Law, we effectively have closed the book on Exam 3 material. Some of the Sample Exam 3s may contain questions on Maxwell's Laws, but we'll save those for the Final Exam this semester.
Tuesday 3/30: The Lorenz Force Law combines the vector forms of FE = qE and FB = q v × B, to find the total force on a charge q in an E-field and a B-field. Maxwell's Equations and Hertz's radio wave LC oscillator -- the spark gap radio. The Marconi wireless telegraph of the RMS Titanic, and the modern cellphone. AM (amplitude modulation) radio versus FM (frequency modulation) and digital radio. For traveling waves in general, a wave is function of both space and time, x and t. The repeat time is T, the Period. Light as a wave. The frequency is f = 1/T. The repeat length is lamda, the wavelength. For all traveling waves, v = frequency × wavelength. For light v = c = 2.998 × 108 m/s (in vacuum).
Wednesday 3/31: Hand back Q12, Q13, Q14. Maxwell's solution gives an electromagnetic wave. The Great 19th Century Debate: Is Light a Particle or a Wave? (Wave-Particle Duality did not seem obvious at the time, so scientists stubbornly stuck to one or the other theory without realizing there was a compromise that embraced ALL the experimental evidence.) Review for Exam 3.
April 4/1: April Fool's Day (Not a WMU holiday). Exam 3.
April 4/2: Good Friday (Not a WMU holiday). Working with 3rd and 4th of Maxwell's Equations to generate partial differential equations of E(x,t) and B(x,t). (see pp. 958-959 in Serway) Looking at the solution to Traveling E-M Wave, with v in x-direction, E in y-direction and B in z-direction. Angular frequency omega, wave number k. c = Emax / Bmax. Derivation of c = E / B. Similar to the relationship between the E-field and the B-field in the velocity selector, where v = E / B. Poynting Vector, S. Rate of energy flow over an area -- SI units (W/m²). Traveling E-M Wave, Poynting Vector.
Monday 4/5: Poynting Vector, S. Traveling E-M Wave, Poynting Vector and Intensity. Energy stored equally in E- and B-fields of the E-M wave. Momentum and Pressure of light waves absorbed or reflected on contact. (Complete absorption like totally inelastic collision; complete reflection like totally elastic collision). Discussion of solar energy -- Serway calculates 160,000 W available on the roof of a house, but we only need about 10,000 W. Even accounting for angles, clouds and night, we don't need 100% capture to significantly reduce power from external sources. Light pressure and momentum transfer, despite the fact that light as mass = zero. NASA used solar panels as solar sails on Mariner 10 near the planet Mercury. Q17 is a Take-Home, handed out Monday 5 April 2010 and due later in the week, in class or by 5pm.
Tuesday 4/6: Light as a particle. The energy of a single photon ("particle" of light) is E = h f, where h = 6.626 × 10 -34 J·s is Planck's constant, a fundamental constant involved in Modern Physics. (If there was only Classical Physics, then h = 0.) Pure colors (EM waves, ROYGBIV for visible light) vs. Perceived colors (RGB, CMY, pinks, browns, "white" light). "Normal" human vision, some types of color blindness. The Electromagnetic Spectrum. Visible light (ROYGBIV=red orange yellow green blue indigo violet). Visible light is 400nm to 750nm (4000 angstroms to 7500 angstroms). Cannot "see" atoms with visable light, because the atom is about 1 angstrom across (1.00E-10 meters). The visible light wave is too large to see something that small. Frequencies LOWER and wavelengths LONGER than visible light (IR infrared, Microwave, Radio waves, ELF extremely low frequency). Frequencies HIGHER and wavelengths SHORTER than visible light (UV ultraviolet, X-rays, Gamma rays). IR perceived as heat. Night vision devices -- visible light amplification vs. IR. Discussion of how microwave ovens "cook" food.
Wednesday 4/7: The Electromagnetic Spectrum. Visible light (ROYGBIV=red orange yellow green blue indigo violet). Visible light is 400nm to 750nm (4000 angstroms to 7500 angstroms). Cannot "see" atoms with visable light, because the atom is about 1 angstrom across (1.00E-10 meters). The visible light wave is too large to see something that small. Frequencies LOWER and wavelengths LONGER than visible light (IR infrared, Microwave, Radio waves, ELF extremely low frequency). 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. This is similar to why we 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. Radio includes what we consider consumer radio (AM and FM), television (analog and digital), WiFi, Bluetooth, cellular/wireless services, etc. 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. Q18 Take-Home, due Friday 9 April 2010, in class or by 5pm.
