Finding equation for rate of change of distance of spiraling electron

In summary, the equation for the rate of change of the radius of an electron's orbit is dr/dt = -(2ka^2)/(3c^3).
  • #1
lausco
6
0

Homework Statement


(This problem concerns an electron orbiting a proton. Ultimately we are trying to find the time for a classical electron to spiral into the nucleus of an atom, which will lead us to a discussion of why classical mechanics gives way to quantum mechanics when discussing things on very small scales or very high speeds, etc.
In this part of the problem, we're looking for an equation for the rate of change of the radius of the electron's orbit.)

The total energy of the electron is E= (-1/2)(k/r),
and it can be shown that when an electron accelerates it radiates energy at a rate
dE/dt = -(2ka^2)/(3c^3)
Assume the electron is always moving in a circular orbit but one whose radius r decreases as the electron loses energy. Find an equation for the rate of change dr/dt of the radius.

Homework Equations


F = k/r^2, the force on an electron from a proton. The force points toward the proton.
I found the acceleration by setting F=ma, and came up with a = 230/r^2 m/s^2

The Attempt at a Solution


For finding dr/dt, I think I need to first find an equation for the radius of the electron, but I'm not sure if just rearranging the given equation for E is the right way to go about that.
If that's correct, then I end up with r = (-1/2)(k/E).
Differentiating, I end up with dr/dt = (-1/2)(k/(dE/dt)) = 3c^3/4a^2.

I'm not totally sure if my approach is correct or not, and I think I might be neglecting a time dependence for k somehow. Any help you guys could offer is greatly appreciated!
 
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  • #2
Are you sure about your equations ?
 
  • #3
lausco said:
Differentiating, I end up with dr/dt = (-1/2)(k/(dE/dt)) = 3c^3/4a^2.
The result looks dimensionally wrong (LT instead of L/T). I get a different expression. Please post your working.
 
  • #4
Thanks for the quick response!
The first two given equations I'm sure about, and I've made sure I typed them correctly.

As for my differentiation, I rearranged the expression for E and got r = (-1/2)(k/E).
Differentiating both sides, I get dr/dt = (-1/2)(k/ (dE/dt)). [This is the bit I suspect could be wrong; this seems overly simple...]
Then I plug in dE/dt, and get dr/dt = (-1/2)(k/ (-3c^3/2ka^2)), and simplify.
 
  • #5
lausco said:
r = (-1/2)(k/E).
Differentiating both sides, I get dr/dt = (-1/2)(k/ (dE/dt)).
Yes, that's wrong. What is (d/dt)(1/E)? It isn't 1/(dE/dt).
1/E = E
-1
; apply the chain rule.
 
  • #6
I understand that I should use the chain rule, my mistake; I'm not sure I understand what you've got in the spoiler tags. This class is more for philosophy students who want to understand quantum mechanics, so I wasn't expecting it to be so heavy on the calculus right off the bat . . .

If I call u = 1/E, then du = -1/E^2dt, I think. I'm not sure how to factor in the dE/dt that I've been given with that result. Am I still totally off?
 
  • #7
Anyone out there that can help me understand this a bit better? I'm still not really sure how to substitute my given expression for dE/dt into the dr/dt that I solved for ...

Thanks in advance for any help you guys can offer!
 

FAQ: Finding equation for rate of change of distance of spiraling electron

1. What is the significance of finding the equation for the rate of change of distance of a spiraling electron?

Finding the equation for the rate of change of distance of a spiraling electron allows us to understand the behavior and movement of electrons in a spiral motion. This can provide insights into the principles of electromagnetism and help us develop new technologies.

2. How is the rate of change of distance of a spiraling electron calculated?

The rate of change of distance of a spiraling electron can be calculated using the formula: rate of change of distance = change in distance / change in time. This can be derived from the basic principles of calculus.

3. What factors can affect the rate of change of distance of a spiraling electron?

The rate of change of distance of a spiraling electron can be affected by factors such as the strength of the magnetic field, the velocity of the electron, and the mass of the electron. Other external factors, such as interference from other particles, can also impact the rate of change.

4. Can the equation for the rate of change of distance of a spiraling electron be applied in real-life scenarios?

Yes, the equation for the rate of change of distance of a spiraling electron can be applied in real-life scenarios, particularly in the field of electromagnetism. It can also be used in the development of technologies such as particle accelerators and MRI machines.

5. How does the rate of change of distance of a spiraling electron relate to the concept of angular velocity?

The rate of change of distance of a spiraling electron is directly related to the concept of angular velocity, which is the rate at which an object rotates around a fixed point. In this case, the fixed point is the center of the spiral, and the electron is rotating around it at a constant rate, determined by its velocity and the strength of the magnetic field.

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