Talk:Rutherford scattering experiments/Archive 2

Archive 1

Archive 2

Maximum deflection in collision with an electron

@Johnjbarton and Headbomb: What do you think of this stuff?

In his treatment of beta particle scattering, Thomson provided the following equation for how a beta particle might be scattered by a single atomic electron:

$\tan {\frac {\theta }{2}}={\frac {kq_{\beta }q_{\beta }}{bm_{\beta }v^{2}}}$

where m_β and q_β are the mass and charge of an electron or beta particle. We will replace m_β and q_β with m_a and q_a and, in not assuming the atomic electron has infinite mass due to atomic binding, we account for conservation of momentum:

$\theta =2\arctan \left({\frac {kq_{a}q_{e}}{bm_{a}v^{2}}}\cdot {\frac {m_{e}}{m_{a}+m_{e}}}\right)$

Kurzon (talk) 18:58, 1 July 2024 (UTC)

In his 1911 paper Rutherford writes:

$$\cot {\frac {\phi }{2}}={\frac {2p}{b}}.$$

Inverting this formula and replacing Rutherford's variable for the impact parameter ($p$) with yours, $b$, while substituting for Rutherford's b:

$$b=-{\frac {NeE}{{\tfrac {1}{2}}m_{\alpha }v_{0}^{2}}}.$$

gives

$$\tan {\frac {\phi }{2}}={\frac {NeE}{bm_{\alpha }v^{2}}}.$$

This looks like your first formula, but there may be a factor of 2 for the reduced mass in electron-beta collisions:

$$\mu ={\frac {m_{\beta }m_{e}}{m_{\beta }+m_{e}}}=2m_{e}$$

For collisions between alpha and electron, $m_{\alpha }$ is replaced by:

$$\mu ={\frac {m_{\alpha }m_{e}}{m_{\alpha }+m_{e}}}\approx m_{e}$$

so your second equation seems incorrect to me. Johnjbarton (talk) 19:28, 1 July 2024 (UTC)

@Johnjbarton: So it really should be

$\theta =2\arctan \left({\frac {kq_{a}q_{e}}{bm'v^{2}}}\right)$

where $m'={\frac {m_{a}m_{e}}{m_{a}+m_{e}}}$

That's what I think it says in Heilbron's paper on page 270, but I get weird results when I punch that formula and values into Desmos. I get a scattering angle of 179° (when b = 7×10⁻¹⁵ m).

Kurzon (talk) 19:45, 1 July 2024 (UTC)

Rutherford writes his equation as a ratio of impact parameter (his $p$) to the minimum approach distance (his $b$):

$$\cot {\frac {\phi }{2}}={\frac {2p}{b}}.$$

That choice was not an accident. The ratio amounts to measuring the impact parameter in units of the minimum approach distance, so much easier to think about.

For electron + alpha, from the formula

$$r_{\text{min}}={\frac {1}{4\pi \epsilon _{0}}}\cdot {\frac {2q_{1}q_{2}}{mv^{2}}}$$

the minimum approach will be 7200 times larger for the ratio of alpha and electron mass but 79 times smaller for the charge ratio.

His minimum approach was 3.4 x 10^-14, so the new minimum is about 300 x 10^-14. If your impact parameter is 0.7 x 10^-14, the ratio is very small, and thus you get 179 degrees (see the Rutherford's table). You basically hit the bullseye and got direct backscatter.

The kinetic energy of an electron at the same velocity as an alpha particle is 7200 times less, and the potential energy due to charge difference is only 79 times less. So the electron can't get as close to the alpha particle as the alpha particle can get to the nucleus. Another way to say this is that the cross section for the electron is large. The difference in mass means that the electron recoil is huge, the alpha particle basically plows through and the electron gets blasted off. Johnjbarton (talk) 21:51, 1 July 2024 (UTC)

That kinda sounds like what I put in the article that you criticized. The electrons are so light compared to the alpha particle that they get blasted out of the way and therefore have negligible impact.

