Magnetic field experiment confusion with error calculations

[Taylor_1989]

Homework Statement

I am having a issue with calulating my errors for this particular experiment the reason I will detail below. I have also print sceend in the section of my lab report to show the experimental setup etc.

Lab Script

So in the lab during the experiment the tesla meter could neve be zero and this was due to interferce, so the readings would go from 0.00mt to 0.03mt so we took an average, and this was the same for all the readings. This is shown more clearly in my table of results:

So of the averages have been rounded or auto corrected by excel which I will need to adjust.

So my throughts are this is a random error and not a systematic as I can't reproduce the same results each time, we did take a repeat of the first couple of values a did get differnt readings for the average. So is it possible to use a SD to calulate the error on this, or is this inccorect? I would then plot this as the Y error bars

My second issue is what they mean but "Compare with theoretical estimates". Dose this mean just take the equation:

$$B=\frac{\mu_{0} I}{2R} [1]$$

And sub in the current 0, 2,...16 with the assumtion there is no error on the equipment, or dose it mean with the error on the equipment if so, I would have to propagte the errors, which would be anouther set of Y axis error bars. But as we have been told to plot on the same axis the error bars would overlap, so I can't see this being correct.

Here is my graph without any error bars at the moment:

I am also having the same issue with the second part of the experiment, where we keep the amp the same at 16A and move the detector along the x-axis in the postive direction by 10cm and the negitive direction by 10cm. Beacuse now in my mind I have an uncerity on the ruler and the fluxuation in the magnetic field. So how can I plott the error bars for this type.

One of my ideas was to do the following.

Calulate the SD on the magnetic field then use the equation:

$$B=\frac{\mu I R^2}{2(R^2+x^2)^{3/2}} [2]$$

and then use propagation with the uncerity on the ruler as $0.5cm$ and combine the two, but then thinking a little more I don't think this would work.

The equation we are told to use for our error calulations is the

$$(\delta f)^2=\left(\frac{\partial \:f}{\partial \:x}\left(\delta \:x\right)\right)^2+\left(\frac{\partial \:f}{\partial \:y}\left(\delta \:y\right)\right)^2 [3]$$

where delta x and y are my estimated uncerity, i.e my ruler being 0.5cm beacuse i did not repeat the give measument.

Just to clarify, we had a very quick and very brife course in stats so much so, that my knowledge is very limited and I am trying to learn as I go.

Homework Equations

The Attempt at a Solution

 

[mfb]

The estimated standard deviations from your sets of measurements should work as statistical uncertainty for the data points.

Your current and magnetic field measurements won’t be exact but it is hard to tell how uncertain the measurements are. R has an uncertainty as well.

Your plot is a comparison with theory - but if it looks as it does you should comment on the discrepancy (whixh is clearly not from statistical uncertainties on individual measurements).The formula you have for error propagation is fine for the second part.
If your ruler has 0.5cm uncertainty, throw it away and get a better ruler.

 

[BvU]

Hello Taylor,

Nice experiment. I couldn't find BiotSavart_2014.pdf, but http://www.21dianyuan.com/_1341/download_bbs_attachment.php?id=20188 seems to be the same thing -- and is a lot better legible.
Not so clear to me what 'highest to lowest' stands for.

I also think that

So in the lab during the experiment the tesla meter could never be zero and this was due to interference

is a premature hypothesis that you seem to treat as a conclusion.

When I look at your results and let excel draw a straight line through all the points 'magnetic field Average', I see a very straight line:

and if I leave out the first point

the line is so perfectly linear, it's astonishing !

I also do not understand what you mean with 'auto corrected' in

So of the averages have been rounded or auto corrected by excel which I will need to adjust.

Does Excel know about the linear relationship ? It should not !

