Download General Instructions

General Instructions
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Introduction
Read the general instruction carefully at the beginning of the course. It tells you about how to
perform the experiments and how to write the lab report. All the manuals for the individual experiments are available in the departmental portal. The manuals elaborately explain both the theory
and the principles behind the experiments. Go through the manual of the particular experiment
you are going to perform before coming to class. This will help you enjoy the experiment.
Use a hard bind laboratory note book with one side ruled and other side plain (no loose pages)
for the lab report. Do not tear off any page from the lab note book. If you find that a particular
reading is taken wrongly and you need to discard that page, strike off the page saying discarded.
A good lab record book also keeps the records of what went wrong during the experiment.
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How to write the lab report
Here are the things that you need to write in your lab report for all the experiments.
Title
Title of the experiment, your name, roll no, the date you have performed the experiment.
Aim of the experiment
In one sentence or two what is the aim of this experiment.
Apparatus used
List the apparatus used.
Working formula
Working formula clearly mentioning what are the different symbols in it with proper
units. Use only SI units. If you are using a circuit, draw the circuit diagram.
Measurements
Tables of the measurements as given in the individual manuals for the experiments. Fill
up the tables as you do the experiment. If there is a mistake and you have to redo
the experiment, do not remove the previous table but strike through it. Make a new
table and write down the new measurements. You need to get the readings checked and
signed by the lab instructor as you take the readings.
Calculations
Necessary calculations and graphs if required. Always mention the units of the different
quantities. (see section 3)
Error estimation
The maximum possible errors of the individual measured parameters in a table and the
maximum possible error in the estimated value(s). (see section 4 and 5)
Result
Quote result with proper unit. Mention the number of significant digits of the measurement. (see section 6)
Discussion
Discussion of the result that you have got, possible source of systematic errors and if
there could be any improvement of the experiment to reduce them.
Make your lab report clean and easily readable.
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Figure 1: The measured values of the current for the different voltages provided by the battery is
shown with blue dots. The red line represents the best fit straight line assuming Ohm’s law. The
estimated value of the resistance is shown.
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Statistics of the measurements
Here we shall discuss about the statistics of the measurements. The question we address here is the
following: We have used some apparatus to get some values of the quantities we have measured.
How do we interpret these measurements. There is a whole lot of statistical theories behind this,
we shall discuss it in very brief here. Interested students may go through the references given at
the end.
Let us use an example of measuring a resistance in a circuit. We perform the experiment in
the following way. We keep the resistance, an ammeter and a variable battery in the circuit. The
battery is such that it can take several cells of 1.5 volts. You can use either one cell, or two cell etc
to produce a 1.5 volt or a 3 volt battery respectively. Let us assume that the set up can take up
to 5 cells to produce a total of 7.5 volts. In the experiment you change the voltage produce by the
battery by inserting the cells successively and measure the current in the ammeter for each case.
We plot the measured current for each value of battery voltage. In an ideal case, the measurements
supposed to lie in a straight line following Ohm’s law. From the slope of the, line we can estimate
the value of the resistance R,
∆V
V = IR,
= R.
(1)
∆I
The method of estimation of parameters following a relation between the measured quantities is
known as the parameter estimation. In this case the parameter is the resistance R. In practice
though, the plot of the measured values look like Figure (1). Clearly, the points do not lie in a
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straight line, but on an average, we may make a straight line to pass through them (giving justice
to all the measurements). This is called the best fit straight line (shown in the figure). There
are many methods to estimate the best fit line, we have briefly outlined one of these methods in
section (6) which you will be using. To calculate the resistance in the circuit, we use the slope of
this best fit line. We call this the estimated value of the measured parameter R. In this
example let us assume that we have got the value to be
R = 1.2345 kΩ.
Next we ask, how accurate is this estimation. As in the plot the data points do not lie in a
straight line, it may be possible that straight lines with different slopes will fit the data. Hence,
best fit value we get for the resistance has an uncertainty associated with it. This uncertainty may
arise because of different reasons:
• We know the value of the resistance changes with temperature. May be, while doing the
experiment, the resistance were subjected to different temperatures. Hence the inherent
value of the resistance was not constant during the experiment.
• We assumed that the cells we have used to constitute the battery are all of 1.5 Volt. Usually,
there is a cell to cell variation and the voltage values may be different for different cells. For a
constant resistance, the value of the measured current (for a battery with a different voltage
than what we expect it to) be will be different.
• The ammeter has an accuracy, may be it is sensitive up to 1 mA of current, any change in
current smaller than this will not be recorded.
Whatever the reason may be, we need to estimate the uncertainty in measuring the value of the
resistance. There are several methods to do so. These are outlined in section (4) and section (5).
We call this as error associated with the estimated value. Let us assume that the error is
0.03 kΩ. Hence, the estimated value with errors will be
R = [1.23 ± 0.03] kΩ.