OK, so what should I go with? Kurzon (talk) 00:34, 2 July 2024 (UTC)

I suggest putting the Thomson scattering discussion in the plum pudding model article.

Use Thomson/Heilbron for beta-electron scattering. Use Beiser/hyperphysics for alpha scattering from positive sphere since Thomson evidently is silent on this subject. That directly eliminates many of my complaints on this article. Johnjbarton (talk) 01:55, 2 July 2024 (UTC)

But what about alpha scattering by the atomic electrons? Kurzon (talk) 02:09, 2 July 2024 (UTC)

Rutherford explicitly ignores this effect on the alpha particle scattering, citing Thomson's work that any single encounter results in small angle scattering. Thomson's results were for beta particles with even less momentum than alpha particles. Rutherford's assumption is ultimately justified by his success in explaining the small but not insignificant large angle scattering. This is the key to Rutherford's paper -- large angle scattering is not insignificant as assumed by Thomson -- and that is why the Geiger-Mardsen experiment is so much the focus of modern explanations.

That is core to my complaint with the use of the Thomson model in an article on Rutherford scattering. The fact that the Thomson model gives only small angle scattering is only in support of ignoring the electrons: a big deal is made of that part of the model that Rutherford completely ignores.

I did add a section to Rutherford scattering based on your question here. Johnjbarton (talk) 15:37, 2 July 2024 (UTC)

So for this article I should say "Here is a scattering of a beta particle by a single encounter with an electron. It is trivially small. Since alpha particles have thousands times more momentum, alpha particle scattering by electron collisions will be even smaller, and there is no need to go into the math for that". Kurzon (talk) 16:50, 2 July 2024 (UTC)

I suppose I should go with the conservation of momentum approach in the Beiser textbook. Kurzon (talk) 00:53, 2 July 2024 (UTC)

"integral above has three unknown variables"

In the integral above the subject phrase, dt is not an unknown. using capital R for a variable is not standard notation. the integral would be much clearer if you write the radius and angle as functions of time. the steps which follow convert to a polar coordinate form, which is where standard treatments start. Johnjbarton (talk) 18:57, 2 July 2024 (UTC)

I have made several attempts to fix the math content in the scattering sections to match the textbook reference that this derivation seems to be based on:

• Beiser (1969). Perspectives of Modern Physics, p. 109

Note that this is the ref that was used by the Hyperphysics site. However that site attempts to condense the entire derivation down to one slide. The missing parts have been filled in incorrectly.

The content is still not correct but @Kurzon keeps reverting my changes. I'm done with this. Johnjbarton (talk) 15:08, 14 July 2024 (UTC)

Ok I will take a closer look at the Beiser book. Kurzon (talk) 18:51, 14 July 2024 (UTC)

The only thing I reverted was you writing R as R(t). I don't feel it's necessary and Beiser doesn't do it. I understand it's frustrating to see your edits reverted but this is overreacting. Kurzon (talk) 19:53, 14 July 2024 (UTC)

The presentation was incorrect about exactly the integration variable. Making the functional dependence explicit is the best way to avoid this. Johnjbarton (talk) 21:34, 14 July 2024 (UTC)

$$\int \limits {\frac {kq_{a}q_{g}}{R^{2}}}\cdot \cos \varphi \cdot \mathrm {\mathrm {d} } t$$

So you're saying that unless you make it clear that R and phi are functions of t, a reader might mistakenly resolve the integral to

$${\frac {kq_{a}q_{g}}{R^{2}}}\cdot \cos \varphi \cdot t+C$$

Is that your complaint? Kurzon (talk) 22:10, 14 July 2024 (UTC)

No, I am saying that an editor may create a version like this one with limits in angles and integration in time. Johnjbarton (talk) 22:14, 14 July 2024 (UTC)

