Introductory Physics Volume Two

7.8 Far Field Approximation 151
§ 7.8 Far Field Approximation
In most cases the detector is placed far from the two sources, far
in the sense that the distance from the sources to the detector r is
much bigger than the distance between the sources, d. There is a
useful approximation for the path difference that can be used when the
detector is far from the source.
First consider the case when the detector
is not far, as pictured in the figure to
Detector
the right. Notice that if we make of a section
of arc, with the center at the detector,
Source
and going through the closest source, and
then consider the section of the path from
the furthest source that is cut off by this
arc. This cut off section (marked as ∆r in
the figure) is equal to the path difference.
Δr
Also notice that the arc is perpendicular to
Source
the path.
Now imagine moving the detector further from the sources, as
pictured in the diagram below. The section of arc that is between the
two paths, subtends a smaller and smaller angle, and thus this section
of arc becomes closer and closer to a straight line. At the same time
the grey region becomes a right triangle.
θ
Let us examining this right triangle
more closely. We see from the diagram
to the right that
sin θ = ∆r −→ ∆r = d sin θ
d
This is a very useful result, that allows
us to find the path difference from the
distance between the sources and the angle
to the detector.
Combining this with our previous result, ∆r = m λ 2
, we find the
d
θ
Δr
θ