A
source of light, at rest
in theframe, emits
radiation uniformly in all directions. (a) We
have to show that the fraction of light emitted into a cone of
half-angle
is given by
We have to
calculate f for.
(b) When the source is viewed from S,
the relative velocity of the two frames being
0.80c,
we have to find the value of
(in frame S) using
the relativistic aberration formula
We have to
repeat the calculation forand.
From our calculations we have to conclude why
the aberration phenomenon is often referred to as the
“headlight effect”.
The area
spanned by a cone of half-angle on
the surface of a sphere of radius r
will be given by the integral
As the source
of light is emitting radiation uniformly in all directions it will
be distributed uniformly in area
at
a distance r from the
source. Therefore, the fraction of light contained in the cone of
half-angle
will be
For
the f will be
(b)
Using the
relativistic aberration formula, we calculate next the angle
for
;
,
and
.
The relativistic
aberration formula is
For
,
we have
We thus find
that for
,
;
for
,
;
and for
,
.
The above calculations
reveal that as the speed of the source increases the width of the
light cone as observed by a stationary observer narrows down, as
happens in a moving “head light”.
frame, emits
radiation uniformly in all directions. (a) We
have to show that the fraction of light emitted into a cone of
half-angle
is given by
.
(b) When the source is viewed from S,
the relative velocity of the two frames being
0.80c,
we have to find the value of
(in frame S) using
the relativistic aberration formula
and
.
From our calculations we have to conclude why
the aberration phenomenon is often referred to as the
“headlight effect”.
at
a distance r from the
source. Therefore, the fraction of light contained in the cone of
half-angle
,
;
;
.