394 Week 12: Lenses and Mirrors
- For a thin lens, the focal length is given by thelensmaker’s formula:
1
f= (n^2 −n^1 )
(
1
r 1 −1
r 2)
(962)
In this expression,n 1 is the index of the surrounding medium (typically air,n 1 = 1)
andn 2 is the index of refraction of the lens itself.r 1 (r 2 ) is the radius of curvature
of thefirst (second) surface struckby the ray, with the sign convention that it is
positive (negative) on the side of the lens refracted light is goingto(comingfrom).
The advantage of using diopters as a measure of lens strength is inherent in this
expression, as you can see that the combined strength of the twolensing surfaces
(in diopters) is equal to thesumof the strength ofeachsurface, in diopters. This
extends to any pair of lenses placed close together – the effective strength of two
lenses closely placed (relative to their focal lengths) in front of oneanother is the
sum of their strength in diopters.- True Facts about the Eye:
The eye is approximately one inch in diameter. Alensin front casts arealimage of
objects being viewed onto itsretina, where rods and cones transform the light into
neural impulses which are then conveyed to the brain for processing by the optic
nerve. Rods and cones are very sensitive to light (and easily damaged) – the light
content is regulated by theirisof the eye, which expands and contracts thepupil–
the aperture through which light passes as it enters the lens.
The focal length of a relaxed lens of an eye withnormalvision is on the retina,
so distant objects are automatically in focus. Given the diameter ofthe eye, this
means that the strength of the lens of a normal eye is approximately 40.00d. The
focal length of a relaxedfarsightedeye isbehindthe retina (too long, strength less
than 40.00d) and is corrected with aconverginglens to make up the difference. The
focal length of a relaxednearsightedeye is infrontof the retina (too short, strength
greater than 40.00d) and is corrected with adiverginglens to take away some of its
strength.
There are muscles that surround the lens of the eye in a ring that contract, making
the lens bulge (to a greater radius of curvature) and therebyshorteningthe focal
length (a process calledaccommodation) to bring nearby objects into focus. The
nearest point one can bring an object to the eye and still bring it intofocus on the
retina is called thenear pointof the eye and is also thedistance of most distinct
vision, representedxnp. In most adults, this distance is around 25 cm (less for small
children, longer for the elderly).
A nearsighted person’s lensalreadyhas too short a focal length to be able to focus
distant objects on the retina, and accommodation only shortens the focal length
still farther. A nearsighted person cannot see anything clearly atdistancesgreater
than some point, called thefar pointfor that person’s eyes. A nearsighted person
is one for whom the far pointxfpis less than infinity. - The simple magnifier is a converging (f >0) lens placed immediately in front of the
eye. An object placed at its focal point therefore forms a virtualimage at infinity
that is automatically brought into focus by the relaxed normal (or vision corrected)
eye. The magnification of the object occurs because one can bringthe objectcloser
to the eye thanxnpand still see it clearly, where it subtends agreaterangle on the
retina (angular magnification). Its magnification is given by:
M=xnp
f(963)
It is very important to understand the simple magnifier, as it forms the eyepiece of
boththe microscopeandthe telescope.