physics 2dl – spring 2010courses.physics.ucsd.edu/2010/spring/physics2dl/lecture 6 slides.pdf ·...

PHYSICS 2DL – SPRING 2010

MODERN PHYSICS LABORATORY

Monday May 3, 2010Course Week 6

Lab Week

Prof. Brian Keating

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2Day in 2DL

• Questions/Announcements

• Error Analysis: Review Ch 6 (rejecting data) Ch 7 weighted averages; NEW! Ch8 least squares fitting.

• HW due in lab this week

• Special Topic: Franck-Hertz: Using Data Acquisition to improve your life.

New Today Ch 8

• Ch 6,7 Review• Ch 8 = Least Squares fitting

Final exampleA student makes several measurements of a resistance R and determines R=3.5Ω and σR=0.2 Ω. Write an expression for the probability of obtaining a measured value between R0 and R0+ΔR.

What is the probability of obtaining a measured value between 3.35Ω and 3.8Ω?

_

R0

R0+ΔR

t=1.5

Then divide by 2 and add to previous

(t=0.75 sigma)

probability.

Chapter 7

New Today Ch 8

• Ch 6,7 Review• Ch 8 = Least Squares fitting

LEAST SQUARES FITTING (Ch.8)

Purpose:

1) Agreement with theory?2) Parameters

y(x) = Bx

χ2 TEST for FIT (Ch 12)

# of degrees of freedom

Least Sq. Fits : Derived from:

LEAST SQUARES FITTING

y=A+Bx+Cx2+Dx3+…+ ZxN

minimize

…

A,B,C…


1.

2. Minimize χ2:

y = f(x)

LINEAR FITy(x) = A +Bx :

velocity at const accelerationOhm’s lawmany other…

x1 y1x2 y2x3 y3x4 y4x5 y5x6 y6A

B

?LINEAR FIT

y(x) = A +Bx

x1 y1

x2 y2x3 y3x4 y4x5 y5x6 y6

y=-2+2x

y=9+0.8x

LINEAR FITy(x) = A +Bx

y=-2+2x

y=9+0.8xAssumptions:

1) δxj << δyj ; δxj = 0

2) yj – normally distributed

3) σj: same for all yj

x1 y1x2 y2x3 y3x4 y4x5 y5x6 y6

LINEAR FIT: y(x) = A + Bx

Assumptions:δxj << δyj ; δxj = 0

yj – normally distributedσj: same for all yj

y3-(A+Bx3)

y4-(A+Bx4)

true value

[yj-(A+Bxj)]Σ 2minimize

LINEAR FITy(x) = A +Bx

y=-2+2x

y=9+0.8xAssumptions:

1) δxj << δyj ; δxj = 0

2) yj – normally distributed

3) σj: same for all yj

x1 y1x2 y2x3 y3x4 y4x5 y5x6 y6

true value of y


Best estimates of A&B max Prob(y1…yN) min [yj-(A+Bxj)]Σ 2


Best estimates of A&B max Prob(y1…yN) min [yj-(A+Bxj)]Σ 2

In Taylor p. 197


ln V=ln(Vo) – x/td

Better to put in linear form:

This is just ok, but hides details of fit

Here’s our “final” example of the general technique when fitting for

See Taylor

Problem 8.18

Don’t Use Linear fit with A=0!


y=eAx y=A+Bx+Cx2+Dx3+…+ ZxN y=f(x)

minimize

…

A,B,C…

q = x + A II q = Bx II q = x + y II general case

prob(x) normallydistributed

prob(q)= prob(x=q-A)


prob(q) = prob(x=q/B)

normallydistributedprob(x)

X_

σx

Bx_

Bσx


x=y=0_ _

prob(x)

prob(y)

prob(x and y) = prob(x)*prob(y)

prob(x&y)

prob(x&y)

prob(x+y and z)


prob(x+y and z)

prob(x+y)

=(2π)

0.5

σx

σy


q(x,y)

fixednumber

fixednumbernormally distributed with σx

How DAQ can simplify your (experimental) life.

e--Atom collisions overhead!

Electronic Measurement using Digital to Analog Conversion

Franck Hertz DAQ

•Program (called a “.vi” file) is on Floppy drive.

•Save data to hard disk, on desktop.

•Email yourself the data from IE

•Save channel 1 data (acceleration voltage)

•Channel 2 is current, measured as a voltage

physics 2dl – spring 2010courses.physics.ucsd.edu/2010/spring/physics2dl/lecture 6 slides.pdf ·...

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