Did you actually calculate these averages from sets of observations ? Some of the averages are very close to an extreme value :

Even if I use the extremes as data points a nice line is the result:

Then: your theoretical relationship is $B = \pi I$ 10^-5 for the field from the 4 cm loop only. Did you notice your lab instructions start with a picture of the Earth magnetic field ? That will have been the same for all measurements: a systematic deviation, not 'interference'.

You measured for currents from 0 to 16 A. Can you still improve your result by measuring from -16 to 0 A as well ? Who knows you may be able to deduce how strong the Earth magnetic field is at the center of your loop -- but then you need to find out the direction as well, and link the result to the orientatioin of your setup.

Will come back to the second experiment later.

 

[BvU]

Still about the first experiment: as mfb makes clear, in the theoretical $B = \pi I$ 10^-5 T there is a factor $1\over R$ common to all observations, so if 2R = 0.04 m has a 1 mm uncertainty you are stuck with a 2.5 % systematic error in your comparison with the theoretical result !

Even though the pdf is very legible, I overlooked the background magnetic field compensation feature: how come you end up with a non-zero magnetic field average for zero current ? Or did you (wisely) not use that feature and that way are able to determine a magnitude of 0.1 gauss for a component of the Earth magnetic field ?

 

[Taylor_1989]

Even though the pdf is very legible, I overlooked the background magnetic field compensation feature: how come you end up with a non-zero magnetic field average for zero current ? Or did you (wisely) not use that feature and that way are able to determine a magnitude of 0.1 gauss for a component of the Earth magnetic field ?

I orginally did it to find the Earth's magentic field, which would be the systematic error but, when doing the other mesurments it is not constant, this is what tripping me up a bit as well, I am still not sure how to plot the y bar error bars.

Looking at the graph you displayed, which I am most greatful for that you took you time to do it, so thank you, I see you have taken the minum vlaue that I messured on the tesla meter and the maximum value. Is this correct?

Did you actually calculate these averages from sets of observations ? Some of the averages are very close to an extreme value :

I did get the average from a set of observations here are my full observations, with the average results. This is just for when x=0

and my graph with the corrected average

My throughts at this moment in time are as follows.
Now beacuse I took an average from the set of observed reults, and just to clarify, the way the observed reults were obtained were as follows, beacuse we are only give at total of 9hrs and there are 4 sections to this lab we decied to say right we time 30 seconds and see what the highet value and lowest value appera on the tesla meter, now these value did repeat within the 30sec thare were not just seen once, so an example would be, for 0 current we timed 30 seconds and

0.00 apeared x3
0.01 appread x2
0.02 appered x4
0.03 appered x3

I did ask my demostartor should we not take the value that appred most times he said just take the average from highes lowest, so that what I did. Now to me beacuse I have take an average from a set of measurments I should be using a statical approch to calculate my error bars but, at the same time I have to consdier the fact the Earth's mag field at the surface. Which i belived you perviously mentioned was a systemtic error, which I understand. Now I did read in a stats book that you can combine the two by doing the following:

$$(T_E)^2=(E_{stat})^2+(E_{sys})^2$$

But then if I were to use the approach for the second part and use

(δf)2=(∂f∂x(δx))2+(∂f∂y(δy))2​

this equation which is really only used for random errors I have the issue where I have a radom error, from taking the average values from the tesla meter, I have the systematic from the Earth's magnetic field and I now also have the error in my measument made when adjusting the dector along the x-axis.

Just to make this a little clearer I will post my results for the second part for the 40mm

Mag Vs Postion: +x

Mag Vs Postion:-x

The graph of Experimental vs Theretical

So I don't really no how to evalulate the error when to me there are two random and one systematic. Also for some reason we were told not to use log plots, I do not know why.

 

[Taylor_1989]

Ok so I have been trying to read up as much as possible today, looking at other reports similar to mine experiment and reading up about stats from various sources and I have done the following to calculate the error in the magnetic field like so, if possible could someone please advise me to tell me if I am even close to being correct:

So for the magnetic field I broke the random error into there componets: which were as follows:
The uncerity from the average results recorded and the uncerity in the amp meter reading (resolution uncertainty), I then also calculated the uncerity on the tesla meter (resolution uncertianty).