Ah, now I understand. Well spotted. Kurzon (talk) 22:29, 14 July 2024 (UTC)

OK, is it better now? Kurzon (talk) 22:40, 14 July 2024 (UTC)

Make things easy

@Johnjbarton: I don't want to offend you but your way of explaining things is hard to understand. I went to pains to lay out all the steps to make things easy to understand for a high school student. We don't have to be faithful to Beiser as long as we produce something that is correct. Kurzon (talk) 08:10, 18 July 2024 (UTC)

I also do not want to offend, but your version was incorrect and also not easier to understand. Consequently we do need to be faithful to Beiser unless we can agree. My suggestion is that you restore my Beiser based version and then let's discuss what parts you think are difficult to follow and find better ways to explain them. Johnjbarton (talk) 15:25, 18 July 2024 (UTC)

Formatting of math blocks.

Unfortunately math formatting has issues. As far as I understand it, the best compromise for web and mobile is to use

• <math display="block">...</math>

This adds the correct space above and below the math if placed in a paragraph. If extra blank lines are added, extra vertical space appears in the article. I assume that the extra blank lines are to make the math stand out in the editor? Maybe a format like

• <math display="block">
• ...
• </math>

with no extra lines would be useful? Johnjbarton (talk) 22:12, 18 July 2024 (UTC)

NeE

@Johnjbarton: I'm confused about Rutherford use of NeE whereas I used kqQ, with k being the Coulomb constant and the charges expressed in Coulombs. How would you rewrite Rutherford's equation to use modern conventional variables? Kurzon (talk) 18:13, 19 July 2024 (UTC)

I added a paragraph to address this, please take a look.

Unfortunately "modern conventional" for electrostatics depends on where you look and what is "modern". For a long time cgs held the field, then MKS. The SI system was changed as recently as 2019. And these are mostly application or engineering-focused works. Physics theory usually adopts natural units. Rutherford's use of the variable 'b' is similar: by expressing lengths in units of 'b', the formula are simpler and the units only come in one time. Johnjbarton (talk) 21:47, 19 July 2024 (UTC)

If we take e = 4.65×10⁻¹⁰ esu, then NeE with a gold atom is 3.41×10⁻¹⁷. But kqQ, where q and Q are in Coulombs, is 1.82×10⁻²⁶. I don't understand. Kurzon (talk) 22:31, 19 July 2024 (UTC)

I guess that 3.4×10⁻¹⁷ will be in dyne-cm², in CGS units. 1 dyne is 1×10⁻⁵ Newtons (per wikipedia anyway) and 1 m is 1×10² cm so I get 3.4×10⁻²⁶N*m²

For kqQ, where q and Q are in Coulombs, 8.987×10⁹ N·m²/C² * 1.26×10⁻¹⁷ C * 3.20×10⁻¹⁹ C so about 3.7×10⁻²⁶N*m² (numbers copied from article)

Did I mention how great it is to have units only come in one time ;-) Johnjbarton (talk) 23:19, 19 July 2024 (UTC)

Units.

@Kurzon I deliberately used Rutherford's formulas as presented in his paper to ensure verifiability. As we discussed in other topics here, the units for electromagnetism are not standardized universally; adopting any one convention makes understanding the sources harder. I'm not against changing the formulas to one consistent approach if we can do it in a way that addresses this concern.

Things that I think would help address this concern would include:

• explicit discussion of units and their appearance in historical work.
• footnotes on each conversion (I generally disagree with footnotes but this is one case where I think they make sense.
• a specific reliable reference or references as the standard agreed, called out.
• limited use of specific values to avoid clutter.
• consistency throughout.

