Gather Information:
Contact lenses are worn directly on the eyes with corneal radius
~0.8
cm (Ref.3), have index
of refraction of 1.42 to 1.52 (Ref.3), and thickness of ~0.2 mm.
Eyeglasses are typically worn 2 cm from the eyes (Walker), have
index of refraction 1.5 to 1.7 (Ref.2), and radius of curvature ~10 cm
(found from
examining several eyeglasses and comparing with balls of similar
curvature),
and thickness of 2 - 5 mm (~10X thicker than contacts).
Organize: As can be seen from the Lens Maker's Equation, for a given refractive power (1/f), the thickness of a lens depends on the index of refraction of the lens material (higher n results in greater refractive power) and the relative difference between the radii of curvature for the front and back surfaces of the lens (bigger difference in radii requires a thicker lens). The thickness of the lens also depends on the type of material used and how rigid it must be to avoid breaking or tearing.
Analyze:
Contact lenses can be made thinner than eyeglasses for the following
reasons:
1) Contacts require slightly less refractive power than eyeglasses to correct a person's vision because a contact lens is worn directly on the eyeball. For example, suppose that a person is farsighted with a near point of 57 cm (as in Example 27-3). The refractive power required for eyeglasses that are 2.0 cm away from the eyes is 2.53 D (f = 39.5 cm). However, as outlined in Active Example 27-4, this same person would require contact lenses with a refractive power of only 2.25 D (f = 44.5 cm). While this ~10% difference is significant, the relative difference is much less for persons with better eyesight, so this effect accounts for part, but not all of the difference in thickness between contacts and eyeglasses.
2) Contacts can be made of material with a higher index of
refraction. While this could be true, most modern eyeglasses
are
made of light-weight plastic material with n =1.5 to 1.7, while the
index for contact lenses is generally 1.4 to 1.5 (Ref.2&3), so it
appears that this explanation is not valid in practice.
3) The average radius of curvature of a contact lens is
about
1/10 that of eyeglasses. To examine the effect of this
average
radius difference, suppose we compare eyeglasses and contacts that both
have the same index (n = 1.5) and refractive power (2.0 D). Using
the Lens Maker's Equation, and assuming air as the sourrounding medium,
we can solve for one radius of curvature
given
the other: 1/r2 = 1/r1 - 1/[f(n-1)]. So for contacts (r1 =
1.00 cm): r2 = 1.04 cm, while for eyeglasses (r1 = 10.0 cm), r2 =
16.7 cm. This means that using material of similar index requires
only a 4% difference in radii for contact lenses to yield the same
refractive
power as a 50% difference in radii for eyeglasses! Clearly, this
is an important factor that explains a majority of the significant
difference
in thickness between contacts and eyeglasses. Note, however that
this effect is not as significant for contact lenses where essentially
all of the refraction must occur from the front surface of the contact
since the back surface is touching the cornea, which has nearly the
same index as the contact lens.
4) Contacts do not need to be as rigid as eyeglasses and therefore can be made thinner. While this factor is rather basic, it is the single most important reason for the difference in thickness between contacts and eyeglasses.
Learn: It is interesting to note
that
the thickness of a lens depends more on the average radius of curvature
than on where it is located relative to the eye. The relatively
small
difference in front and back radii required for contact lenses means
that
these lenses must be manufactured with much greater precision than
eyeglasses.
This is yet another marvel of precision engineering that we take for
granted.
Grading Rubric (10 points maximum)
7 - Proper justification using correct physical qualitative and
quantitative
reasoning.
3 - Good organization, explanation, and citation of sources as
appropriate