Thursday 4/8: X3 returned. (Exam 3 curve here) Note that we have run out of textbook, so we will finish the course touching on a number of interesting topics in optics, relativity and modern physics, which may be useful either because this is your last Physics course or because you might get interested enough to continue in the 3rd semester course, PHYS-3090. Optics: Geometric Optics (ray tracing, light as stream of particles), Physical Optics (wave nature of light).
Friday 4/9: Light rays from a point source radiate outward, much like E-field lines from a point charge. From a large distance away, the light rays look parallel. 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".) 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.) First set of Sample Final Exams handed out. (Click here and here for a copy.)
Monday 4/12: Q16 returned. The Law of Reflection. 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". The Law of Refraction - Snell's Law. As light travels through an interface, if there is a change in the index of refraction, the light will be bent at the interface. If going from an high index of refraction media to a lower index media ONLY, 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. This is due to dispersion, a change in the speed of light for each wavelength. Q19 Take-Home, due Wednesday 14 April 2010, in class or by 5pm.
Tuesday 4/13: Corner (2 perpendicular mirrors) and Corner Cube (3 perpendicular mirrors) reflectors. Corner cube reflectors were placed on the Moon and used to measure the distance to within a cm (working now to get it down the mm). 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. This offset d is based on the thickness t of the glass and the angles of the refracted ray, so the offset increases as the angle in air increases. 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. Thin Lenses. Simplest lens surfaces are spherical (convex = bows out, concave = bows in) and flat (plano). So some lenses might appear to be biconvex, plano-convex, biconcave, convex-concave. A biconvex lens is also called a positive or converging lens. Parallel light rays coming into such a lens will all pass through (converge on) the focal point, a distance f from the center of the lens. By itself, could use as a magnifying lens. Concentrating sunlight: burning paper or popping ants? A biconcave lens is also called a negative or diverging lens. Parallel light rays will diverge after passing through the less, coming together at the near focal point, not the far focal point -- you can only see the bright spot by looking through the lens. 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!
Wednesday 4/14: 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. 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 the two reflections are still in phase with each other and we still get the same quarterwave anti-reflection solution. If both reflections are different (one is low to high and one is high to low), then the two reflections begin with a ½ wavelength phase shift, and we get halfwave anti-reflection coatings. Use the wrong quarterwave or halfwave coating yields a solution with maximum reflections. Back to Anti-Reflection and Max-Reflection coatings with 0, 1 or 2 half-wavelength shifts upon reflections. Q20 Take-Home, due Friday 16 April 2010, in class or by 5pm.
Thursday 4/15: Tax Day (Not a WMU holiday). The focal length f of a lens depends on the index of refraction of the material. But we said that the index of refraction is slightly different for different wavelengths, i.e. the speed of light in the material varies slightly. This is called dispersion. That means there is a slightly different f for each wavelength, which means that the different colors of visible light will focus to slightly different places. For long lenses with large pieces of glass in front, this can become a problem taking color pictures. We get around this by using more exotic materials such as fluorite crystals (calcium fluorite) or ED (extra low dispersion) glass which has less dispersion than more ordinary, cheaper optical grade glass. Modern Physics -- goes to size/time/length scales far outside our normal experience. Classical Relativity (two observers, two frames of reference), Special Relativity (speed constant), General Relativity (accelerations or gravity). Einstein's postulates: (1) All observers see the same Physics laws. (2) All observers measure the speed of light in vacuum as c. Beta, gamma, Length Contraction and Time Dilation. Alpha Centauri is 4.20 LY from Earth (proper length). Those on a starship see a different distance and experience a different time than the observer left on the Earth. But both think the other observer is moving at v < c. No preferred observer in Special Relativity. Two observers cannot agree on what they see, distance or time. They can only agree that the speed of light in vacuum is c. One sees the proper length: a length measurement where both ends are measured at the same time. One sees the improper length: a length measurement made at two different times. Neither observer is preferred -- that is one is not "more right" than the other. They are both right. These differences in time and length measurements have been confirmed by experiment. Experimental confirmation of Special Relativity: put atomic clocks on aircraft, spacecraft. Difference in time with identical clocks left on the ground. Second set of Sample Final Exams handed out. (Click here and here for a copy.) FIRST DAY to turn in Topic 1 Book Reports.
