<p>I asked this question on physics.stackexchange, but as it is a space/magnetosphere question maybe someone here can help me out:</p>

<p>Taking the earth as an idealized dipole and the E field from the sun uniform (at least for a few earth radii) can the energy of particles at distinct radii from earth, in the equatorial plane, be compared as just the fall off of the B field? For example, a particle at geosynchronous orbit and one twice as far away. Would the difference in their energy just be: <span class="math-container">$\frac{1}{2^{3}}=\frac{1}{8}$</span> as the E field is constant and contributes equally to each?
To be clear Im using:
<span class="math-container">$$\Delta KE\propto \Delta B_{r_0}=\frac{\mu_0 M_E}{4\pi}\frac{1}{r_0^3}[r2\cos\theta + \theta\sin\theta]$$</span></p>

CC BY-SA 4.0
2020-04-17T18:03:49.287Z
No, that's the radial component of the background magnetic field you're computing there. If "KE" means kinetic energy, that's as wrong as is gets. You know that forces and energies are not the same thing. Furthermore, gravity plays a role, depending on where you are in the magnetosphere.
AtmosphericPrisonEscape
CC BY-SA 4.0
2020-04-17T18:17:00.503Z
@AtmosphericPrisonEscape  I guess I should have said the change in KE is proportional to the change in B? And I'm neglecting g at the moment, just trying to understand going from most simple to less. So a changing B with constant E
NotSoSN
CC BY-SA 4.0
2020-04-18T00:45:18.367Z
It is strongly discouraged in Stack Exchange to cross-post the exact same question in [multiple sites](https://physics.stackexchange.com/q/544512/83380), one reason is that it can result in answer fragmentation. I'd recommend that you delete the copy in Physics SE.
uhoh
CC BY-SA 4.0
2020-04-18T00:48:42.697Z
You've asked "Would X be proportional to Y?" without suggesting why you would think it might or might not be. The real distribution of proton or electron energies in the magnetosphere is complicated because there there are sources and sinks, it's a dynamic process. I recommend that you do some reading first and once you get stuck on something explain what you've read and where, and ask a more specific question about it.
uhoh
CC BY-SA 4.0
2020-04-18T01:47:05.310Z
@uhoh I've done reading and am working through Bittencourt's Fundamentals of Plasma Physics. I understand the magnetosphere is extremely dynamic. I realize this is an extremely oversimplified example but that is what I want. We know a charged particle will lose energy as it gets farther away if we treat the Earth as a perfect dipole with adiabatic invariants. My question is, does a uniform dusk to Dawn E field the amount of energy lost at larger radii?
NotSoSN
CC BY-SA 4.0
2020-04-18T01:57:58.540Z
okay but I'm explaining how to best write your Stack Exchange Question post in order to keep within best practices here. Asking "Would X be proportional to Y?" without suggesting why you would think it might or might not be is not a good Stack Exchange question. Please consider adding to your question itself, down voters and close voters may not take the time to read through comments before voting!
uhoh
CC BY-SA 4.0
2021-10-27T15:06:17.857Z
Why would this matter?  If you take a particle from far out in the magnetosphere and "drag" it in closer, it can gain energy if the process is slow enough (slow is relative here).  However, some of the most energetic electrons are found in the outer radiation belt, not closer to Earth.
honeste_vivere
<magnetic-field><plasma-physics>
Energy of a charged particle in the magnetosphere
NotSoSN
# Energy of a charged particle in the magnetosphere

(Asked by: username_0 on 2020-04-17)

I asked this question on physics.stackexchange, but as it is a space/magnetosphere question maybe someone here can help me out:

Taking the earth as an idealized dipole and the E field from the sun uniform (at least for a few earth radii) can the energy of particles at distinct radii from earth, in the equatorial plane, be compared as just the fall off of the B field? For example, a particle at geosynchronous orbit and one twice as far away. Would the difference in their energy just be: $\frac{1}{2^{3}}=\frac{1}{8}$ as the E field is constant and contributes equally to each?
To be clear Im using:
$$\Delta KE\propto \Delta B\_{r\_0}=\frac{\mu\_0 M\_E}{4\pi}\frac{1}{r\_0^3}[r2\cos\theta + \theta\sin\theta]$$

**Tags:** magnetic-field, plasma-physics

> **Comments:**
> **username_1**: No, that's the radial component of the background magnetic field you're computing there. If "KE" means kinetic energy, that's as wrong as is gets. You know that forces and energies are not the same thing. Furthermore, gravity plays a role, depending on where you are in the magnetosphere.  
> **username_0**: @username_1  I guess I should have said the change in KE is proportional to the change in B? And I'm neglecting g at the moment, just trying to understand going from most simple to less. So a changing B with constant E  
> **username_2**: It is strongly discouraged in Stack Exchange to cross-post the exact same question in [multiple sites](https://physics.stackexchange.com/q/544512/83380), one reason is that it can result in answer fragmentation. I'd recommend that you delete the copy in Physics SE.  
> **username_2**: You've asked "Would X be proportional to Y?" without suggesting why you would think it might or might not be. The real distribution of proton or electron energies in the magnetosphere is complicated because there there are sources and sinks, it's a dynamic process. I recommend that you do some reading first and once you get stuck on something explain what you've read and where, and ask a more specific question about it.  
> **username_0**: @username_2 I've done reading and am working through Bittencourt's Fundamentals of Plasma Physics. I understand the magnetosphere is extremely dynamic. I realize this is an extremely oversimplified example but that is what I want. We know a charged particle will lose energy as it gets farther away if we treat the Earth as a perfect dipole with adiabatic invariants. My question is, does a uniform dusk to Dawn E field the amount of energy lost at larger radii?  
> **username_2**: okay but I'm explaining how to best write your Stack Exchange Question post in order to keep within best practices here. Asking "Would X be proportional to Y?" without suggesting why you would think it might or might not be is not a good Stack Exchange question. Please consider adding to your question itself, down voters and close voters may not take the time to read through comments before voting!  
> **username_3**: Why would this matter?  If you take a particle from far out in the magnetosphere and "drag" it in closer, it can gain energy if the process is slow enough (slow is relative here).  However, some of the most energetic electrons are found in the outer radiation belt, not closer to Earth.  

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## 0 answers

