Our initial populations can be described by a phase density for electrons
and ions as follows:

(4) |

(5) | |||

where we specify the density of the beam as , and the density of the cold plasma as ; the first invariant is ; the beam value is ; the total energy ; the beam energy ; and the cold plasma temperature, . Generalizing from Whipple's equation 18, we allow to take on values other than zero.

If we carry out the zeroeth moment integral to find the quasi-neutrality
condition according to the Whipple prescription, we arrive at a similar
equation,

By the appropriate asymptotic expansions, we can characterize the solution
as a function of . Let be small and positive, corresponding to a
few volt (few ) potential needed to shift the thermal plasma such that
the ion beam is neutralized.

(10) | |||

(11) |

Taylor expanding the exponential and truncating gives:

(12) | |||

(13) |

From this expansion, we see that when we are at the equator, and the potential is zero, as defined by Whipple. Furthermore, since and is positive, the denominator is always positive definite. Thus the potential quadratically increases with away from the equator, which means a proton will be confined to the equator, but an electron will be accelerated away from the equator. This solution produces the well-known potential of several along the field line [15] as documented by Whipple [5]. Heuristically, this is all the voltage needed to shift massive numbers of cold electrons to the ion mirror point and shield the ion charge.