Rutherford's 1911 Experiment: Electric Field & Deflection Formula

[D.Hilbert]

Hi,

My question is about the famous paper by E. Rutherford on the scattering of alpha particles (1911). The paper is easy to find on internet. Rutherford gives the formula for the electric field inside an atom, at a distance r from the nucleus (here reduced to a point):

X = N e (1/r^2 - r/R^3)

Here N e in the electric charge in the nucleus and R is the radius of the atom. After he says

It is not difficult to show that the deflection (supposed small) of an electrified particle due to this field is given by

theta = b/p (1 - p^2/R^2)^(3/2)

where p is the perpendicular from the center on the path.

I can obtain X but I don't see where the formula for theta comes from.

Any suggestion?

Thanks DH

 

[BvU]

If I were old Ernest, I would integrate the vertical component of the force over the section of the trajectory through the nucleus. If theta is small enough, pretending it is straight may be OK, and it's easier.

 

[D.Hilbert]

Hi,
Thanks for the answer.
Let the trajectory of the alpha particle be on the x-axis.

Are you suggesting to calculate

\int_{t1}^{t2} Fy dt

where Fy is the y-coordinate of the force?

DH

 

[BvU]

I found Rutherford-1911 where your formulas feature. There also is an explanation of what b stands for, which you would have included in your problem statement if you would have used the template. Please use it from now on. PF has a simple rule: no template, no assistance. It would have saved me some time that I could have used for others. Now I have to do some errands, so I am short on time.

But yes, (read: atom in my post, the nucleus is considered pointlike, somewhat incongruent in this context: it still had to be discovered. Better to speak of the center of the atom, but never mind). t₁ and t₂ can be related to R and p and the speed of the $\alpha$. So you change from dt to dx. Some $\beta$ comes in with $\sin\beta = {x\over R}$; perhaps you go from dx to d$\beta$. All constants go into b and there you are!

 

[paranormal]

Hello DH,

Thank you for your question about Rutherford's 1911 experiment. Rutherford's experiment was a groundbreaking discovery in the field of atomic physics and has been studied extensively since its publication.

To understand where the formula for theta comes from, it is important to first understand the experiment itself. Rutherford's experiment involved firing alpha particles (positively charged particles) at a thin sheet of gold foil. The expected result was that the alpha particles would pass straight through the foil with little to no deflection, as the prevailing theory at the time suggested that the atom was a uniform, positively charged sphere.

However, Rutherford observed that some of the alpha particles were deflected at large angles, and a few even bounced back in the direction they came from. This led him to conclude that the atom must have a dense, positively charged nucleus at its center, with most of the atom being empty space.

To explain this observation, Rutherford developed a formula for the electric field inside an atom, which you mentioned in your question. This formula takes into account the electric charge of the nucleus and the radius of the atom. However, this formula does not fully explain the deflection of the alpha particles.

To understand the formula for theta, we must consider the motion of the alpha particles as they approach the nucleus. The perpendicular distance from the center of the atom (where the nucleus is located) to the path of the alpha particle is given by p. As the alpha particle approaches the nucleus, it experiences a force due to the electric field of the nucleus. This force causes the alpha particle to deviate from its original path and be deflected.

Using principles of classical mechanics and the formula for the electric field, Rutherford was able to derive the formula for theta, which describes the deflection of the alpha particle as a function of its initial momentum and the distance p from the center of the atom.

I hope this explanation helps you understand where the formula for theta comes from. Rutherford's experiment and subsequent discoveries revolutionized our understanding of the atom and paved the way for further advancements in atomic physics.

Best,