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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: $$\Delta KE\propto \Delta B_{r_0}=\frac{\mu_0 M_E}{4\pi}\frac{1}{r_0^3}[r2\cos\theta + \theta\sin\theta]$$

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    $\begingroup$ 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. $\endgroup$ – AtmosphericPrisonEscape Apr 17 '20 at 18:03
  • $\begingroup$ @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 $\endgroup$ – NotSoSN Apr 17 '20 at 18:17
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    $\begingroup$ It is strongly discouraged in Stack Exchange to cross-post the exact same question in multiple sites, one reason is that it can result in answer fragmentation. I'd recommend that you delete the copy in Physics SE. $\endgroup$ – uhoh Apr 18 '20 at 0:45
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    $\begingroup$ 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. $\endgroup$ – uhoh Apr 18 '20 at 0:48
  • $\begingroup$ @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? $\endgroup$ – NotSoSN Apr 18 '20 at 1:47

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