My calculations as follows

1. Taking the average value of mag field for each amp
2. I then found the range
3. Divided the range by 2
4. The divied the value in 3, by the square root of how many values I recorded.

So an example of my calculation would be for my first measurment:

So for ecach section I did the following:
Average Value mT
$$\frac{0+0.01+0.02+0.03+0.04}{5}$$

Range
$$0.04-0.00$$

Range/2
$$\frac{0.04-0.00}{2}$$

Unceritanty mT
$$\frac{0.02}{\sqrt{5}}=\Delta B_1$$

For the resolution uncerity I did the following:
On the tesla meter we had a value to 2dp so I did $\frac{0.005}{\sqrt{3}}=\Delta T$

On the Amp meter we had a value to 1dp so I did $\frac{0.05}{\sqrt{3}}$

So once I had calulate these I want to find the uncerity on the mag field due to the uncertainty in the current to which I did the propagation equation, but as there only one varible and all the rest are constants I just took the deriavte with repect to I and muliplt by uncerity I calculated in the amp meter:

$$\frac{d}{dI}\left(\frac{\mu _{0\:}\cdot \:I}{2R}\right)\cdot \Delta I=\frac{\mu _0}{2R}\cdot \Delta I=\Delta B_2$$

Also as a side note the R is constant as I do not know what they measured the radius so I have assumed no error for now.

I then took all these unceritainces and combined them to make a total uncerincerty in the magnetic field

$$\Delta B_T=\Delta B_1+\Delta B_2+\Delta T$$

My graph with now look of the forum

Have I gone about this correctly?

 

[BvU]

Still about experiment 1:

my full observations

Still trying to understand. Looks as if this is already ordered somehow and to me it seems as if you did not use all of your information

0.00 appeared x3
0.01 appeared x2
0.02 appeared x4
0.03 appeared x3

(no 0.04 ?)
because this would give you an average 1.58 with an estimated error of 0.34 ( estimated sigma / sqrt(N) = 1.16 / sqrt(12) )

Your 0 A 'average' says 2.00 -- as you tell, obtained from (0 + 4)/2So one can not conclude that e.g. in the 10 A series the measurements 31, 32, 33, 34, 35 and 36 all have the same weight.

And it is strange that the 8 A measurement does not stick out in post #1, but it sure does in post #6. What happens here ?

Nevertheles I did regression (in https://pdfs.semanticscholar.org/0815/66656b9e5975e7034750e641c2e81ae027a7.pdf: Data tab | Data Analysis | regression) with your 'full observations', just to see what the error in the slope could be. The result was a slope 3.16 $\pm$ 0.06 (so 2%) and an intercept 1.7 $\pm$ 0.6 . I expect that the error is smaller when you include the frequencies of the 'full' data points.

So for the magnetic field I broke the random error into there components: which were as follows:
The uncertainty from the average results recorded and the uncertainty in the amp meter reading (resolution uncertainty), I then also calculated the uncertainy on the tesla meter (resolution uncertainty).

You make a plot for $B$ as a function of $I$. Uncertainties in $B$ end up in the vertical eror bars, in $I$ in the horizontal error bars. Usually one of the two is a lot smaller than the other (linear regression usually assumes the error in $x$ can be neglected). A calibration error in the current meter is NOT a statistical error: it is common to all measurements !
The background from the Earth field is constant; the fluctuations in the readings of the Teslameter may come from stray fields in your lab (mains, power cords, etc.). These make that your equipment (that probably digitally samples some Hall voltage) shows different readings. It is not a very good idea to then work with the extremes of the observed values (they are noise) - an average of relative frequency $\times$ value would be better. But it isn't a disaster either.

