--class 1-- read ch 1 coordinateS.pdf Here you will investigate the basic integral theorems given on p.18: Eqs. 1-47, 1-45, 1-37. As detailed below you will separately calculate the left hand side (lhs) and the right hand side (rhs) of each equation and (hopefully) confirm that they are in fact equal. In order for you to do this I must specify what functions to use in the equations and what region you should integrate over. Eq 1-47; phi=x^2+4xy+2yz^3; select yourself two different paths that connect (0,0,0) to (1,1,1). Remember to report/document the parameterization of paths you are using. Eq. 1-37; F=; cube of side 2, aligned with axes, running between (0,0,0) and (2,2,2). Note that the cube has 6 faces requiring 6 surface integrals. Notation: the above reports the x, y, and z (respectively) components of the vector F Eq. 1-45; same F as above; isosceles right triangle in the zy plane that connects the points: (0,0,0), (0,2,0), (0,0,2) --class 2-- read up to section 2-4 old exam: emt112.pdf #1 and #2a textbook problem 2-5a --class 3-- read thru p.48, skipping (for now) the material of Gauss' Law (pp. 37-42) textbook problem 2-5a again, but this time find the potential on the z axis textbook: 2-6, 2-25, 2-27 ALSO: Consider a uniformly charged stick running along the z axis from -1 to 1. Find and ContourPlot the potential in the xz plane. Note: you would quite naturally think about using Mathematica with something like: Integrate[1/Sqrt[x^2+(z-z1)^2],{z1,-L,L}] but this produces a huge ConditionalExpression. Instead try something like the indefinite integral: Integrate[1/Sqrt[x^2+u^2],u] and then evaluate between top and bottom: Evaluate[% /. u->z+L]-Evaluate[% /. u->z-L] Finish off with something like: phi=% /. L->1 ContourPlot[phi, {x, -2, 2}, {z, -2, 2},ContourShading ->False,Contours -> 16,PlotRangePadding ->None] ContourPlot[phi, {x, -10, 10}, {z, -10, 10},ContourShading ->False,Contours -> 16,PlotRangePadding ->None] Required: write down the steps needed transform the generic form of Coulombs Law (i.e., like Eq. 2-8, but with lambda dz') into the integral that you typed into Mathematica. This is the main part of this problem since the only additional part is to run the above commands through Mathematica. Turn in a copy of Mathematica commands and one hardcopy ContourPlot. (Please avoid huge printouts of Mathematica notebooks: a copy&paste of commands is all I really want to see. Use of the command line version of Mathematica: "rlwrap math" is recommended but not required.) --class 4-- Finish reading all of ch.2, including Gauss's law and delta functions There are 3 highly symmetric charge distributions for which we can find the electric field just using Gauss's Law: 1) Uniformly charged infinite plane 2) Uniformly charged infinitely long rod 3) Uniformly charged sphere Write down the steps/explanation that allows us to find E for these 3 cases. Include sentences that describe how the symmetry forces E to be in a particular direction and have the same magnitude on the Gaussian surface you use. In particular: 1) infinite plane with surface charge density sigma (both above and below) 2) textbook problem 2-16 (inside and out) 3) just like above but this time a sphere with radius R and uniform charge density rho For each of these 3 cases also write down the potential phi that would produce the E field you calculated from Gauss's Law NOTE: You really need to be able to run these arguments in your head instantaneously for Exam 1 old 200 exam: http://www.physics.csbsju.edu/200/200t105.pdf problems: 6,7,12 --class 5-- Read thru p.69 handout: legendre.pdf 3-1, 3-2, 3-3 old exam: emt112.pdf #2 (all this time; note: azimuthally symmetric!) --class 6-- more on solutions to Laplace in spherical and cylindrical coordinates start images legendre.pdf homework #1, #3 textbook 3-12 Eq. 1 of legendre.pdf reports the orthogonality of Legendre polynomicals which we can check with Mathematica: Integrate[LegendreP[5,c] LegendreP[7,c],{c,-1,1}] Out[1]= 0 Integrate[LegendreP[7,c] LegendreP[7,c],{c,-1,1}] 