Electron sprials from r_i to r_f

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In summary: Other than that, your approach seems reasonable. In summary, the time it takes for an electron to spiral in from an initial radius ##r_i## to a final radius ##r_f## can be expressed as ##\Delta t = \left[\frac{c^3m_ee^2}{32E^3}\right]_{E_i}^{E_f}##, where ##E_i = E(r_i)## and ##E_f = E(r_f)##. This approach is based on using the Larmor formula, potential energy, magnitude of force, and the velocity and energy calculations for a circular motion.
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Homework Statement


Find an expression for the time it takes for an electron to spiral in from an initial radius ##r_i## to a final radius ##r_f##. Write your answer in terms of ##r_i##, ##r_f##, ##m_e##, e, and c.

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 Attempt at a Solution


I tried solving it as a separable differential equation.
Most of the terms on the right were constants, so I focused on ##e^2a^2##.
Since it was circular motion, ##F = e^2/r^2 = m\frac{v^2}{r} = ma##, so ##a = \frac{v^2}{r}##.
So, ##a^2 = \frac{v^4}{r^2}##, using the formula for v(r),
$$a^2 = \frac{(\sqrt(\frac{e^2}{m_er}))^4}{r^2} = \frac{(\frac{e^2}{m_er})^2}{r^2} = \frac{e^4}{r^4m_e^2}$$
Since ##E =E = -\frac{1}{2}\frac{e^2}{r}##, ##\frac{e^2}{r} = -2E##, so
$$a^2 = \frac{(-2E)^2}{m_er^2}$$
So,
$$e^2a^2 = \frac{(-2E)^2e^2}{m_er^2} = \frac{(-2E)^3}{m_er} = \frac{-8E^3}{m_er}$$
Also, ##\frac{1}{r} = \frac{e^2}{re^2} = \frac{-2E}{e^2}##
So $$e^2a^2 = \frac{-8E^3}{m_e}\frac{-2E}{e^2} = \frac{16E^4}{m_ee^2}$$
Plugging this into the initial equation:
$$\frac{dE}{dt} = -\frac{2}{3}\frac{16E^4}{c^3m_ee^2} = -\frac{32}{3}\frac{E^4}{c^3m_ee^2}$$
Rearranging the terms:
$$\frac{3c^3m_ee^2}{32E^4}dE = -dt$$
Integrating both sides:
$$\int_{E_i}^{E_f} \frac{3c^3m_ee^2}{32E^4} dE = \int_{t_i}^{t_f} -dt$$
Solving the integral:
$$-\Delta t = \left[\frac{-c^3m_ee^2}{32E^3}\right]_{E_i}^{E_f}$$
or
$$\Delta t = \left[\frac{c^3m_ee^2}{32E^3}\right]_{E_i}^{E_f}$$

From there I could plug in ##E_i = E(r_i)##, and ##E_f = E(r_f)##

My concern is that the problem stated we would need differential equations, so I was unsure if this was a good approach.
 
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Your work looks good to me, except I think you dropped the square on the electron mass when going from the first to the second equation in your solution.
 
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1. What is an electron spiral from ri to rf?

An electron spiral from ri to rf refers to the movement of an electron from an initial position (ri) to a final position (rf) in a circular or helical path. This can occur in various physical systems, such as in atoms, molecules, and particle accelerators.

2. How does an electron spiral from ri to rf occur?

An electron spiral from ri to rf occurs when an electron experiences a force that causes it to deviate from a straight path and instead follow a curved path. This force can be due to the presence of an electric or magnetic field, or interactions with other particles.

3. What factors influence the trajectory of an electron spiral from ri to rf?

The trajectory of an electron spiral from ri to rf is influenced by several factors, such as the strength and direction of the applied force, the initial velocity and position of the electron, and the properties of the medium through which it is moving.

4. Can an electron spiral from ri to rf be controlled?

Yes, the trajectory of an electron spiral from ri to rf can be controlled through the manipulation of the influencing factors. By adjusting the strength and direction of the applied force, for example, scientists can alter the path of an electron and guide it to a desired final position.

5. What are the applications of electron spirals from ri to rf?

Electron spirals from ri to rf have various applications in science and technology. They are essential for understanding the behavior of particles in atoms and molecules, and are also used in technologies such as electron microscopy and particle accelerators. Additionally, the manipulation of electron spirals has potential applications in fields such as quantum computing and nanotechnology.

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