| Re .0:
Consider that if you are far enough away from any configuration with a
net charge (that is, a total charge that is not zero), the field is
hard to distinguish from the field due to a charge at the center of the
configuration (where the center is computed for the charges in the same
way that centers of masses are computed). The field varies closely
with the inverse square of distance, and the components of the field
that do not vary inversely with the square are small with respect to
the size of the entire field.
But if the total charge is zero, the normally dominating component (the
unipole moment) is also zero. In this case, the configuration may be
modeled with a dipole. The strength of the field varies inversely with
the cube of distance, and other components are drowned out at
sufficient distances. (If the dipole moment turns out to be zero, you
can go on to the quadrupole moment, and so on.)
So, at sufficient distances, the more complex moments are not
important. But at moderate distances, you might model the
configuration as a unipole with a dipole to make it more accurate than
a unipole alone. To find the dipole moment of the first configuration,
first find its unipole moment (a charge of Q at (.2,.2)). Then add the
inverse of the unipole (a charge of -Q at the origin) to the
configuration to remove it. Then find the dipole moment of the new
configuration.
We can find an estimate of the field for several dipoles by considering
each dipole separately, finding its field, and adding the fields. If
the sum of fields of dipoles looks like a single field of a different
dipole, then the configuration can be modeled with a single dipole. I
would guess this is the case, but I am too lazy to check it out at the
moment. If so, you can find the dipole moment just by finding the
dipole moments of the individual dipoles and adding them (as vectors).
This is the same as finding the dipole moment of a dipole consisting of
the total negative charge at the center of the negative charges and the
total positive charge at the center of the positive charges.
-- edp
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