Resolving Power

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LETTERS
The editor welcomes letters, by e-mail to ped@iop.org or by post to Dirac House, Temple Back, Bristol BS1 6BE, UK.
592 PHYSICS EDUCATION November 2005
Collecting bubbles
In the article Fizziology in
the January 2004 issue (Phys.
Educ. 39 65–8), Gorazd Planinˇsiˇc
described a method of collecting
CO2dissolved in a fizzy drink.
I would like to describe a small
improvement to his method. The
problem is to get a PVC hose
inside a closed bottle (and not lose
the carbon dioxide). I used a
syringe (with an inner diameter of
0.8 mm, green in the upper photo)
that fits easily into the end of a
4mm PVC hose. I then carefully
pushed the syringe through the top
of the bottle.
All the CO2leaves the bottle
through the hose. If you then shake
the bottle the gas leaves the drink
much faster. It will expel the water
from a second identical bottle
(lower photo).
Vaclav Piskac
Gymnazium tr.Kpt.Jarose, Brno,
Czech Republic,
E-mail: piskac@jaroska.cz
Resolving power
Warren et al [1] described
how to measure the resolving
power of the eye using checker-
board patterns. The students on
my course for non-science majors
enjoyed the activity, especially
how it related to the size of objects
visible on the Moon. However, the
explanation of what determines
the eye’s resolution was flawed.
Warren et al suggested that a
reduction in pupil size improves
(lowers) the resolving power
because aberration is reduced and
the light is ‘concentrated near the
optical axis’ so that it ‘stimulates
the cones, resulting in high reso-
lution of fine detail’. On the other
hand they argued that resolution
is worse with wider pupils since
there is more aberration and ‘due
to the stimulation of the rods which
are not present on the optical axis’.
It is true that a smaller pupil
reduces the effects of aberration
and increases the depth of field,
but according to Falk et al [2,
p149] ‘the few aberrations that are
not well corrected turn out to be
of little consequence’. Although
light may be near the optical axis
as it passes through a contracted
pupil, the size of the image pro-
duced by the lens is not affected
by the size of the aperture. There-
fore, the stimulation of cones ver-
sus that of rods is not a factor in
determining the resolution.
The important factor that Warren
et al failed to mention is that dif-
fraction limits resolution more as
the size of the pupil decreases.
Resolving power was defined as
R=d/D, where dis the distance
between squares in the checker-
board pattern and Dis the distance
from the observer at which
squares can be distinguished.
Since the ratio is usually very
small, it is about equal to the angu-
lar separation θ (in radians)
between the edges of the square.
Since light diffracts as it passes
through the pupil, objects sepa-
rated by angles that are too small
will not appear distinct.
According to the Rayleigh cri-
terion, the minimum angular sep-
aration that can be resolved with
a lens is θ ≈λ/bwhere λ is the
wavelength of the light and bis
the size of the lens’ aperture. Under
well-lit conditions, the diameter
of the pupil is about 3 mm, so
θ ≈0.01◦≈2×10−4radians
for λ=550 nm [3].
The resolving powers measured
by my students were fairly con-
sistent with that value. Interest-
ingly, the angular separation of
cones in the fovea is about the
same as the resolving power, so
there are not more cones than
would be useful [2, pp337–9].
It is simple to demonstrate that
the resolving power of the eye
becomes worse (increases) due to
diffraction as the size of its aper-
ture decreases. Hold the gap
between the jaws of a vernier
caliper in front of the eye to view
a pattern of evenly spaced verti-
cal stripes, then decrease the width
of the opening [3].
References
[1] Warren T H, Henriksen P N and
Ramsier R D 2003 Phys. Educ.
38 413–7
[2] Falk D, Brill D and Stork D
1986 Seeing the Light (Wiley)
[3] Miles C L 1990 Am. J. Phys.
58 552–4
Alan J DeWeerd
University of Redlands,
Redlands, CA 92373, USA.