Friday 4/16: 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. Another confirmation of Special Relativity: Muons (a form of heavy electron) are created in the upper atmosphere -- they're unstable and will decay. Muons measured at mountaintop -- by sea level, nearly all should have already decayed. But you detect almost as many at sea level as on the mountaintop, because the muon lifetime is measured in the muon's rest frame not while we are watching it moving. The Correspondence Principle -- at some point our Classical Physics results need to match the Modern Physics results. So when do we need Special Relativity? For eyeball measurements, we have trouble distiguishing the size of things that are only off by 10%. That would correspond to a gamma = 1.10, and a beta = 0.417 c. Q21 Take-Home, due Tuesday 20 April 2010, in class or by 5pm. SECOND DAY to turn in Topic 1 Book Reports.
Monday 4/19: Classic Farmer vs. Pole Vaulter problem. Each believes that they win the bet, both are entitled to their opinion. Cannot realistically do it though -- cannot build a mechanism that would work as the farmer wants it to work. Relativistic momentum, prel = gamma mv, and relativistic Kinetic Energy, KErel = (gamma - 1) mc². Total Energy, Etotal = gamma mc². The Einstein Relation, E = m c², and conversion between energy and matter (mass). Pair creation of an electron-positron pair from two high energy gamma rays. Once we used to talk of a "relativistic mass", to try to explain why an object of matter cannot be accelerated up to the speed of light in vacuum, c. You can just as easily use the Work-Energy theorem to show that it would take infinite work to get a matter object up to c. (British SF writer Charlie Stross talks about the difficulties of space travel, including traveling to another star here.) LAST DAY to turn in Topic 1 Papers.
Tuesday 4/20: The neutrino -- Fermi's "little neutral one" -- was necessary to carry a missing piece of momentum in some nuclear and particle reactions. It seems to travel at v = c, but seems also to be a particle and not a photon. How can this be? Turns out it has a very small, but non-zero mass, and therefore travels almost, but not quite at v = c. Faster than the speed of light? The tachyon -- a hypothetical particle, whose time properties are very confusing. What we can do is have c > v > cm, so that we can have a particle traveling faster than the local speed of light in a material. Get Cerenkov Radiation -- typically a blue glow which is the optical equivalent of a sonic boom in air. The deBroglie wavelength -- Wave/Particle Duality for Matter. Planck's constant -- a very small number, but it is NOT zero ( h = 0 in Classical Physics). So the deBroglie wavelength only matters for very small objects, not Buicks. A slowly moving electron is more wavelike, while an electron moving at 99.99% of c is very particle-like. The Heissenberg Uncertainty Principle, means that there are limits to how well we can know (measure) pairs of certain quantities at the same time. delta-p and delta-x, also delta-E and delta-t -- where delta means the uncertainty or error in measuring the quantity. Q22 Take-Home, due Thursday 22 April 2010, in class or by 5pm.
Wednesday 4/21: Some comments about the Periodic Table of Elements. 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. 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 "a0", 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. NOTE: This material is covered more fully in an Atomic & Nuclear Physics handout, along with the Periodic Table with equations sheet which will be handed out on Wednesday. (Click here for the Atomic & Nuclear Physics handout (not given in class) and here for the Periodic Table handout -- with today's derivation on the back.)
Thursday 4/22: Takeaway from Bohr Atom: Equations for orbital radius, rn, and energy, En, are quantized -- electrons can only exist in certain orbits. Energy of emitted or absorbed photon due to electron changing from orbit i to orbit j is E = hf = |Ei - Ej|. Summary of topics covered since January. See Week 15 Finals REVIEW Checklist. Third set of Sample Final Exams handed out. (Click here for a copy.) There is also another Sample Final put online only.
Friday 4/23: Last Day of Class.