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PHYSICS 2CL – SPRING 2009 Physics Laboratory: Electricity and Magnetism, Waves and Optics Prof. Leonid Butov (for Prof. Oleg Shpyrko)

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Presentation on theme: "PHYSICS 2CL – SPRING 2009 Physics Laboratory: Electricity and Magnetism, Waves and Optics Prof. Leonid Butov (for Prof. Oleg Shpyrko)"— Presentation transcript:

1 PHYSICS 2CL – SPRING 2009 Physics Laboratory: Electricity and Magnetism, Waves and Optics Prof. Leonid Butov (for Prof. Oleg Shpyrko) Mayer Hall Addition (MHA) 3681, ext 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 WeekLecture TopicExperiment 1 Mar. 30 Introduction NO LABS 2 Apr. 6 Error propagation; Oscilloscope; RC circuits 0 3 Apr. 13 Normal distribution; RLC circuits 1 4 Apr. 20 Statistical analysis, t-values; 2 5 Apr. 27 Resonant circuits 3 6 May 4 Review of Expts. 4, 5, 6 and 7 4, 5, 6 or 7 7 May 11 Least squares fitting,  2 test 4, 5, 6 or 7 8 May 18 Review Lecture 4, 5, 6 or 7 9 May 25 No Lecture (UCSD Holiday: Memorial Day) No LABS, Formal Reports Due 10 June 1 Final Exam NO LABS

4 Labs Done This Quarter 0. Using lab hardware & software 1.Analog Electronic Circuits (resistors/capacitors) 2.Oscillations and Resonant Circuits (1/2) 3.Resonant circuits (2/2) 4.Refraction & Interference with Microwaves 5.Magnetic Fields 6.LASER diffraction and interference 7.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 LINEAR FIT y(x) = A +Bx : A – intercept with y axis B – slope x1y1 x2y2 x3y3 x4y4 x5y5 x6y6 A  where B=tan 

7 ? LINEAR FIT y(x) = A +Bx x1y1 x2y2 x3y3 x4y4 x5y5 x6y6 y=-2+2x y=9+0.8x

8 y(x) = A +Bx y=-2+2x y=9+0.8x Assumptions:  x j <<  y j ;  x j = 0 2)y j – normally distributed  j : same for all y j x1y1 x2y2 x3y3 x4y4 x5y5 x6y6 LINEAR FIT

9 LINEAR FIT: y(x) = A + Bx y 3 -yfit 3 y 4 -yfit 4 Yfit(x) [y j -yfit j ]  2 Quality of the fit Method of linear regression, aka the least-squares fit….

10 LINEAR FIT: y(x) = A + Bx y 3 -(A+Bx 3 ) y 4 -(A+Bx 4 ) true value [y j -(A+Bx j )]  2 minimize Method of linear regression, aka the least-squares fit….

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” w 1, w 2 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  (x 1 =100 ,  1 =1  ) Student B measured resistance 105±5  (x 2 =105 ,   =5  ) Or in this case calculate for i=1, 2: with “statistical” weights: BOTTOM LINE: More precise measurements get weighed more heavily!

15 How good is the agreement between theory and data?    TEST for FIT (Ch.12)

16 d = N - c # of degrees of freedom # of data points # of parameters calculated from data # of constraints

17 y 3 -(A+Bx 3 ) y 4 -(A+Bx 4 ) true value LEAST SQUARES FITTING Minimize    … 3.  A in terms of x j y j ; B in terms of x j y j, … 4. Calculate   5. Calculate 6. Determine probability for x j y j y=f(x) y(x)=A+Bx+Cx 2 +exp(-Dx)+ln(Ex)+…

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 Fitting Parameter Space Best fit (lowest  2) Sometimes the fit gets “stuck” in local minima like this one. Solution? Give it a “kick” by resetting one of the fitting parameters and trying again

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  2landscape fit

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


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