PHYSICS 2CL

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PHYSICS 2CL SPRING 2009 Physics Laboratory
Electricity and Magnetism, Waves and Optics
Prof. Leonid Butov (for Prof. Oleg
Shpyrko) oshpyrko_at_physics.ucsd.edu Mayer Hall
Addition (MHA) 3681, ext. 4-3066 Office Hours
Mondays, 3PM-4PM. Lecture Mondays, 200 p.m.
250 p.m., York Hall 2722
Course materials via webct.ucsd.edu (including
these lecture slides, manual, schedules etc.)
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Todays Plan

• Chi-Squared, least-squared fitting
• Next week
• Review Lecture (Prof. Shpyrko is back)

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Long-term course schedule
Week Lecture Topic Experiment
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, c2 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
Schedule available on WebCT
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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 weeks lab(s), 3 out of 4
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LEAST SQUARES FITTING (Ch.8)
Purpose
1) Agreement with theory?
2) Parameters
y(x) Bx
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LINEAR FIT
y(x) A Bx A intercept with y axis B
slope
x1 y1
x2 y2
x3 y3
x4 y4
x5 y5
x6 y6
q where Btan q
   
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LINEAR FIT
?
y(x) A Bx
x1 y1
x2 y2
x3 y3
x4 y4
x5 y5
x6 y6
y-22x
y90.8x
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LINEAR FIT
y(x) A Bx
x1 y1
x2 y2
x3 y3
x4 y4
x5 y5
x6 y6
y-22x
y90.8x

• Assumptions
• dxj ltlt dyj dxj 0
• yj normally distributed
• sj same for all yj

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LINEAR FIT y(x) A Bx
Method of linear regression, aka the
least-squares fit.
Yfit(x)
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Quality of the fit
y4-yfit4
y3-yfit3
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LINEAR FIT y(x) A Bx
Method of linear regression, aka the
least-squares fit.
true value
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minimize
y4-(ABx4)
y3-(ABx3)
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What about error bars? Not all data points are
created equal!
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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)
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More precise data points should carry more
weight! Idea weigh the points with the inverse
of their error bar
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Weight-adjusted average
How do we average values with different
uncertainties? Student A measured resistance
1001 W (x1100 W, s11 W) Student B measured
resistance 1055 W (x2105 W, s25 W)
Or in this case calculate for i1, 2
with statistical weights
BOTTOM LINE More precise measurements get
weighed more heavily!
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c2 TEST for FIT (Ch.12)
How good is the agreement between theory and
data?
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c2 TEST 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)
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LEAST SQUARES FITTING
true value
xj yj
yf(x)
y(x)ABxCx2exp(-Dx)ln(Ex)
y4-(ABx4)
y3-(ABx3)
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
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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
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Example fitting datapoints to yAcos(Bx)
Perfect Fit
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Example fitting datapoints to yAcos(Bx)
Stuck in local minima of c2landscape fit
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Next on PHYS 2CL Monday, May 18,  Review
Lecture