Most modern physicists use natural units because all the extra k's and $1/4\pi \epsilon _{0}$ stuff is not physics. But I understand that textbooks are fascinated with units so I'm ok with picking one. Johnjbarton (talk) 15:49, 20 July 2024 (UTC)

Possible references:

• Kibble 5th ed uses SI units, $F={\frac {q_{1}q_{2}}{4\pi \epsilon _{0}r_{12}^{2}}}$ (not k)
  • Kibble, T. W. B.; Berkshire, F. H. (2004). Classical mechanics (5 ed.). London : River Edge, NJ: Imperial College Press ; Distributed by World Scientific Pub. ISBN 978-1-86094-424-6. OCLC 54415965. Page 8.
• Hand and Finch use natural units.
  • "It is common practice in physics to chose units to simplify the formula..." page 85.
  • Hand, L. N., & Finch, J. D. (1998). Analytical Mechanics. Cambridge: Cambridge University Press.
• Goldstein 3rd edition uses cgs $F={\frac {ZZ'e^{2}}{r^{2}}}$ Page 109
  • Goldstein, Herbert; Poole, C. P.; Safko, J. L. (2001). Classical Mechanics (3rd ed.). Addison-Wesley. ISBN 978-0-201-65702-9.

By the way, the use of k is common force and potential problems as meaning "whatever constants". So it's not a standard notation for Coulomb's constant AFAIK. Johnjbarton (talk) 16:38, 20 July 2024 (UTC)

I proposed to use Kibble as the reference for units. Two options however:

1. Put $\left({\frac {q_{Au}q_{\alpha }}{4\pi \epsilon _{0}}}\right)$ in front of most equations or
2. define $F={\frac {k}{r^{2}}},\ k=\left({\frac {q_{Au}q_{\alpha }}{4\pi \epsilon _{0}}}\right)$ and use k everywhere.

Johnjbarton (talk) 22:07, 20 July 2024 (UTC)

Legacy + Reception.

An edit by @22merlin made me realize that we did not properly integrate the Legacy section during the merge. I think we want the Summary to have a wrap up eg Legacy, but the Summary now needs to include the scattering topic a bit more. So I will move some of the Reception content towards Legacy rather than the other way around. Johnjbarton (talk) 21:07, 20 July 2024 (UTC)

Ok I think this is mostly fixed up. The Reception section is merged into Legacy. The Legacy section needs a few more references. Also

• The astronomer Arthur Eddington called Rutherford's discovery the most important scientific achievement since Democritus proposed the atom ages earlier.

is unclear: Which achievement? Johnjbarton (talk) 21:59, 20 July 2024 (UTC)

Rutherford's scattering experiments with light nuclei.

Around 1917 Rutherford did more more experiments with alpha particle scattering, ultimately leading to the discovery of the proton. I guess these are covered in Proton but we need to mention this aspect in Legacy. A good ref is already used in the article:

• Barrette, J. (2021). Nucleus-nucleus scattering and the Rutherford experiment. Journal of the Royal Society of New Zealand, 51(3–4), 434–443. https://doi-org.wikipedialibrary.idm.oclc.org/10.1080/03036758.2021.1962368 Johnjbarton (talk) 22:48, 20 July 2024 (UTC)

Cloud chamber image.

The cloud chamber image of Rutherford scattering was removed in an edit summarized: This image screws up the layout on desktop Instead of removing something like this for a minor reason, please post the issue and we can find a way to fix it.

In general wikipedia pages do not render perfectly. Removing content to fix layout will on be a temporary fix and it won't impact most readers.

I added the image back with more concise text. The rendering is fine for the moment. Johnjbarton (talk) 00:53, 24 July 2024 (UTC)

Is that cloud chamber image even useful? Does it tell us anything the diagram above it can't? Kurzon (talk) 11:52, 24 July 2024 (UTC)

Yes, the image does three things:

1. Historically significant independent experimental evidence
2. Direct visual experimental demonstration of scattering angle
3. Introduction of alternative particle physics arenas beyond gold foil scattering.

