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




Module 2 – Atomic Physics



Wave bath
Young’s Slit Experiment
Interferometry
Franck-Hertz Experiment
Photoelectric Effect
Module 3 – RCL Circuits


Linear response in time domain and Frequency domain
Fourier transform
1
2
3
Introduction

4
5
6
7
8
9 10 11 12 13 14 15
Presentations!!
Presentations!!
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!!!!!

2 weeks to prepare your report.

Final mark is given on the 5 best reports out
of a total of 6
http://labwrite.ncsu.edu
Safety
1. Special care should be taken to avoid unintentional
reflections from mirrors
2. 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?
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
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.
BAD EXAMPLE
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
GOOD EXAMPLE
The Conservation of Linear momentum
“In an elastic collision the total momentum before and after are equal”
m1v1
m2v2
Fbefore   mi vi  m1v1  m2v2
i
Fafter   mi vi  m1v1  m2v2
i
Fbefore  Fafter
m2
v2  v2 
v1  v1 
m1
Prediction
Repeatable
of objective
results
quantities
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
Measurement
How do we measure a quantity?
This may sound like a stupid question but
understanding the limits of measurement is
the heart of experimentation
A
B
Uncertainties in aligning the ruler lead to ERRORs in
our measurement.
Errors and their Analysis

Systematic errors
An error that remains constant through out an experiment and cannot
therefore be detected. For instance:

Hardware
Lets 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!
px  


Machine tolerance
Random Errors
measuring the area of a plate 2.5 cm x 5 cm to an accuracy of 100 μ2
w
l
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
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 tolerance
of the measurement
device
x  x [units]
x
In terms of fractional uncertainty
x
For example our ruler has a tolerance of…?
x  1 / 32 inch
Comparison of two measurements
Consider our earlier example of linear momentum
Uncertainties are additive
m1v1
p1
p1 [kg∙m ∙s -1] p2 [kg∙m ∙s -1]
±0.04
±0.06
p2
1-p
2=0
p1p-p
2
1.49
1.56
-0.07
2.10
1.16
2.12
1.05
-0.02
What is the uncertainty
in the difference?
0.11
 p1  p2   p1  p2 
p1  p1
p2  p2
The propagation of uncertainties
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?
f  F x, y, z,...
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
f  f  F x  x
On the understanding that δx is small compared to x
we can expand F(x) as a Taylor’s series
F
F  x  x   F x  
x
x
If we consider a further parameter, y, then
F
F
F x  x, y  y   F x, y  
x 
y
x
y
In other words our result, f=F(x,y,..), has an error estimated by:
F
F
f 
x 
y
x
y
This result works very well for sums and multiplication of
errors. However it is limited for a more general class of
functions.

F  Fx xˆ  Fy yˆ
F
F

xxˆ 
yyˆ
x
y
 
F  F  F
2
2
F
F
2
2

x 
y
x
y
2
F
F
x  Fx
y  Fy
x
y
F(x,y)


F  δFx

F


F  δFy
 
F  δF
y
δx
x
δy
The General Expression for the
Propagation of Error
f  F xi 
f  
2
i
2
F
2
xi
xi
Examples
Box Volume
V  xyz
V 
 yz  dx  xz dy   yx dz
2
2
2
2
2
2
Statistical Treatment of Errors
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?
xi 
1
x
N
Measured set of values
N
 xi
The best estimate
i 1
1 N
2
xi  x 


N i 1
Standard deviation
If the measurements are normally
distributed then there is a 70% chance that
a measurement will be within σ of the
actual value
The Standard deviation of the mean
From our set of measurements, xi , the average x represents our best guess
of the correct value. What is the error in this estimate?
x
10
x   x   x / N
8
6
f(x)
For example if we measured the spring
constant of a spring 10 times and got
the following readings
4
x 
x 
2
0
-2
0
2
4
6
8
10 12 14 16 18 20 22 24 26 28 30 32
k  85.9 N / m
  1.9 N / m
δk=0.6 N/m
x
N/m
k
86 85
84
89
86
88
88
85
83
85
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.
20
C
Linear Fit of DATA1_C
f(x)
15
10
5
0
0
5
10
15
X
20
25
30
Fitting Data to a Straight Line
Consider 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
Experimental Data
Linear regression
Y=B*X
Weight given by Data1_E error bars.
100
Parameter Value
Error
------------------------------------------A
0
-B
10.19094 0.08383
------------------------------------------
Voltage [V]
80
60
R
SD
N
P
-----------------------------------------0.9994 0.82239 10 <0.0001
------------------------------------------
40
20
0
0
2
4
6
Current [A]
8
10
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
Experimental Data
B=9.99
B=10.59
B=9.21
100
x=6.7
Voltage [V]
80
y=61.7
60
y=79.9
y=64.6
40
The black line is the best
line drawn by eye and
passing the closest to
the experimental plots
The 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.
20
x=6.1
x=8.0
0
0
2
4
6
8
10
B=9.99±(10.59-9.21)/2
=10.0±0.7 Ω
Current [A]
Compared to the fitting program this is an over estimation.
Summary
f  F xi   f 2  
Propagation of errors:
i
Statistical Errors:
xi 
2
F
xi 2
xi
Measurement set of supposedly identical measurements
1
x
N
N
x
i 1
i
Averaging
x   x   x / N
1 N
2



x

x
 i
N i 1
Error in average.
Note σx can be replaced by machine error