Graphing

[Brock369]

I'm having troubling with finding the function of the table below. I'm sure that it can be solved using calculus...
Let Length/Diameter be the x values, and H the y values. Having (0.70,0.47) being the first pair, and (50,5.8) the last.

 

[topsquark]

What have you tried? I strongly suspect that if you graph the data points it might give you a clue.

-Dan

 

[Dr.Peterson]

I put the first column of these into a spreadsheet, and it looks linear. You can either do as I did, and do regression (have Excel find a trendline), or just pick two points and find the equation of the line they determine.

[Edit]
I entered the rest of the data, having noticed odd things at the end, and it's only approximately linear down to about 1.5 or so. What you may want to do is to find that line, then subtract that quantity from each data point and see if you can approximate the residue by another simple function.

We'll need to hear back from you about what you've tried in order to make any more suggestions.

 

[Brock369]

What have you tried? I strongly suspect that if you graph the data points it might give you a clue.

-Dan

I have graphed it, have worked it out mathematically, have even used a calculator, and it is not linear, nor quadratic.

 

[Brock369]

I put the first column of these into a spreadsheet, and it looks linear. You can either do as I did, and do regression (have Excel find a trendline), or just pick two points and find the equation of the line they determine.

[Edit]
I entered the rest of the data, having noticed odd things at the end, and it's only approximately linear down to about 1.5 or so. What you may want to do is to find that line, then subtract that quantity from each data point and see if you can approximate the residue by another simple function.

We'll need to hear back from you about what you've tried in order to make any more suggestions.

It is not linear or quadratic... I have used a few calculators after giving it a stab, and cannot find anything.

 

[lev888]

Looks like 1/x shifted up.

 

[Dr.Peterson]

I found that it is modeled reasonably as a straight line (something like 0.11x + 0.22, but moved up a little further so that all residues are positive) plus a power, not x^-1 but around 0.26x^-0.44. (There are several things to play with to improve the fit.) Your version of the graph looks almost like a circle near zero.

The next question is, what is it that you are measuring, so that an analysis of the system might explain the apparently hybrid nature of the relationship. It makes me think of a weak force and a strong force.

 

[Brock369]

I found that it is modeled reasonably as a straight line (something like 0.11x + 0.22, but moved up a little further so that all residues are positive) plus a power, not x^-1 but around 0.26x^-0.44. (There are several things to play with to improve the fit.) Your version of the graph looks almost like a circle near zero.

The next question is, what is it that you are measuring, so that an analysis of the system might explain the apparently hybrid nature of the relationship. It makes me think of a weak force and a strong force.

The graphing and table is simply showing coil capacitance, with consideration of length to height ratio (x), matching to a specific value. The table says it all... I just need to find the function between the length/diameter column, to the H value column.

 

[topsquark]

Perhaps this link will help. It even has the exact same data set that you listed. Creepy, huh? ?

It would help in the future if you told us all of the information in the problem. Please do so next time.

What I'm seeing here is that there is a linear relationship to the capacitance (as per the web page) that is linear. What I don't understand is why there is a "tail" from the first few data points. Probably the linear relationship is breaking down in this region so the approximation \(\displaystyle C_s = HD\) due to a ptich that is too large.

-Dan

 

[Dr.Peterson]

I wouldn't say "the table says it all"; that's not how science works. Data alone can't tell you the formula behind it; typically, we need a model to explain the data and suggest an appropriate form even for an empirically determined formula (and the data then confirms or corrects the model). I imagine there are competing effects here that account for the different behavior for large and small inputs.

It looks like you're getting your data from a source similar to this: https://www.teslascientific.com/products/coil-capacitance-calculator/

I took some hints from that page, and found this paper that includes (page 18) "Medhurst's formula",

CL / D = 0.1126 (ℓ/D) + 0.08 + 0.27 √(D/ℓ ) [ pF / cm ],​

which produces data very close to your table when I add it to my spreadsheet. This, or some variation on it, may well be the source of your H. I haven't tried to read through the paper, but it may show you both how the model was arrived at, and why it is not quite right. (Nothing ever is.)