We mentioned in one of the points above that the actual voltage provided by the cells may
be different than 1.5 volt. But does the voltages of the individual cell vary randomly such that
the mean of the voltages of the five cells are zero, or otherwise ? We can check this by doing the
following experiment.
Let us assume that the temperature of the resistance does not vary during the experiment time.
We further assume that by the previous experiment we have already measured the value of the
resistance. We insert one cell at a time in our battery and take five measurements for the current.
In an ideal case, all these five measurements should be same. But we see they vary. As we know
the resistance and have measured the current, we may now find out the voltages corresponding to
each measurements. Let us assume we find them to be
V = [1.48, 1.46, 1.46, 1.47, 1.45] V.
Clearly, all the cells provide voltages less than 1.5 V. In fact, the mean of the five voltages comes
out to be 1.46 V, which is 0.04 V less than the expected value. This is common for the commercially
available cells. The statistical quantity that gives the offset between the expected value and the
actual mean value is known as bias. Here the voltages of the individual cells have a negative bias of
0.04 V. In practice, we try to estimate the bias and subtract it from our measurements. Sometimes,
it is not straight forward to estimate the bias and the experiment is designed in a way that a non
zero bias do not change the estimated values.
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Maximum possible error estimate (using the working formula)
In this section we discuss one of the methods to estimate uncertainty or error associated with the
estimated value in an experiment. In this method we try to estimate the maximum possible error
or uncertainty associated with the estimate of our desired parameter in an experiment. The idea
behind the maximum possible error is to provide an uncertainty with the best measured value in our
experiment, such that, any future measurement of the same parameter with better methodology
and more accurate instruments would lie within the limit provided by the present estimated value
of the parameter ± the maximum possible error.
Let us assume that in an experiment we measure different quantities say X = A, B, C, D etc,
which are related to the quantity we want to estimate E by
E = Aa B b C c D d .
(2)
We measure X = A, B, C, D individually and then using above equation we estimate the value of
E. Let us assume that the maximum possible error in measuring any of the X be ∆Xmax , then the
maximum possible error in E is given by
∆Amax
∆Bmax
∆Cmax
∆.Dmax
∆Emax = E | a
|+|b
|+|c
|+|d
|
(3)
A
B
C
D
To estimate the individual ∆Xmax , we note the following:
• We use an instrument to measure the quantity X. The instrument may have an accuracy of
∆X. Let us assume that we have performed five measurements. If the linear dimension you
are measuring does not vary across these five measurements, then the maximum error will be
given by the accuracy of the instrument. If the different measurements give different values,
then the maximum error would be given by the difference in the two extreme measured values.
• To measure the quantity X, we may need to take two measurements with the same instrument.
Then the maximum possible error in measuring X would be given by twice the least count of
the experiment.
For example if we are measuring the length of a metal bar using a meter scale, the accuracy
will be 1 mm. The length of the metal bar usually do not vary more than 1 mm. Hence, all the
five measurements will give same values. On the contrary, if we are using a slide caliper to measure
the width of a metal bar, the accuracy to which we can measure may be 0.1 mm. The width of
the bar may change at different places more than 0.1 mm and the five measurements will give five
different results. Hence for the case of the length of the metal bar the maximum possible error
would be 2 mm, while for the width of the metal bar, the maximum possible error will be given by
the difference between the two extreme values of the five measurements.
Sometimes we estimate the value of E from a graph. In this case we can not use the above
method. The method to estimate the maximum possible error from the graph is explained in the
next section.
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Maximum possible error estimate (from the graph)
Quite often the functional relation between the measured quantities x and y in an experiment is
either linear or we can find a function of y which is linear in x. In these cases we can plot x vs y and
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find out the parameters those we want to measure from the slope (we call m) and the y-intercept
(we call c) of the best-fit straight line to the data points.
y = mx + c
In an undergraduate lab the plotting is usually done by means of a graph paper. In the next section
we have outlined how to find the best fit line using linear least square regression technique. Here
we discuss how to estimate the maximum possible error form the graph.
Let us assume that we know (see previous section) the maximum possible errors in x and y.
The maximum possible error in x is most often just the least count of the apparatus that is used
to measure it and have fixed values for all x. There can be examples where it will vary. The errors
in y however, most of the cases can be different for different y.
Let us ask, now how can we estimate the maximum possible error on the best fit values of m
and c given the informations above.
To estimate the maximal errors in m and c we proceed the following way. We have demonstrated
the procedure using a mock data set. In our data set we have assumed that the values of x
are changed regularly and the error in x is independent of x. These assumptions simplify the
demonstration.
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• We plot, in a graph paper the values of x and y ( or the linear function of y). These are
shown by black circles in Figure (2).
