Physics Laboratory for Engineering

“No amount of experimentation can ever prove me right; a single experiment can prove me wrong.”

Albert Einstein (1879-1955)

What do we want to do?

Illustrate important concepts in Physics

The History of Science

The practical theory of Experimentation

An introduction to Experimental Science

How is the Lab designed Module 1 – Waves

Wave bath Young’s Slit Experiment Interferometry

Module 2 – Atomic Physics Franck-Hertz Experiment Photoelectric Effect

Module 3 – RCL Circuits Linear response in time domain and Frequency domain Fourier transform

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Presentations!! Presentations!!Introduction

Each Lab takes a 2 weeks period!!!

Presentation of Work

Tutorial Pages

Colloquium

Experiment

Lab report

Lab Report Every experiment will be judged on the Lab

report presented. No Report --- No Mark!!!!!

http://labwrite.ncsu.edu

2 weeks to prepare your report.

Final mark is given on the 5 best reports out of a total of 6

Safety1. Special care should be taken to avoid unintentional

reflections from mirrors2. Where possible the laser beam should terminate on a

material which scatters the light diffusely after the beam has passed along its intended path. The colour and reflection properties of the material should enable the beam to be diffused, so keeping the hazards due to reflection as low as possible.

3. Eye protection is necessary if there is a possibility of either direct or reflected radiation entering the eye or diffuse reflections can be seen which do not fulfil the conditions in b.).

4. The entrances to supervised laser areas should be identified with the laser warning symbol

What is an Experiment?

A test under controlled conditions that is made to demonstrate a known truth, examine the validity of a hypothesis, or determine the efficacy of something previously untried.

Multiple choice

A. To prove what I know is right

B. To prove that he is wrong

C. To test an idea

D. To see what happens

E. All of the above

In our context an Experiment is a procedure to test the validity of theoretical idea – a hypothesis.Is every idea given to experimentation?

“If a black cat crosses your path you will have bad luck!!”

“Bad Luck” is a subjective idea which relies on comparison to “Good Luck”, another subjective idea, and so cannot really be quantified

BAD EXAMPLE

GOOD EXAMPLE

2211 vmvmvmFi

iibefore

The Conservation of Linear momentum

“In an elastic collision the total momentum before and after are equal”

m1v1 m2v2

2211 vmvmvmFi

iiafter 221

211 vv

m

mvv

FF afterbefore

Prediction of objective quantitiesRepeatable results

The Outline of Scientific Proof

Perception (The unfortunate tale of the blind elephant trainers)

Theory (Flat earth or round ball) Hypothesis – a Prediction Validation (or negation) of the Hypothesis

Experiment (A physical interpretation of the prediction) Quantifiable results Repeatable results

MeasurementHow do we measure a quantity?

This may sound like a stupid question but understanding the limits of measurement is the heart of experimentation

AA BB

Uncertainties in aligning the ruler lead to ERRORs in our measurement.

Errors and their Analysis

Systematic errorsAn error that remains constant through out an experiment and cannot therefore be detected. For instance:

HardwareLets look at our ruler again.

Errors and their Analysis

Experimental errors Protocol

Doing an experiment in the wrong order can lead to errors

Nature (Uncertainty)Imagine you have to measure the position of a quantum particle!

Machine tolerance Random Errors

measuring the area of a plate 2.5 cm x 5 cm to an accuracy of 100 μ2

xp

measured Value [mm] Average

l [mm] 50.32, 50.36, 50.35, 50.41, 50.37, 50.36, 50.32, 50.39, 50.38, 50.36, 50.38

50.368

w [mm]

24.25 ,24.26, 24.22, 24.28, 24.24, 24.25, 24.22, 24.26, 24.23, 24.24

24.245

Area is 50.368 x 24.245 = 1221.172 mm2

l

w

But is this accurate? What is the error in the calculation?

Conclusion – “There are sadistic scientists who hurry to hunt down errors instead of establishing the truth” -Marie Curie

If anyone is going to accept the results of your experiment then the errors involved must be accurately assessed.

A result from an experiment must be presented with its error

Everything possible must be done to reduce systematic errors in the setup

Presentation and Calculation of Errors

If x is a measured value: where δx is the toleranceof the measurementdevice

][unitsxx

In terms of fractional uncertainty x

x

For example our ruler has a tolerance of…?

inchx 32/1

Comparison of two measurements

p1 [kg∙m ∙s-1]

±0.04

p2 [kg∙m ∙s-1]

±0.06

1.49 1.56

2.10 2.12

1.16 1.05

11 pp

Consider our earlier example of linear momentum

m1v1 p1 p2

p1-p2=0 What is the uncertainty in the difference?