I thought about also adding an image related to particle accelerators. The primary legacy of Rutherford's scattering experiments was in the physics of scattering. Johnjbarton (talk) 16:47, 24 July 2024 (UTC)

If you can pad out the Legacy section a bit, that would solve the layout issues. Kurzon (talk) 17:56, 24 July 2024 (UTC)

The Geiger Marsden section in Summary

The section "Rutherford, Geiger, and Marsden" is, as I understand it, intended to explain the the gold-foil backscattering result of 1909. Due to recent changes it no longer does this. It ends with a comment about uranium sources. Johnjbarton (talk) 16:41, 24 July 2024 (UTC)

Johnjbarton's writeup of Rutherford's math

@Johnjbarton: I'm confused. In a diagram you use $\varphi$ as one of the polar co-ordinates of the alpha particle but later on you use $\phi$ for the scaterring angle. Your diagram also uses the Greek letter rho for the impact parameter, not p. Double-check Rutherford's paper. Kurzon (talk) 13:02, 27 July 2024 (UTC)

Rutherford uses $\phi$ for the "angle of deviation". He never uses polar coordinates for the particle because he jumps directly to "the eccentricity is $\sec \theta$". I used another source as cited to try to explain this jump, and that source used polar coordinates with $\varphi$. To further distinguish these I used $\varphi '$.

To me Rutherford's diagram looked like a rho but once I went through the math detail I realized it was p. I'll fix that. Johnjbarton (talk) 03:26, 28 July 2024 (UTC)

Oh Christ, this terrible and I'm fed up. The notation you use is inconsistent with the rest of the article and inconsistent with modern convention. I suggest you translate all your equations from Rutherford's notation to modern notation. We don't have to be that faithful to Rutherford's paper as long as we get the physics right. Use $\theta$ for the scattering angle and $\varphi$ for the angle between $r$ and $r_{A}$ and b for the impact parameter. Fix this yourself or I'll do it myself and you might not like how I butcher things. Kurzon (talk) 11:07, 28 July 2024 (UTC)

I changed the URLs for Rutherford's 1911 paper to a PDF scan of the original. The re-typed versions we previously used have some copy errors. Kurzon (talk) 17:03, 27 July 2024 (UTC)

Thanks, good move. Johnjbarton (talk) 03:11, 28 July 2024 (UTC)

Newtonian model for scattering from Thomson atomic model.

Once again I removed the Newtonian model for scattering from the Thomson atomic model.

1. The content appears in the article on Thomson's model, Plum pudding model.
2. Historically the case against Thomson was not made using these approaches.
3. Any discussion of a Newtonian mechanics approach to scattering is in my opinion bad physics. Force based models have not been used in atomic physics since before the time of Rutherford; energy based models are used. Force based methods are more complex (because the energy approaches already include the integrals) and do not work in quantum problems.
4. The description is lengthy and detracts from the topic of the article, which is scattering from Rutherford's model.

I think the material works well in Plum pudding model because Thomson did not provide a detailed account of scattering from the positive sphere. In addition, the source for this material, Beiser, presents it as Thomson-model scattering. A short WP:SUMMARY section could be included or we could expand the discussion of the historic case against Thomson's model.

I was originally against any presentation of a force based scattering. I did not oppose the presentation of the force-based model of Rutherford scattering as a compromise, but now I wonder if that was a good choice. Johnjbarton (talk) 03:59, 28 July 2024 (UTC)

I am fine with deleting that stuff once we fix the notation issues in your stuff. Kurzon (talk) 16:45, 28 July 2024 (UTC)

Request for comment: presenting Rutherford's 1911 paper in Rutherford's way.

