High-Latitude Electrodynamics - The Earth's Ionosphere: Plasma Physics and Electrodynamics - page 388

12 hr

60°

70°

80°

E pc E a

E a

18 6

24

Figure 8.3a Representation of ionospheric electric fields in the northern hemisphere
polar cap and auroral zone, as well as the plasma flow due to those fields.

encompassing a width w in the current sheet and extending a distance d / 2in

each direction perpendicular to the current sheet. We further take d dx . For

such a loop, the steady-state integral form of Ampère's law can be written

>>

= μ 0
S 1

1 δ

B · d I J · d a

When the surface S 1 is far from the magnetosheath and ionospheric ends of
the current sheet, we may assume that the magnetic perturbation, δ B x , is zero at

the surface edges. We may also assume from symmetry that δ B y along part (a)

of the loop is equal and opposite to B y along part (h) of the loop. Thus, evalu-

ating both sides of the previous equation gives

δ

2 δ B y w = μ 0 J z wdx

Hence,

δ B y = μ 0 J z dx / 2 (8.16)

Note that the magnetic perturbation amplitude is independent of the distance
from an infinite current sheet. Thus the magnetic perturbations from the two
equal and opposite current sheets shown in Fig. 8.4 will add together in the region
between the two sheets and exactly cancel each other in the regions outside. The
result will be a magnetic perturbation, δ B y = μ 0 J z dx , confined completely to

the region between the current sheets.