Rutherford’s α-Particle Scattering Experiment

In Ernest Rutherford’s α-particle scattering experiment, it is well-known that the solid angle density of α-particle flux

$$frac{ddot{N}}{dOmega}$$

on the inner surface of a sphere of which the centre is the point of impingement of a beam of α-particles on a (extremely) thin gold foil is proportional to

$$csc^4frac{phi}{2}$$

where $phi$ is the angle away from the pole of the apparatus – the point on the sphere upon which an udeflected beam would impinge. This proportionality relation incurs the problem of having no normalisation: integration of it over even the smallest non-zero region of the sphere comprising the pole will yield an infinite flux.

But there are various mechanisms operating to blur the distribution of flux over the sphere and ‘wash-out’ the singularity at the pole, such as the non-zero width of the beam and the non-zero spread of directions in the beam. Clearly, at the pole flux-density is not going to exceed the flux density of the beam, and

$$frac{ddot{N}}{dOmega}leq frac{a^2dot{N_0}}{pi b^2}$$

where $dot{N_0}$ is the flux in the beam, $a$ is the radius of the sphere, & $b$ the radius of the beam.

I do not find in Rutherford’s paper that there is any attempt to hardwire this blurring mathematically into the propotionality relation – it probably wasn’t necessary to do so in order sufficiently to evince that the nucleus is an extreme concentraction of charge and mass; and besides, the detection medium at the point on the sphere antipodal to the α-particle source was almost certainly utterly saturated: but I am curious anyway as to how it would be hardwired in. I fairly sure the greatest contribution to the blurring would be the finite width of the beam, followed by the spread of directions in the beam. I think the contribution from the finity of cross-section about a nucleus, in that more than a certain distance away another nucleus would be encountered, would be a small one.

So the question is primarily “how would the finite cross-sectional area of the beam be hardwired into the expression for solid-angle-density of flux (the $csc^4frac{phi}{2}$ expression) so that it can be normalised?”. It’s almost certainly going to be an integration over a disc centred on point at polar-angle $phi$ (I know $phi$ normally denotes azimuth these days, but Rutherford used it for polar-angle); but I can’t quite catch the particular details of how to do it.

Does this experiment show that Kirchhoff’s Law hold when there’s a changing magnetic field involved in a…

In this video, the electrical engineer and youtuber Mehdi Sadaghdar (ElectroBOOM) disagrees with another video from professor Walter Lewin.

Basically, professor Lewin shows in an experiment that if we have two different resistances connected in a closed loop, and if we generate a changing magnetic field using a coil, the voltage at the endpoints of the two resistances will be different, contrary to the expectations from Kirchhoff’s Voltage Law (KVL).

schematic

simulate this circuit – Schematic created using CircuitLab

According to the experiment, the left voltmeter VM1 shows a voltage different from the second voltmeter VM2. Lewin then concludes that KVL does not hold when there’s a changing magnetic field. The mathematical reason that he gives is that the magnetic field is non-conservative, and KVL can be derived from Maxwell’s equations only when the field is conservative. He then says that this experiment is a proof of his claims.

Mehdi, on the other hand, points out two things: first, that the way the probing was done is incorrect. The changing magnetic field has an effect on on the probe wires, and that’s one of the reasons why the voltmeters change value depending on the position.

Second, he says that because there’s a loop, then the loop is behaving like an inductor, and together with the coil it’s forming mutual inductor:

schematic

simulate this circuit

I understand Lewin’s derivation of the KVL, so I understand that there’s an issue with the non-conservative magnetic field, but at the same I think Mehdi is right: that loop is an inductor, and the way Lewin is probing the circuit looks wrong to me. So where is the mistake here?

  • Does KVL hold in the circuit above?
  • Is the probing being done right?
  • Does the circuit have a mutual inductor that should not be ignored?

Did Galileo perform an experiment at the Leaning Tower of Pisa?

Galileo’s pupil Viviani said that Galileo dropped unequal weights from the Leaning Tower of Pisa and observed them to take equal times to hit the ground. Galileo’s own writings do not describe such an experiment, but it is clear that he did quantitative experiments with balls rolling down inclined planes, which would make it natural to extrapolate to the same result for falling objects. (I suppose it’s not a matter of pure extrapolation, since the acceleration of a sphere rolling without slipping is strictly less than $gsintheta$.) Wikipedia says that most historians of science think there was never any more than a thought experiment (e.g., Koyre seems to have thought this), but that Drake disagrees.