As of this version the section Atomic model in Rutherford's crucial 1911 paper presented the 1911 paper using Rutherford's notation. @Kurzon has changed the formulas to arbitrary and confusing notations. I proposed to revert these changes and leave the original notation. I am asking for the input of other editors so we don't go back and forth on this issue. Johnjbarton (talk) 22:29, 28 July 2024 (UTC)

Use Rutherford 1) Any notation is arbitrary, 2) This paper is a classic, 3) Using Rutherford's notation allows a simple narrative, 4) Using Rutherford's notation makes verification straightforward. Johnjbarton (talk) 00:57, 29 July 2024 (UTC)

How many readers are going to even read the original paper? If they make that effort, it shouldn't be too hard for them to translate the notation. Kurzon (talk) 06:29, 29 July 2024 (UTC)

 Done I concede the issue. Johnjbarton (talk) 22:16, 30 July 2024 (UTC)

@Headbomb: Could you comment on this? Rutherford used p for the impact parameter, but p is more often used for momentum these days. Kurzon (talk) 11:54, 29 July 2024 (UTC)

I don't get this insistence on following Rutherford's paper. I find the article has become an overall downgrade from what it used to be just a month or so ago. And I find impact parameter b to be much clearer (and modern) because it can't be confused with momentum. Headbomb {t · c · p · b} 12:09, 29 July 2024 (UTC)

Although I respect Johnjbarton's expertise, his sense of presentation leaves something to be desired. Kurzon (talk) 12:55, 29 July 2024 (UTC)

Per the Wikipedia code of conduct WP:CIVIL keep your personal comments to yourself and discuss the article content here. Johnjbarton (talk) 15:51, 29 July 2024 (UTC)

There is nothing incivil about Kurzon's comments. Headbomb {t · c · p · b} 16:05, 29 July 2024 (UTC)

I disagree. I have ignored his insults in the past but seems to have been a mistake. There is no reason to post an insulting personal opinion about my abilities. Johnjbarton (talk) 16:50, 29 July 2024 (UTC)

You're a little sensitive, I meant no offense. Kurzon (talk) 20:18, 29 July 2024 (UTC)

Rather than more unfounded claims about my character, the traditional response is an apology. Johnjbarton (talk) 21:30, 29 July 2024 (UTC)

Alright I'm sorry. Kurzon (talk) 03:05, 30 July 2024 (UTC)

Thanks. Johnjbarton (talk) 16:02, 30 July 2024 (UTC)

A month ago the article was factually incorrect and based on WP:OR. If you have specific concerns, please be specific. Johnjbarton (talk) 15:53, 29 July 2024 (UTC)

I've changed p to b, swapped phi and theta to be consistent with other diagrams in this article and others, and made a few more changes. Tell me what still remains incorrect. Once we've sorted this all, I'll be happy to delete the stuff I adapted from Hyperphysics. Your historically-relevant stuff is better (it also gave me a reason to brush up on hyperbolic geometry, thanks for that). Kurzon (talk) 15:05, 30 July 2024 (UTC)

Using b for impact parameter is common in my experience, and using p for momentum is of course a widespread convention. Physicists are also more inclined to use $\theta$ for the angle from the axis, and $\phi$ for the azimuthal angle (as noted in our Spherical coordinate system article). I think it makes sense to follow the modern conventions; anyone taking the time to look up Rutherford's original paper will be competent enough to make a few letter substitutions. XOR'easter (talk) 21:46, 30 July 2024 (UTC)

r min

@Johnjbarton: In one part of the article, $r_{\text{min}}$ is defined as

$r_{\text{min}}={\frac {kq_{a}q_{g}}{{\tfrac {1}{2}}mv_{0}^{2}}}$

but in another part it's

$r_{\text{min}}=-{\frac {kq_{a}q_{g}}{{\tfrac {1}{2}}mv_{0}^{2}}}$

Which is it? Kurzon (talk) 16:25, 31 July 2024 (UTC)

Well r_min can't be negative. I removed the minus. But this copyedit fail means we may have a sign error in the other equations. Johnjbarton (talk) 16:31, 31 July 2024 (UTC)

Did RUTHERFORD make mistake? Look, I'll retype what he put in his own paper:

