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PHYSICS 2CL – SPRING 2009 Physics Laboratory: Electricity and Magnetism, Waves and Optics

PHYSICS 2CL – SPRING 2009 Physics Laboratory: Electricity and Magnetism, Waves and Optics. Prof. Leonid Butov (for Prof. Oleg Shpyrko) oshpyrko@physics.ucsd.edu Mayer Hall Addition (MHA) 3681, ext. 4-3066 Office Hours: Mondays, 3PM-4PM.

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PHYSICS 2CL – SPRING 2009 Physics Laboratory: Electricity and Magnetism, Waves and Optics

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  1. PHYSICS 2CL – SPRING 2009Physics Laboratory: Electricity and Magnetism, Waves and Optics Prof. Leonid Butov (for Prof. Oleg Shpyrko) oshpyrko@physics.ucsd.edu Mayer Hall Addition (MHA) 3681, ext. 4-3066 Office Hours: Mondays, 3PM-4PM. Lecture: Mondays, 2:00 p.m. – 2:50 p.m., York Hall 2722 Course materials via webct.ucsd.edu (including these lecture slides, manual, schedules etc.)

  2. Today’s Plan: Chi-Squared, least-squared fitting Next week: Review Lecture (Prof. Shpyrko is back)

  3. Long-term course schedule Schedule available on WebCT

  4. Labs Done This Quarter 0. Using lab hardware & software • Analog Electronic Circuits (resistors/capacitors) • Oscillations and Resonant Circuits (1/2) • Resonant circuits (2/2) • Refraction & Interference with Microwaves • Magnetic Fields • LASER diffraction and interference • Lenses and the human eye This week’s lab(s), 3 out of 4

  5. LEAST SQUARES FITTING (Ch.8) Purpose: 1) Agreement with theory? 2) Parameters y(x) = Bx

  6. A LINEAR FIT y(x) = A +Bx : A – intercept with y axis B – slope q where B=tanq

  7. LINEAR FIT ? y(x) = A +Bx y=-2+2x y=9+0.8x

  8. LINEAR FIT y(x) = A +Bx y=-2+2x y=9+0.8x • Assumptions: • dxj << dyj ; dxj = 0 • yj – normally distributed • sj: same for all yj

  9. LINEAR FIT: y(x) = A + Bx S [yj-yfitj] Method of linear regression, aka the least-squares fit…. Yfit(x) 2 Quality of the fit y4-yfit4 y3-yfit3

  10. LINEAR FIT: y(x) = A + Bx S [yj-(A+Bxj)] Method of linear regression, aka the least-squares fit…. true value 2 minimize y4-(A+Bx4) y3-(A+Bx3)

  11. What about error bars? Not all data points are created equal!

  12. Weight-adjusted average: Reminder: Typically the average value of x is given as: Sometimes we want to weigh data points with some “weight factors” w1, w2 etc: You already KNOW this – e. g. your grade: Weights: 20 for Final Exam, 20 for Formal Report, and 12 for each of 5 labs – lowest score gets dropped)

  13. More precise data points should carry more weight! Idea: weigh the points with the ~ inverse of their error bar

  14. Weight-adjusted average: How do we average values with different uncertainties? Student A measured resistance 100±1 W (x1=100 W, s1=1 W) Student B measured resistance 105±5 W (x2=105 W, s2=5 W) Or in this case calculate for i=1, 2: with “statistical” weights: BOTTOM LINE: More precise measurements get weighed more heavily!

  15. c2TEST for FIT (Ch.12) How good is the agreement between theory and data?

  16. c2TEST for FIT (Ch.12) # of degrees of freedom d = N - c # of data points # of parameters calculated from data # of constraints (Example: You can always draw a perfect line through 2 points)

  17. LEAST SQUARES FITTING true value xj yj y=f(x) y(x)=A+Bx+Cx2+exp(-Dx)+ln(Ex)+… y4-(A+Bx4) y3-(A+Bx3) 1. … 2. Minimize c2: 3. A in terms of xj yj ; B in terms of xj yj , … 4. Calculate c2 5. Calculate 6. Determine probability for

  18. Usually computer program (for example Origin) can minimize as a function of fitting parameters (multi-dimensional landscape) by method of steepest descent. Think about rolling a bowling ball in some energy landscape until it settles at the lowest point Best fit (lowest c2) Sometimes the fit gets “stuck” in a local minimum like this one. Solution? Give it a “kick” by resetting one of the fitting parameters and trying again Fitting Parameter Space

  19. Example: fitting datapoints to y=A*cos(Bx) “Perfect” Fit

  20. Example: fitting datapoints to y=A*cos(Bx) “Stuck” in local minima of c2landscape fit

  21. Next on PHYS 2CL: Monday, May 18,  Review Lecture

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