22 pp

2121 pppp

Uncertainties are additive

p1 [kg∙m ∙s-1]

±0.04

p2 [kg∙m ∙s-1]

±0.06

p1-p2

1.49 1.56 -0.07

2.10 2.12 -0.02

1.16 1.05 0.11

Let’s consider only one of the parameters, x, and its attendent error δx. Clearly this will lead to an error in f given by

The propagation of uncertainties

,...,, zyxFf

What happens when our result is a function of a number of measured quantities, for instance the linear momentum in the previous slide? How do we calculate the error in our answer?

xxFff

On the understanding that δx is small compared to x we can expand F(x) as a Taylor’s series

xx

FxFxxF

If we consider a further parameter, y, then

yy

Fx

x

FyxFyyxxF

,,

In other words our result, f=F(x,y,..), has an error estimated by:

yy

Fx

x

Ff

This result works very well for sums and multiplication of errors. However it is limited for a more general class of functions.

xFxx

F

yFyy

F

y

x

F(x,y)

F

δyδx

FδF

yFδF

xFδF

yyy

Fxx

x

F

yFxFF yx

ˆˆ

ˆˆ

2

2

22

2

yy

Fx

x

F

FFF

The General Expression for the Propagation of Error

2

2

2i

i i

i

xx

Ff

xFf

Examples

xyzV Box Volume

222222 dzyxdyxzdxyzV

Statistical Treatment of Errors

ix

The exist systems that we may want to measure where there are fluctuations about a mean value of measured quantity which are greater than the accuracies of our equipment. Obviously the average of the measurements gives the best estimate of the variable. But what of the errors?

Measured set of values

N

iix

Nx

1

1The best estimate

N

ii xx

N 1

21Standard deviation

If the measurements are normally distributed then there is a 70% chance that a measurement will be within σ of the actual value

From our set of measurements, xi , the average represents our best guess of the correct value. What is the error in this estimate?

The Standard deviation of the mean x

-2 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32

0

2

4

6

8

10

f(x)

x

x

x x

Nx xx / For example if we measured the spring constant of a spring 10 times and got the following readings

N/m

k 86 85 84 89 86 88 88 85 83 85

mNk /9.85mN /9.1

δk=0.6 N/m

Fitting a Function to a set of Data

One of the strongest tools in the experimentalist’s toolkit is the concept of data fitting.If we have a set of measurements depending upon a parameter, for instance Resistance of an electrical wire against length of wire, then our data can be modeled by mathmatical function whose values depend on a set of fitting parameters. Varying the parameters can bring the model into agreement with our data. The process of varying the model parameters is known as fitting.

0 5 10 15 20 25 300

5

10

15

20

C Linear Fit of DATA1_C

f(x)

X

Fitting Data to a Straight LineConsider an experiment to validate Ohm’s Law. We measure the voltage across a resistor as a function of current. We have an error due to machine tolerance in our measurement of voltage and current and we decide to plot a graph

0 2 4 6 8 100

20

40

60

80

100

Experimental Data Linear regression

Vol

tage

[V]

Current [A]

Y = B * XWeight given by Data1_E error bars.

Parameter Value Error-------------------------------------------A 0 --B 10.19094 0.08383------------------------------------------

R SD N P------------------------------------------0.9994 0.82239 10 <0.0001------------------------------------------

Clearly the gradient of the graph is the resistance. The fitting programs used provide error estimation for the parameter B of the straight line. But what happens if you have no program. Consider the graph again

0 2 4 6 8 100

20

40

60

80

100

Experimental Data

Vo

ltag

e [V

]

Current [A]

y=79.9

x=8.0

x=6.1

y=64.6

y=61.7

x=6.7

B=9.99B=10.59B=9.21

The black line is the best line drawn by eye and passing the closest to the experimental plotsThe red and green lines are the worst lines that still pass through the error bars. They give an estimation of the error in the gradient.

B=9.99±(10.59-9.21)/2 =10.0±0.7 Ω

Compared to the fitting program this is an over estimation.

SummaryPropagation of errors: 2

2

2i

i ii x

x

FfxFf

Statistical Errors: ix

N

iix

Nx

1

1

N

ii xx

N 1

21

Nx xx /

Measurement set of supposedly identical measurements

Averaging

Error in average. Note σx can be replaced by machine error