ANSWER:
The average energy flux from a magnetic dipole antenna, Eq. (11.39), is
ANSWER:
Problem 11.12 gives the total power radiated by a magnetic dipole
generated by a time-varying current in a circular loop:
.
Problem 11.4 describes an electric dipole rotating
about the
axis as a superposition of two oscillating
dipoles in the
and
directions,
out of phase:
.
You are then invited to find the intensity as a function of
the polar angle
and calculate the total power radiated,
explaining why the power seems to satisfy the superposition
principle even though it is quadratic in the fields.
A more conventional way to represent precession of a dipole
is to make the component real and the
component imaginary:
which amounts to the same thing as above. In the Earth's case
it is only the transverse component
that precesses; the axial component
just adds a constant magnetic dipole field.
Thus
and we expect3
.
ANSWER:
Again we use
to get
= 1027 Am2. This is only some 8000 times
bigger than the Earth's magnetic moment, but the frequency
s-1 is a lot bigger,
and
is . . . well . . . huge. Thus
(assuming the star's magnetic moment is perpendicular to its
axis of rotation, which gives the biggest result) we get
!
This is about a factor of two larger than the value
predicted by Griffiths. No doubt this is because of
the probability distribution of angles
between the star's magnetic moment and its axis of rotation.
If we assume all values of
between 0 and
are equally likely, then we should multiply our result by
, giving
.
But this seems a little silly in two respects:
first, we are just making an estimate for a "typical"
neutron star. A 20% change of
would have the same effect.
Second, it seems improbable that the formation of neutron stars
from supernovae of spinning suns would indiscriminately orient
the star's magnetic moment relative to its axis or rotation;
naively one might expect
to be more likely,
which would bias our estimate toward much smaller values of
. A more realistic estimate
would require a deeper knowledge of astrophysics
than I (for one) possess.4