2 Out[2]= -- 15 Pick out five different pairs of values of (n,m) and confirm using Mathematica Eq. 1 of the handout. Note: select only (even,even) or (odd,odd) cases as the orthogonality of (even,odd) is trivial. p. 62 of the textbook reports the formula for a few Legendre polynomicals. "By hand" check the cases (1,3) amd (2,2) (this is just integrating polynomials). --class 7-- Note: approximately 1 week to Exam 1 finish reading ch 3 20,23 Hints on 23: Clearly the image of the dipole is a dipole, note carefully the direction of the image dipole. Method 1: Eq 2-36 gives the E field produced by a dipole (in this case the image dipole) and Eq 2-44 gives the PE that the real dipole experiences in the image dipole field. Recall that Force =-grad(PE) Method 2: Make the dipole out of charges q and -q separated by some distance d. In the end take the limit d->0, where p=qd. Expect the force to be proportional to p^2 computed ("numerical") solutions to Laplace #1: Use spreadsheet to solve Laplace inside a 2d box with 10V on right side and the other three sides grounded. Find in this folder the file: relax.gnumeric. Download it and open it with the linux spreadsheet "gnumeric". If you really must use excel, you can probably do a "save as" and transfer it to a windoze box (I don't know much about windoze, but feel free to ask me for help; I have tested it in gnumeric). The file should be a 27x27 box with 0s on three sides and a column of 10s to the right. The cell AE10 contains the function that calculates the average of the 4 next-door cells; you will want to copy that into all of the currently blank cells inside the box, but first you must fill the box with some initial guess as to voltage value. One option would be a sort of average weighted according to how far a box-side is from the current cell (start with cell B2): =10*column()/27*(1-abs(row()-14)/13) worse (but OK) options would be fill with all 0s or all 5s (do not change the values on the box sides). No matter what initial guess you select, in the end you should get the same stable pattern. Once you've filled the cells, copy and paste the average formula from cell AE10 to each cell within the box walls (start with B2). Every time you hit function key F9, the array will update. Repeat until you have a stable pattern. select, copy, and paste special (as value) the contents in cell N14 somewhere Hit F9 50 more times, and, without overwriting the previously saved value, select, copy, and paste special (as value) the contents in cell N14 newwhere Hit F9 50 more times, and, without overwriting the previously saved value select, copy, and paste special (as value) the contents in cell N14 newwhere Hit F9 50 more times, and, without overwriting the previously saved value select, copy, and paste special (as value) the contents in cell N14 newwhere Print out your spreadsheet showing the array values and the 4 saved values of N14. Copy and paste special (as value) the array values into the same spot on Sheet2 (just array not saved N14, etc); Save as file out.csv: Data->Export Data->Export as CSV File In Mathematica run the commands: v=Import["out.csv"] ListContourPlot[v,ContourShading->False,AspectRatio->Automatic,Contours->30] NOTE: you must make sure the file out.csv is in the same folder that Mathematica will seek it...in the GUI version it looks like Insert->File Path will help with this. Print out a copy of the plot (but don't waste paper on the list-of-values Mathematica may generate). #2 Use a given fortran program to solve Laplace for a finite-size parallel plate capacitor. The capacitor you will be solving is 2d; the problem is symmetric so we just need to concern ourselves with above the mid-plane and just the rhs half of the capacitor. In a terminal on your linux account, type: bethe 117% cap enter tolerance, relaxation parameter, max iterations "tolerance" = what the text calls "iteration parameter Dm" [example: .001] "relaxation parameter" = w in text [example: 1.5] "max iterations" = control against the program running forever; upper