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I would like to know if the drift due to the gradient of the earth's magnetic field could be balanced by the sum of the drifts due to corrotation and conviction of the earth's magnetosphere.

To my understanding the gradient drift can be written as:

$\vec{V^{\nabla B}_{D}} = \frac{c e_{\perp}}{q B} \frac{3}{R}\hat{\phi}$

and the drift due to corrotation:

$\vec{V^{cor}_{D}} = \frac{\vec{E^{cor}}\times \vec{B_{earth}}}{B^{2}}$ where $E^{cor} = - \frac{R \Omega_{earth}}{c} \hat{R}$ and thus I would expect the drift due to corroation to be in the to be in the $-\hat{\phi}$ direction so why would one need the sum of the two ( convection and corrotation ) to balance the gradient dirft since the convection field (which to my undertanding is in the $\hat{R}$?

Jokerp
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  • First does the following help: https://physics.stackexchange.com/a/556682/59023. Second, take a bar magnetic and put it in vacuum then rotate it. Wait a while for any things to settle into a steady state and then try to measure the field. Will you be able to tell if the bar magnet is rotating? – honeste_vivere Mar 05 '21 at 15:23
  • @honeste_vivere i have updated my question n case you want to check ! Thanks – Jokerp Mar 05 '21 at 15:25
  • If you look down upon the Earth sitting in the ecliptic plane (i.e., look toward ecliptic south), the Earth rotates in a counterclockwise manner. If we orient so that in this plane the sun is up, then dawn would be to the right and dusk to the left (still looking down on plane). The grad-B drift is charge-dependent so electrons go to the right/dawn, ions go to the left-dusk. The effect of corotation does matter and causes some initial drift to the right for ions but eventually grad-B wins. – honeste_vivere Mar 05 '21 at 15:29
  • @honeste_vivere Thanks for the reply! Is there a chance however for a certain enrergy the sum of the two drifts( corrotation and convection) t balance the gradient drift? – Jokerp Mar 05 '21 at 15:31
  • Yes, as I said there is part of the period during inflow from the geomagnetic tail where the ions don't go left, they go right but eventually go left. That means that at some point the corotation and grad-B balance. I do not know what you mean by convection, however. – honeste_vivere Mar 05 '21 at 15:33
  • @honeste_vivere I mean the electric field tha occurs because of the solar wind. SOlar wind has a solar wind speed and and IMF and because of that occurs a convection electric field – Jokerp Mar 05 '21 at 15:36
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    That is not seen by the particles within the magnetosphere in the way I think you are implying. It is implemented in MHD codes, but it's physically not realistic to impose a $\mathbf{V}{sw} \times \mathbf{B}{IMF}$ on all particles in the system because one parcel of solar wind has no idea about the convective electric field of another parcel. Most of the time $\mathbf{V}{sw}$ and $\mathbf{B}{IMF}$ are coplanar nearly in the ecliptic, so the E-field would be north/south. – honeste_vivere Mar 05 '21 at 15:54

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