PHYSICS 2CL – SPRING 2009 Physics Laboratory: Electricity and Magnetism, Waves and Optics
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Transcript of 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)[email protected]
Mayer Hall Addition (MHA) 3681, ext. 4-3066Office Hours: Mondays, 3PM-4PM. Lecture: Mondays, 2:00 p.m. – 2:50 p.m., York Hall 2722Course materials via webct.ucsd.edu(including these lecture slides, manual, schedules etc.)
Today’s Plan:
Chi-Squared, least-squared fitting
Next week: Review Lecture (Prof. Shpyrko is back)
Long-term course schedule
Schedule available on WebCT
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, 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
Labs Done This Quarter
0. Using lab hardware & software1. Analog Electronic Circuits
(resistors/capacitors)2. Oscillations and Resonant Circuits (1/2)3. Resonant circuits (2/2)4. Refraction & Interference with
Microwaves5. Magnetic Fields6. LASER diffraction and interference7. Lenses and the human eye
This week’s lab(s), 3 out of 4
LEAST SQUARES FITTING (Ch.8)Purpose:
1) Agreement with theory?
2) Parameters
0 5 10 15 20 25x
0
10
20
30
y =
f(x
)
y(x) = Bx
LINEAR FIT y(x) = A +Bx :
A – intercept with y axisB – slope
0 5 10 15 20 25x
0
10
20
30
y(x)
x1 y1
x2 y2
x3 y3
x4 y4
x5 y5
x6 y6A
where B=tan
?LINEAR FIT y(x) = A +Bx
0 5 10 15 20 25x
0
10
20
30
y(x)
x1 y1
x2 y2
x3 y3
x4 y4
x5 y5
x6 y6
y=-2+2x
y=9+0.8x
y(x) = A +Bx
0 5 10 15 20 25x
0
10
20
30
y(x)
y=-2+2x
y=9+0.8xAssumptions:
1) xj << yj ; xj = 0
2) yj – normally distributed
3) j: same for all yj
x1 y1
x2 y2
x3 y3
x4 y4
x5 y5
x6 y6
LINEAR FIT
LINEAR FIT: y(x) = A + Bx
0 5 10 15 20 25x
0
10
20
30
y(x) y3-yfit3
y4-yfit4
Yfit(x
)
[yj-yfitj] 2Qualityof the fit
Method of linear regression, aka the least-squares fit….
LINEAR FIT: y(x) = A + Bx
0 5 10 15 20 25x
0
10
20
30
y(x) y3-(A+Bx3)
y4-(A+Bx4)
true va
lue
[yj-(A+Bxj)] 2minimize
Method of linear regression, aka the least-squares fit….
What about error bars?Not all data points are created equal!
0 5 10 15 20 25x
0
10
20
30
y(x)
Weight-adjusted average:
N
xxx
N
xx Ni
...21
N
NN
i
ii
www
xwxwxw
w
xwx
...
...
21
2211
Reminder:Typically the averagevalue 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:
%205*%12%20
%20%12%20
FINALLABSGRADE
Formal
Weights: 20 for Final Exam, 20 for Formal Report, and 12 for each of 5 labs – lowest score gets dropped)
More precise data points should carry more weight!Idea: weigh the points with the ~ inverse of their error bar
0 5 10 15 20 25x
0
10
20
30
y(x)
Weight-adjusted average:How do we average values with different uncertainties?
Student A measured resistance 100±1 (x1=100 , 1=1 )Student B measured resistance 105±5 (x2=105 , =5 )
21
2211
ww
xwxwx
21
1
1
w
22
2
1
w
N
NN
i
ii
www
xwxwxw
w
xwx
...
...
21
2211
Or in this case calculate for i=1, 2:
with “statistical” weights:
BOTTOM LINE: More precise measurements get weighed more heavily!
0 5 10 15 20 25x
0
10
20
30
y(x)
How good is the agreementbetween theory and data?
TEST for FIT (Ch.12)
) )
N
j j
jj xfy
12
2
2
0 5 10 15 20 25x
0
10
20
30
y(x)
TEST for FIT (Ch.12)
NN
y
y 2
2
d
22~
d = N - c
# of degrees of freedom
# of datapoints # of parameters
calculated from data
# of constraints
1
) )
N
j j
jj xfy
12
2
2
0 5 10 15 20 25x
0
10
20
30
y(x) y3-(A+Bx3)
y4-(A+Bx4)
true v
alueLEAST SQUARES FITTING
1.
2. Minimize 02
A
0
2
B
…
3. A in terms of xj yj ; B in terms of xj yj , …
4. Calculate 5. Calculated
20~
6. Determine probability for20
2 ~~
xj yj y=f(x)
y(x)=A+Bx+Cx2+exp(-Dx)+ln(Ex)+…
) )
N
j j
jj xfy
12
2
2
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
22
Fitting Parameter Space
Best fit (lowest 2)
Sometimes the fitgets “stuck” in local minima like this one.
Solution? Give it a “kick” by resetting one of the fitting parameters and trying again
Example: fitting datapoints to y=A*cos(Bx)
“Perfect” Fit
Example: fitting datapoints to y=A*cos(Bx)
“Stuck” in localminima of 2landscape fit
Next on PHYS 2CL:
Monday, May 18, Review Lecture