limit of updates [example:10000] enter half-width and half-separation-distance these both must be whole numbers; both a "half" as by symmetry only first quadrant needs to be displayed half-width is half the parallel plate length [example: half-width<40] half-separation-distance is half the parallel plate separation [example: 10] program prints out the charge on the bottom of the top plate, on the top of the top plate, the total charge (which is proportional to the capacitance, since the voltage=1), the value of the voltage in the middle of the array (just as a typical value) and the number of iterations required to converge. decide on a capacitor geometry and run: for w=1: Dm=.001,.0001,.00001 record the total charge, the value of the voltage in the middle of the array, and the number of iterations required for w=1.25: Dm=.001,.0001,.00001 record the total charge, the value of the voltage in the middle of the array, and the number of iterations required for w=1.5: Dm=.001,.0001,.00001 record the total charge, the value of the voltage in the middle of the array, and the number of iterations required for w=1.75: Dm=.001,.0001,.00001 record the total charge, the value of the voltage in the middle of the array, and the number of iterations required Recording above in a spreadsheet via copy&paste might be the easiest way, but hand-written is OK verify and note: w affects number of required iterations; the actual differences between Vs is much more than you would expect from the tolerance (small change for a iteration does not mean as small change for 100s of iterations) For *one* of these, write the array to a file, and do a mathematica contourplot: v=Import["math.dat"] ListContourPlot[v,ContourShading->False,AspectRatio->Automatic,Contours->30] Print out the plot --class 8-- Note: Help for Exam 1 Tues 7:15 Read thru 4-6 legendre3.pdf homework #4 textbook 4-2, 4-3, 4-6 Hint 4-3: what are the equivalent volume/surface charges? Note: E=electric field; D=electric displacement. I will try to refer to these fields by letter not these names. --class 9-- Exam covers thru p 112 no homework --class 10-- Exam 1 --class 11-- finish reading ch 4 4-10, 4-15, 4-18 --class 12-- read ch 5 old exam emt212.pdf problem #3 Find the force between a point charge q located on the positive z-axis and a dielectric (dielectric constant K) that fills the space z<0. (Hint: images; you should just be able to write down the answer) problem #3 from handout (see center) problems 4-7 and/or 4-8 from handout. Note the "epsilon" in these problems is the dielectric constant we've called K. I assign numerical problems like this occasionally as they seem more real than typical algebraic problems. In this case we need to find some data for N = molecules/m^3 for a real substance under real conditions measured in the units found in practical books...here perhaps there is a bit more reality than you might want: the cited table does not have entries that exactly match the needed values (e.g., values for 10 psi and 40 psi but not 20 psi, and all at temperature 300K not 292K). I'll remove a bit of your fun by doing the table interpolation for you in the form of a formula (which I fit to table data via WAPP+). The ideal gas law says P/NkT = 1 (P=pressure, N=number density, k=Boltzmann's constant, T=temperature, all in good SI units) for this real gas I find P/NkT = 1.0005 -0.5209E-03*x+0.3021E-05*x^2 where x=pressure in atm. Thus you can use Eq. 5-10 to calculate alpha at different pressures...spreadsheet is way-to-go. According to our theory alpha is an atomic constant: it should not depend on pressure. Since it should all be in a spreadsheet, I see no reason to test the simple version of Clausius-Mossotti for pentane: use the correct formula to find alpha For 4-8 express dipole moment in SI units of C-m; you can immediately make an inverse-X plot with WAPP+ to get the slope EXCEPT WAPP is not happy with numbers like 1e-40. Multiply your calculated alpha by 1e38 before you fit FYI: nist.gov reports water's electric dipole