ANSWER:
The text gives the result
for the electric field due to a line of charge, where the
is a vector from the line to the point at which
is to be
evaluated, perpendicular to the line. (By symmetry, there is no other
possible direction for
to point!) Thus the field
due to the first line of charge points directly away from the x axis
[at the specified point this will be
in a direction
]
and has a magnitude
with
m, so
N/C. The second line of charge is the same
distance away from our test point
(r2 = r1) and the
magnitude of the charge per unit length is the same
so the magnitude of the second contribution to the electric
field is the same as that of the first
(E2 = E1) but in
this case the charge is opposite (negative) so
points
toward the other line:
where
. Thus in total
or
ANSWER:
The N excess electrons repel each other and so ``try'' to get
as far from one another as possible; they therefore accumulate
uniformly over the surface of the Earth (radius RE).
By the SHELL THEOREM (which works just as well for
the electrostatic force as for the gravitational force, since
both obey inverse square laws) the resultant electric field will be
the same as if all the charge were concentrated at the Earth's centre.
Thus the electrical force on any one electron will be
(away from the Earth)
whereas the gravitational force on that same electron
will be
(toward the Earth). When we have as many ``extra'' electrons
as possible, the two forces will just cancel (FE = FG),
giving
or
or
(Note that this is only
C!
Since it is possible to pick up as well as lose electrons
from the solar wind, the Earth remains electrically neutral
to uncanny precision!)