${\frac {1}{2}}mV^{2}={\frac {1}{2}}mv^{2}-{\frac {NeE}{SA}}$

$v^{2}=V^{2}(1-{\frac {b}{SA}})$

and on a following page $b={\frac {2NeE}{mu^{2}}}$

That's his stuff. Now I'm going to do my own rearranging:

${\frac {1}{2}}mV^{2}+{\frac {NeE}{SA}}={\frac {1}{2}}mv^{2}$

$V^{2}+{\frac {NeE}{{\frac {1}{2}}mSA}}=v^{2}$

Help me I'm going mad. Kurzon (talk) 17:06, 31 July 2024 (UTC)

@Johnjbarton, Headbomb, and XOR'easter: Did Rutherford make a mistake in his conservation of energy equation? This is what he wrote in his 1911 paper:

${\frac {1}{2}}mV^{2}={\frac {1}{2}}mv^{2}-{\frac {NeE}{SA}}$

V is the velocity of the alpha particle at the start and v is it's velocity at the point of closest approach A. I figure that as the alpha particle approaches A, it should LOSE energy, not gain it as the equation suggests. Should it instead be like this?

${\frac {1}{2}}mV^{2}={\frac {1}{2}}mv^{2}+{\frac {NeE}{SA}}$

Kurzon (talk) 19:37, 31 July 2024 (UTC)

I agree with your reasoning if we assume $NeE>0$ and SA positive. Rutherford did not now the charge on the nucleus and the value of SA could depend on the branch of the hyperbola. Earlier in the paper he has head-on energy balance (where SA is b)

$${\frac {1}{2}}mu^{2}=NeE\cdot \left({\frac {1}{b}}-{\frac {3}{2R}}+{\frac {b^{2}}{2R^{3}}}\right)$$

If we neglect the 2nd and 3rd terms as he does, then the potential energy term is positive:

$${\frac {1}{2}}mu^{2}={\frac {NeE}{b}}$$

BTW we should either use $u$ (initial) in our form of this equation or $V$ (approaching nucleus from afar) but not $v$ (apse velocity) as currently set. Johnjbarton (talk) 19:53, 31 July 2024 (UTC)

OK, I chose to change the energy equation rather than redefine r_min. Kurzon (talk) 20:19, 31 July 2024 (UTC)

I think this change is fine, but Rutherford was not mistaken. If you use the other sign and follow through the derivation you get an equation relating $b/r_{\text{min}}$ to $-\theta$, which is just the other half of the angles in the table or the bottom of the diagram. Both sets of angles are observed in the experiment: the two results are indistinguishable. This is what Rutherford meant when he said "The deductions of the experiment so far considered are independent of the sign of the central charge..." Johnjbarton (talk) 23:51, 31 July 2024 (UTC)

The eccentricity of a hyperbola can be calculated

Two or three times I have tried to fix this. Each time @Kurzon removes it.

We have a paragraph that starts

• The eccentricity of a hyperbola can be calculated ...

My immediate reaction is "wait what"? What is this "eccentricity" thing? Where did it come from? How is this all related to the problem?

My solution was simply to start the paragraph with a sentence about eccentricity, hyperbola, and the geometry of the problem:

---

Any hyperbola can be written in polar coordinates $(r,\varphi )$ with the origin at the center as

$$r={\frac {1}{\sqrt {e^{2}\cos ^{2}\varphi -1}}}.\,$$

where $e$ is the eccentricity. Comparing Fig. 1 to the geometry of a hyperbola shows that SO is the focal distance and OA is the length of the semi-major axis. The eccentricity, e, is the focal distance divided by the length of the semi-major axis or ⁠SO/OA⁠.

---

I think something like this is essential. Johnjbarton (talk) 22:44, 31 July 2024 (UTC)

But you don't use that formula for anything here, it serves no purpose. All we need is ⁠SO/OA⁠ = sec Phi. Kurzon (talk) 01:47, 1 August 2024 (UTC)