moment is 1.85 debyes --class 13-- read ch 6 6-2,4,13,17 --class 14-- more ch 6 6-18, 6-19, 6-24 --class 15-- read ch 7 Note: we'll spend more time with Kirchhoff when we do AC circuits. old 200 exam: http://www.physics.csbsju.edu/200/200t205.pdf problems: 1-9 7-1,7-5,7-8,7-12 RE: 7-8: Treat the wire as an infinitely long cylinder with voltage V=0 @r=a (for all z) and voltage V @r=b. What is the formula for the voltage between a and b (Hint: Laplace)? What is the total current moving radially thru a length L of the cylinder (Hint: Jr)? RE: 7-12: finding the voltage in the plate via cylindrical approximation to Laplace should be usual; Fixing up the approximate solution by finding the average voltage on the plate sides and the total current through those sides will be novel=interesting. --class 16-- ch 8 looks nastier than it is; we'll mostly be concerned with Eq. 8-26 note: exam 2 in two weeks current_loop.pdf: bx=Integrate[ XXX ,{theta,-Pi, Pi},Assumptions->R>0&&Element[{x,z},Reals]] bz=Integrate[ ZZZ ,{theta,-Pi, Pi},Assumptions->R>0&&Element[{x,z},Reals]] b={bx,bz} /. R->1 Show[VectorPlot[b,{x,-2,2},{z,-2,2},VectorPoints->16,PlotRangePadding->None, RegionFunction->Function[{x, z, vx, vy, n}, (Abs[x]-1)^2+z^2>.3], PlotRange->{{-2.2,2.2},{-2.2,2.2}},VectorScale->{.4,.1,Automatic}]] current_sheet.pdf old exam emt212.pdf #4 (ignore the last part as we have not yet discussed vector potential A) Note: take a look at problem 8-27; see that you could use the above result to put together the magnetic fields due to the 4 sides. --class 17-- read section 8-5 to end-of-chapter B2.pdf 1-3 (not #4) Looking ahead (for class 19): Take a look at problem 8-2 which uses the "Hamiltonian" from last semester in mechanics (I hope). Find in your mechanics book how you go from a Hamiltonian to the equations of motion. FYI: Hamiltonians play the leading role in quantum mechanics; the the motion of an electron in a magnetic field is a standard problem in quantum mechanics. Note: canonical momentum not equal to "normal" (191) momentum --class 18--- more ch 8 solid angles, scalar potential Note: exam 2 in 8 days (delay: the beginning part on forces in particular) B2.pdf #4 Find Homework #2 from legendre.pdf which talks about the scalar potential for Helmholtz coils Show why the scalar potential should satisfy Laplace's equation (27) Using the formula for the scalar potential of a loop derived in class find the scalar potential for Helmholtz coils (coil pair spaced by R). Note Eq. (28) is missing an overall factor (it lacks I and mu_0 for example). Continue on to Eq. (29) for the current-reversed Helmholtz pair; Write down the correct scaler potential (overall factor); find the b for a maximally pure quadrupole field. --class 19--- exam: Wednesday help:6:30 Monday Ardolf 142 or 121 finish ch8: magnetic forces/torques 8-2 (go lookup Hamiltonian eqs of motion from mechanics) 8-6 Laboratory magnets (typically superconducting) commonly produce fields of 10 Tesla. Approximating (spherical cow) this as and infinite solenoid, what value of NI is required? What is the force/area (pressure) on those turns? (You might want to think about the magnetic equivalent of Eq. 4-57a) --class 20--- read thru p. 242 Note: B=magnetic induction; H=magnetic intensity. I will always try to refer to these fields by letter not these antique names if I forget and talk about "the magnetic field" I mean B due in class 22: 9-1, 9-8, 9-9, 9-14 B3.pdf --class 21--- exam 2 --class 22-- finish reading ch 9 old exam: emt312.pdf #1 & #2 BC.pdf 9-6 --class 23-- read chapter 10 9-13, handout (36-21), and 9-21 BUT: assume exactly as in Fig 9-19 (i.e., no 800 A turns) follow the method of section 9-11 note Fig.9-9 for Alnico 5=Alnico V and further note that the x-axis of this plot shold be mu_0 H (not mu_0 B) --class 24-- read thru p 281 old exam emt312.pdf #4ab old 200 exam: 200t305.pdf: 1-8, 13 11-18 I prefer to think of my loops stacked vertically, so to be specific, let the top loop (loop1) have radius a and be a distance z above the horizontal xy (z=0) plane which contains loop2 of radius b. So the center of loop1 is (0,0,z) and the center of loop2 is (0,0,0). Note that over a complete dl2 integration, every dl1 must produce the same result (why?), so all that is required is to do the dl2 integration for the easiest possible dl1, and then multiply that result by 2 Pi a (which is the sum of all the dl1). As usual clearly report what you are using for r1, r2, dl2 etc. Mathematica defines the elliptic integrals in terms of m = k^2, so you might want to compare to this answer instead of the one in the textbook: M12=-((mu0*(((a+b)^2 + z^2)*EllipticE[(4*a*b)/((a+b)^2 + z^2)] - (a^2 + b^2 + z^2)*EllipticK[(4*a*b)/((a+b)^2 + z^2)]))/Sqrt[(a+b)^2 + z^2]) Additional problem: Calculate: Limit[M12/(Pi a^2), a->0] and provide a simple explanation for the resulting formula Plot[Evaluate[M12/mu0 /. {a->1,b->1}],{z,0,4}] Explain why this quantity goes infinite as z goes to zero. Plot[Evaluate[M12/mu0 /. {a->1/10,b->1}],{z,0,4}] Explain why this quantity stays finite as z goes to zero. The numerical value of this quantity at z=0 is 0.0158. Provide an argument that gives this result (approximately). Calculate: Series[M12,{z,Infinity,3}] and provide a simple explanation for the resulting formula. Note: eq 8-38 may be of use for some of these problems --class 25 -- read thru p 302 11-7, 8, 15 old exam emt312.pdf #4c Consider the Helmholtz coils used in 200 lab, in the context of the mutual inductance calculation of class 24. Find the force (in N) between these coils, using magnetic energy (and Mathematica). Basic data: a=b=z=.15m; I1=I2=130*2 A Compare the above actual force to the following crude methods of estimating the force: If we "straighten out" the circles, we have two wire segments of length 2 Pi r, separated by a distance r. If we make one of the wire segments infinitely long we can easily calculate its B, and then the resulting force on the remaining wire segment. Calculate that force! Do you expect this to be an over or under estimate for the actual force? If you go back to Class 16.pdf, you'll find a calculation of the B of a current segment (emt212.pdf #4). The upshot was the B field of the segment was that of an infinite wire times: .5*(Cos[theta1]+Cos[theta2]) look the old exam for definitions of theta1 and theta2 if you've forgotten. In our two wire segment approximation for the problem, the smallest B will occur at the ends where theta2=90 degrees. Find theta1. Again calculate the force between the two wire segments pretending that the B field of loop1 was always as small as it is at its end. Do you expect this to be an over or under estimate for the actual force? --class 26 -- read ch 13 exam3 18-april-2013 (2 weeks: Thursday) Find the equivalent resistance of the xkcd.circuit.png circuit (use nodes!) problemsC4.pdf (there is a lot here, but at least get the first two done ASAP) For an infinite vacuum-filled solenoid we know: B=mu0 N I where NI=j is the surface current with N turns per length each turn carrying current I A was given (and checked) in B2.pdf 3d Consider (and calculate) three different ways of finding the energy per length of the solenoid: eq 12-11 (where L is calculated from the flux) eq 12-13b (converted to surface currents) eq 12-15 All should of course produce the same answer! --class 27 -- more circuits! 13-15 (the algebra was unpleasant enough that I used Mathematica) 13-20 Circuits provide a chance to revisit ordinary differential equations. Note for example that in a series LRC circuit, driven with a cosine source at frequency f and amplitude A, satsifies the differential equation: L q''[t]+R q'[t]+q[t]/c==A Cos[2 Pi f t] Given initial conditions Mathematica can numerically solve and plot the solution with code: solution=NDSolve[{L q''[t]+R q'[t]+q[t]/c==A Cos[2 Pi f t],q[0]==q0,q'[0]==i0},q,{t,0,t0}] Plot[Evaluate[q[t] /. solution],{t,0,t0}] Consider component values L=.82 mH; c=2.2 micro F, R=.2 Ohms, f=4000 Hz, A=1 V The capacitor is charged to a voltage of 1 V, and the circuit switch is closed (of