What would be the arguments for and against the actual existence of the experiment? What did Viviani claim Galileo actually did? (Apparently in some versions of the story, Galileo dropped his objects near the heads of professors emerging from their lectures!)

Is the only argument about whether the experiment took place at the Leaning Tower, or is there doubt about whether it happened at all? Galileo does describe such an experiment, in a line spoken by Simplicio:

But I, Simplicio, who have made the test, can assure you that a cannon
ball weighing one or two hundred pounds, or even more, will not reach
the ground by as much as a span ahead of a musket ball weighing only
half a pound, provided both are dropped from a height of 200 cubits.

There seems to be room for doubt, since it’s fictionalized as a dialog, and the words are spoken by Simplicio rather than by Salviati, who is Galileo’s mouthpiece.

As a side issue, there appears to have been some controversy about what Aristotle had really claimed about this. Apparently there is a case to be made that he was only talking about terminal velocities. For translations of the relevant passages from Aristotle see this link.

Single slit experiment and the uncertainty principle – contradiction?

I have read for example here that the single slit experiment can be seen as a visualisation(?) of Heisenberg’s Uncertainty principle. Basically, the photons passing through the slit are given a fixed value for the standard deviation of position. Therefore, the standard deviation for momentum is “fixed” and, given a statistically large amount of photons, the standard deviation for their momentum is made “visible” as the width of the central fringe, where the central fringe is the expected value of momentum for the photons.

If this is correct, why isn’t the intensity pattern on the screen distributed according to the normal distribution, but shows individual fringes?

The double slit experiment – methods used to observe single photons prior to striking the target

I can accept that when single photons are used in the double slit experiment that a diffraction pattern results at the target due to their wave property.

What I am puzzled about is exactly what practical device is used to try to measure which slit the photon travels through in a non-intrusive manner. Why is it such a mystery that this observation disturbs the diffraction pattern? Surely any measurement we take will spoil the experiment.

Still are physicists using fields, other photons or what to detect the photon before it travels through the slit?

Identifying the effect of multiple interventions in a randomized experiment

Suppose I have a design in which persons are randomly assigned to a intervention and a control group. The persons in the intervention group can receive several treatments, most likely not randomly assigned and depending on unobserved variables. The intention to treat effect is easily found and in case of only one treatment the effect of the treatment is also easily found by 2SLS using assignment as a instrument. But how to identify the effects in case of multiple treatments and only one instrument?

Gedankenexperiment (thought experiment)

In the English written scientific literature it is common practice to use gedanken experiment instead of the English translation thought experiment. E.g.

https://scholar.google.gr/scholar?hl=el&as_sdt=0%2C5&q=gedanken+experiments+poroelasticity&btnG=

In French (university level and higher) peut-on employer expérience gedanken au lieu de expérience de pensée ? E.g.

Because of this limiting upper bound for the stress, it is to be kept in mind that all plasticity experiments (including gedanken experiments) are done with displacement control.

En raison de cette limite supérieure limitant pour la contrainte, il convient de garder à l’esprit que toutes les expériences de plasticité (y compris les expériences gedanken) sont effectuées avec un contrôle de déplacement.

Blind experiment to evaluate upvote and downvote tendency by author reputation [on hold]

I have a sneaking suspicion that the decision whether to upvote / downvote content relies heavily on the reputation of the author of the question.

This phenomenon is known as the fallacy of origins.

I am looking for volunteers with high reputation to help me conduct this experiment.

The experiment design would be a classic A/B testing, in which we would ask the same question from 2 different users.

User A: Would have a low reputation <100

User B: Would have a high reputation >5k

We would measure the amount of upvotes and downvotes per view, and apply a statistical significance test to evaluate the impact of the effect.

Do I have volunteers ?

Please do not post a comment to indicate whether you volunteer, as it would break the “blind” assumption when users will see your comment.

Blind experiment to evaluate upvote and downvote tendency by author reputation

I have a sneaking suspicion that the decision whether to upvote / downvote content relies heavily on the reputation of the author of the question.

This phenomenon is known as the fallacy of origins.

I am looking for volunteers with high reputation to help me conduct this experiment.

The experiment design would be a classic A/B testing, in which we would ask the same question from 2 different users.

User A: Would have a low reputation <100

User B: Would have a high reputation >5k

We would measure the amount of upvotes and downvotes per view, and apply a statistical significance test to evaluate the impact of the effect.

Do I have volunteers ?