• We draw rectangles around each of the x − y data point such that the rectangles are centered
at the data point and have x-width twice as the error in x and y-width twice as the error in
y. These rectangles are marked by the black boxes around each data points in Figure (2).
• We find the best fit values of the parameters m and c (see next section). In our example
figure these values are 2.0 and 1.0 respectively. The best fit straight line is shown as a black
line in the figure.
• We now try to find a straight line that has a lowest value of the intercept and highest value
of the slope which at least touches all the rectangles. Though it seems quite difficult to do, it
can be done quite trivially with a transparent ruler. The slope and intercept of this line gives
us the minimum possible value of c, i.e cmin and maximum possible value of m, i.e mmax from
the data.
• Similar to the last step above we can also find the maximum possible value of c, i.e cmax and
minimum possible value of m, i.e mmin by a second straight line. These lines are shown in
the figure as red dashed lines.
• Maximum possible error in m then can be calculated by finding the extremum of m − mmin
and mmax − m, similarly the extremum of c − cmin and cmax − c gives the maximum possible
error in c. The final result of the estimation is shown in the figure.
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Linear least square regression
Following the example in the previous section, let us assume that we are interested to find the
best fit values of the parameters m and c, when the measured values in an experiment x and y are
related by a linear relation
y = mx + c.
(4)
Here we further assume that the measurement errors associated with x and y are either negligible
or we can not estimate the measurement error for x and y. Here we discuss a standard statistical
method to estimate the two parameters m and c from the observed data. The procedure we
outline here is called linear least square regression. This is an example of a more general class
of parameter estimations called the maximum likelihood estimation. Here we assume that the
parameters inherently have some definite values and try to approximate that from the observed
data. We shall not go into the statistical ‘theories’ behind this, we shall rather outline the procedure.
Lets the observations we have of x and y are xi and yi respectively and the actual values of the
parameters be m and c. We measure the mean square deviation of the measured values from the
actual values and call it R.
N
X
R=
[mxi + c − yi ]2
(5)
i=1
Clearly, we can estimate the value of R for a given data if we assume some values of m and c. The
value of R will change, for a given data, with the values of m and c we use to calculate it. For a
given data, R(m, c) will have a minima for the best estimate of the m and c. We differentiate R
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Figure 2: An example of linear least square fit is shown with the best fit straight line and the
estimated values of the parameters with errors from least square fit.
partially with respect to m and c and set that to zero to get:
m=
2
σxy
2
σxx
µx =
2 =
σxx
N
P
i=1
[xi − µx ]2
c = −mµx + µy ,
and
N
P
xi
i=1
N
P
2 =
σyy
µy =
N
P
with
(6)
yi
i=1
[yi − µy ]2
i=1
2 =
σxy
N
P
[xi yi − N µx µy ]
i=1
and N is the number of observed data points. We quantify the efficacy of the fitting by the quantity
r=
2
σxy
σxx σyy
(7)
Expression for m and c given in eqn. (6) and (7) can be used to estimate their best fit
values.
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We can also estimate the errors/uncertainties in the estimated values of m and c as
s
2 σ2 − σ4
σyy
σ
1
µ2
xx
xy
σm =
,
σc = σ
+ x2
where σ 2 =
.
2
σxx
N
σx
(N − 2)σxx
(8)
The Figure (3) shows an example of fitting seven data points to a straight line using linear-leastsquare technique discussed above. The blue points are the sample data points. The best fit line
using the technique is shown in red. The best fit values of the parameters with the error values are
also shown at the bottom of the figure.
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Quoting the result with errors
Let us assume that, by using the above methods, we have estimated the value of a quantity to
be E = 1.234 and its error as ∆Emax = 0.0341. Here we discuss how to quote this result in the
decimal system 1 .
• We first determine the most significant digit in the error. The leftmost nonzero digit is the
most significant digit. In our example, the most significant digit is 3.
• We note the position of the most significant digit with respect to the decimal point, let us
call it as d. We round off the value of the error to the position d, that is ∆Emax = 0.03.
• We round off the value of E to the position d, that is E = 1.23. After doing this the number of
digits in the estimated value E is called the number of significant figure of this measurement.
In this case we have this number as 3.
• We now quote our result as follows
E = [1.23 ± 0.03] Unit,
where E is measured with three significant figures.
References
1. An Introduction to Error Analysis: The Study of Uncertainties in Physical Measurements by
J. R. Taylor, University Science Books, 2nd Edition.
2. Statistical Treatment of the Experimental Data: Lecture Notes Collection by H. D. Young,
Mcgraw-Hill Book Company, available online from
https://archive.org/details/H_D_Young__Statistical_Treatment_of_Experimental_Data
3. A Studentś Guide to Data and Error Analysis by Herman J. C. Berendsen, Cambridge
University Press (2011)
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Decimal system is what we use in all practical cases. Interestingly, the way you quote the result depend on which
number system you use. I leave it to you to think about it.
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