How Do Electrons Avoid Spiraling into the Nucleus in Rutherford's Atomic Model?

[GuitarDean]

Ok, I just posted this in the advanced forum, but looking at some of the topics on there my question might belong in here instead..

I understand Rutherford proposed that electrons orbit around a central nucleus. However, since accelerating charges produce electromagnetic radiation, the orbiting electron should lose energy via E&M and spiral into the nucleus.

But my question is: How do I calculate the time it takes for the electron to spiral into the nucleus, given the rate of energy loss (as a function of acceleration) and the initial electron-nucleus distance?


The power loss equation is: P = (e^2 a^2 ) / (6 pi epsilon c^3)

So far I've thought of calculating the initial energy of the system and integrating the power, and then equating the lost energy to the initial energy; however the final energy is negative inifinity, so this doesn't seem to work.

Algebraic manipulation of circular motion equations didn't get me anywhere either; I'm not really sure how else to proceed now.

 

[anathema]

Thank you for your question. This is an interesting problem to consider when examining Rutherford's model of the atom. To answer your question, we need to take into account a few key factors.

First, it is important to note that Rutherford's model of the atom is a simplified representation and does not account for the complexities of quantum mechanics. In reality, electrons do not orbit the nucleus in a classical sense, but rather exist in probability clouds around the nucleus.

That being said, let's assume we are dealing with a classical system where an electron is orbiting a nucleus. As you correctly pointed out, according to classical electrodynamics, an accelerating charge will emit electromagnetic radiation and lose energy. This would cause the electron to spiral into the nucleus, as you mentioned.

To calculate the time it takes for this to happen, we need to consider the initial energy of the system, as you suggested. The initial energy will consist of the kinetic energy of the electron and the potential energy of the electron-nucleus interaction. As the electron spirals into the nucleus, its kinetic energy will decrease while its potential energy will increase. At some point, the electron will have lost enough energy to fall into the nucleus.

To determine the time it takes for this to happen, we can use the conservation of energy principle. The initial energy of the system will be equal to the final energy of the system, which will be the sum of the kinetic and potential energies at the point where the electron falls into the nucleus.

Using the power loss equation you provided, we can calculate the rate at which the electron is losing energy. From this, we can determine the time it takes for the electron to lose enough energy to fall into the nucleus. Keep in mind that this calculation is based on the classical model and does not take into account quantum effects.

I hope this helps answer your question. Please let me know if you have any further questions or if you need clarification on any of the points I mentioned.

 

[ogb p]

I would like to first clarify that Rutherford's atomic model has been updated and refined over the years, but it is still an important contribution to our understanding of the atom. That being said, the issue you have raised about the potential energy loss of the electron in orbit is a valid concern.

One possible explanation for this is the concept of quantized energy levels in the atom. This means that the electron can only exist in certain allowed energy states, and it cannot continuously lose energy and spiral into the nucleus. Instead, it would jump down to a lower energy level and emit a photon of specific energy, maintaining its overall energy balance.

Another factor to consider is the role of the strong nuclear force in holding the nucleus together. This force is much stronger than the electromagnetic force and would counteract the electron's spiral motion towards the nucleus.

In terms of calculating the time it would take for the electron to spiral into the nucleus, it would depend on the specific conditions of the atom and the energy levels involved. This is a complex calculation and would require knowledge of quantum mechanics and the specific atom in question.

In conclusion, Rutherford's atomic model is a valuable contribution to our understanding of the atom, but it has been refined and updated over the years. The issue of energy loss in orbiting electrons can be explained by the concept of quantized energy levels and the role of the strong nuclear force. Calculating the time for an electron to spiral into the nucleus would require a detailed understanding of quantum mechanics and the specific atom involved.