Minimum energy of an electron with the Larmor formula

In summary, the previous parts of the problem involved using the Larmor formula to find the energy lost per revolution, calculating the velocity for different radii, and determining the time it takes for the electron to spiral from one radius to another. In analyzing the minimum value for the total energy of the electron, it was found that as the radius approaches infinity, the energy reaches a maximum of 0, but as it approaches 0, the energy decreases towards -infinity, indicating that the electron would eventually crash into the proton in a finite amount of time. This highlights the need for quantum mechanics to explain this phenomenon.
  • #1
doggydan42
170
18

Homework Statement


Is there a minimum value for the total energy of the electron (in this analysis)?

The previous parts:
Use Larmor formula to find ##\frac{|\Delta E|}{K}##, where ##|\Delta E|## is the energy lost per revolution.
the result is ##\frac{8\pi v^3}{3c^3}##.

##\frac{v(r)}{c}## was caluclated for r = 50 pm, 1pm, and 1 fm.

Lastly, ##t(50 pm \rightarrow 1 pm)## was calculated.

Homework Equations


Larmor Formula:
$$\frac{dE}{dt} = -\frac{2}{3}\frac{e^2a^2}{c^3}$$

Potential Energy:
$$V = -\frac{e^2}{r}$$

Magnitude of the force:
$$F = \frac{e^2}{r^2}$$

In another part of the problem, the velocity for radius r is calculated:
$$v(r) = \sqrt(\frac{e^2}{m_er})$$

The energy is also calculated to be:
$$E = -\frac{1}{2}\frac{e^2}{r}$$

The time it takes for the electron to spiral from ##r_i## to ##r_f## is:
$$t(r_i\rightarrow r_f) = \frac{m_e^2c^3}{4e^4}((r_i)^3-(r_f)^3)$$

The Attempt at a Solution


I have been trying to look at the formulas and see how if one variable increase what happens to dE/dt. If there was a minimum energy, there would be some r such that dE/dr = 0. This would be at negative and positive infinity radii. Pluggin the infinity into ##E = \frac{-e^2}{2r}## gives E = 0. So as the radius approaches infinity, the energy reaches a maximum, 0.

As the radius approaches 0, however, E approaches -infinity from the right, and infinity from the left. So would the would we be able to say there is a minimum energy at the limit of E as r approaches 0+?

Is there a better way to go about solving this problem?
 
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  • #2
Would the fact that E goes to -infinity mean that that there is no minimum energy? The electron would keep getting closer to the center of the orbit, never reaching, but decreasing in energy all the while?
 
  • #3
doggydan42 said:
Would the fact that E goes to -infinity mean that that there is no minimum energy? The electron would keep getting closer to the center of the orbit, never reaching, but decreasing in energy all the while?
I did not double check all your calculations but the conclusion is that yes, classically the electron would crash into the proton. That was one of the key difficulties with classical physics that led to the need for quantum mechanics.

By the way, you say that it would never reach r=0. But your formula shows that it would reach r=0 in a finite amount of time, as we see by just plugging rf=0 in the equation (I am assuming that ri is not infinite).
 

1. What is the Larmor formula?

The Larmor formula is a mathematical equation that describes the minimum energy of an electron moving in a circular orbit around a nucleus. It takes into account the electron's mass, charge, and the strength of the magnetic field it is moving in.

2. How is the Larmor formula used in science?

The Larmor formula is used in many areas of science, including nuclear physics, quantum mechanics, and astrophysics. It is particularly useful in understanding the behavior of electrons in a magnetic field, such as in MRI machines and particle accelerators.

3. What factors affect the minimum energy of an electron with the Larmor formula?

The minimum energy of an electron with the Larmor formula is affected by the strength of the magnetic field, the mass and charge of the electron, and the radius of its orbit. The formula also takes into account the speed of light and the Planck constant.

4. Can the Larmor formula be used for other particles besides electrons?

Yes, the Larmor formula can be used to calculate the minimum energy of any charged particle moving in a circular orbit around a nucleus or in a magnetic field. However, it is most commonly used for electrons due to their small mass and charge.

5. Is the Larmor formula an exact calculation?

No, the Larmor formula is an approximation that assumes the electron is moving at non-relativistic speeds. It does not take into account the effects of special relativity. For more accurate calculations, the full relativistic version of the formula, known as the Lienard-Wiechert formula, must be used.

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