There are a number of systematic errors in your experiment:
current meter calibration -- probably can be ignored
loop diameter -- probably the biggest contribution. Optimistic estimate 0.5 mm, so around 1% for the 40 mm loop.
position of the sensor -- as you see in the second experiment it's important to be 'right on top of the peak' -- hopefully negligible (?)

And the statistical errors
mainly the error in the B field measurement that you reduce by doing a large number of observations at various current settings
current meter readings -- hopefully negligible

All in all you have an experimental result (according to my example) of 3.16 with a statistical error of 2% (maybe less if frequencies are used) and a systematic error of 1%.

You an report this as "3.16 $\pm$ 0.06 (statistical)$\pm$ 0.03 (systematic)" as is done often in high energy physics , or you can combine the errors:
since they can be considered to be independent, adding in square gives 3.16 $\pm$ 0.07 (with the proper dimension/units etc.). Not bad at all !

from here:

Many times you will find results quoted with two errors. The first error quoted is usually the random error, and the second is called the systematic error. If only one error is quoted, then the errors from all sources are added together. (In quadrature as described in the section on propagation of errors.)

 

[Taylor_1989]

Ok let me clear some stuff up, I appologise for my way of explaining things so far.

Still trying to understand. Looks as if this is already ordered somehow and to me it seems as if you did not use all of your information

So what happens was this at the very begging of our experiment I count how many times in the 20seconds we saw certain values. I then ask my demonstrator is this was correct, he said don’t worries about building a frequency table for the results as we only have 9hrs to complete a total of 4 section of this lab, so I have to assume that all the values are of equal weight which is why I used the method to calculate the uncertainty on the magnetic field. After I posted my last post I did come to relaise that the uncerity in the amp meter etc are not statitical, the problem that was throwing me was that the uncertainty in the amp meter is $\pm 0.02a$

And it is strange that the 8 A measurement does not stick out in post #1, but it sure does in post #6. What happens here ?

For this I do not know why in post one it did not have the same value, I have re-done the graph and get the same results as post 7, maybe it was an input error by me.

As perviosuly mentioned I have a hard time differentiating between random errors and systematic.

So for example as in this experiment we use a ruler to measure the distance the probed moved along the x-axis so in my mind I think this is a random uncertainty because my interpretation and best of the measurement could be $\pm 0.5mm$, so say I moved the probe from 10cm to 11cm and took the following 10.5,11,11.5,10,10,5 etc and workout the SD then, the SD should be 0.5.

However my point of confusion is this because I am not actually measuring it multiple times I am just moving the rod from 10cm to 11cm it dose not matter how many times I move it I will still believe it to move 1cm so surely this means that this is a systematic error as the error will be the same thought all my measurements along the x-axis

Also as seen from my second graph I think what has cause the massive peak in my grpah, I keep thinking that it is to do with the postion of the sensor , but that can't be true as the mag field is always strongest at the centre of the loop, so I am not sure what has caused the massive fluxuations. I mean it could be down to the postion of the probe along the x-ais that would contribute to some but not as big as that, i should not think

 

[haruspex]

believe it to move 1cm so surely this means that this is a systematic error as the error will be the same thought all my measurements along the x-axis

If you were measuring a distance that happens to change, the error would be neither purely random nor purely systematic. If the variations in the readings are large compared with the precision of the scale then you could treat it as random, but suppose e.g. the actual values are bunched between 5.4mm and 6.1mm with a granularity of 0.5mm (i.e., you read the scale to the nearest mm). Then there would be some systematic error, since only values between 5.4mm and 5.4999mm would round down to 5mm, while everything from there up to 6.1mm will round to 6mm. On average, you will round up by 0.1mm.
There may also be a systematic parallax error in the way you read the scale.

But I understand that here you are positioning something at whole multiples of the scale granularity. There will be a small random error from the limits of your dexterity, but again a chance of some systematic parallax error. Which dominates depends your characteristics as operator.