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. 4-3066Office 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

(Example: You can always draw a perfect line through 2 points)

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 a local minimum 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