course the intial current is then 0). Calculate and plot the first .01 sec of the charge. Explain the result. --class 28 -- exam3 in 8 days; help? read ch 16 (aim is for exam3 to cover thru [including] section 17.1) ch16.pdf #1 and #2 16-7, 16-10 --class 29 -- read thru [including] 17.1 Help: Tuesday 4pm 17-2 [most of the problem is to show how the units produce ohms], 5, 15 --class 30-- (not on exam3) Help: Friday 4:15pm (exam now on Monday) Note: exam 3 ch. 9 thru 17-3 (except ch 14+15) allowed: single-sided cheat sheet + mathematica (on your own laptop) Read thru 17-4 Read 18-1, skim 18-2 thru 18-4 sub communications: At low frequency sea water is a "good conductor" (g=3.3 (Ohm m)^-1), so radio signals are attenuated when traveling thru sea water. Assume to communicate with a submerged sub, the sea water skin depth must be more than 100 m. What is the highest frequency we can use? Fresnel: Figure 18-3 shows the reflection coefficient for light entering glass (n2=1.5) from vacuum (n1=1). (Note: s=TE, p=TM) Eqs. 18-28 and 18-33 (which can also be found in the chapter summary) give the electric field ratios: r=E'/E from which R is calculated (R=|r|^2). Use Mathematica to plot rs and -rp (together) with something like: Plot[{rs,-rp},{t1,0,Pi/2}] Note that the equations involve both theta1 and theta2, so you must express theta2 in terms of theta1 using Snell's Law (Eq. 18-19). Your results can be checked by comparing with the corresponding plot of R. Figure 18-4 shows the reflection coefficient for light leaving glass (n1=1.5) and entering vacuum (n2=1). Exactly the same equations apply, but now we have total internal reflection at theta1=ArcSin[n2/n1] where E' becomes complex (and hence hard to plot on a real axis). R really depends on |r|^2, so we can plot Abs[r] for the range that includes total internal reflection. So you'll be doing something like: Plot[{rs,-rp},{t1,0,ArcSin[n2/n1]},PlotRange->All] Plot[{Abs[rs],-Abs[rp]},{t1,ArcSin[n2/n1],Pi/2}] Show[%%,%,PlotRange->All] By hand, mark on your plots the Brewster angle and the angle of total internal reflection. Why do I have you plotting -rp? I like a sign convention that has near-normal (theta1 near 0) incidence for both cases agreeing. In the book's sign convention, at normal incidence, rp=-rs. None of this affects R=|r|^2 of course. --class 31 -- (not on exam3) HW due in class 33 stress.pdf solar sailing: crude estimates; momentum density and flux You aim to accelerate a 100 kg spacecraft from orbit velocity (8000 m/s) to escape velocity (11000 m/s) in 1 month using the 100% reflectivity of a solar sail in the 1300 W/m^2 sunlight. What sail area is required? Note: we will probably have about 3 class periods to discuss applications of Maxwell's equations, i.e., material beyond ch 17. My preference is to talk about radiation and relativity; alternatives would include optics and dispersion relations. Feel free to make suggestions. --class 32 -- exam3 ch. 9 thru 17-3 (except ch 14+15) allowed: single-sided cheat sheet + mathematica (on your own laptop) --class 33 -- read sections: 20-1,20-2,20-5, Summary HW: 20-1,20-2,20-13 #9 mDipole.pdf --class 34 -- skim: chapter 22 R1b.pdf #1,4,5 --class 35 -- m3_Fuv.pdf Note: I did these problems with Mathematica because I'm lazy, but perhaps you seek a work-out. Note: "usual S' frame" has beta in x direction equal to .99 At any point in space-time in the S frame the D 4-vector field can be calculated from: D=(cos(x),sin(y),0,i ct) Find D in the usual S' frame (i.e., D') at the space-time location: (x',y',z',ict')=(1,2,3,i4) Report D'.D' and D.D at this space-time point. At any point in space-time in the S frame the G 4-vector field can be calculated from: G=(x,y,z,0). Note that the 4-divergence of G is 3 for all space-time. Find the formula for G in the usual S' frame (i.e., G') at the general space-time location: (x',y',z',ict') Calculate the 4-divergence of G' in the S' frame. --class 36 -- course evaluation bring a laptop! E,B of moving charge Help: Tuesday 7pm Final Exam: Thursday 10:30am