##### Focal Point of Submerged Eye

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A human eye has a fixed distance d_eye = 2.40 cm between its lens and retina. The eye muscles adjust focus by changing the focal length f_eye of the lens.

Consider an eye under water, which causes its focal length to become
about f_eye = 72.0 cm. What focal length corrective lens placed 2.00 cm in front of the eye
will bring objects at infinity into focus?

If two lenses are mounted one after the other, then the image formed by the
first lens becomes the object for the second lens. One of the simplest and most
useful lens combinations is the astronomical telescope. The lens at the left is
called the *objective* and the lens at the right is called the *eyepiece*
(the one you would put your eye up to). The object is at infinity, and the
image is also at infinity! What good is that? One may wonder. The telescope
magnifies angles -- and if you think about how we see, you will realize that
this is what is meant intuitively by magnification.

Two
lenses which are at the same place -- essentially on top of each other -- have
an effective focal length *f* which obeys

1/f = 1/f1+ 1/f2

Another
way to think about this is to add an ‘eye’ at far right to look through the
telescope. The eye is a *third* lens and a "retina". The object
for this third lens is the image formed by the second lens, so if that image is
at infinity, it will be focussed nicely onto the retina (the relaxed eye can
easily form images of things far away). The eye is seeing the image of distant
stars as points and their angular separation is magnified.

How far
apart should the two lenses be to make a telescope? Find *q _{o}*
and

*p*(in the notation of the thin lens equation) where the subscripts

_{e}*o*and

*e*refer to the objective and the eyepiece respectively. The distance between the lenses is just their sum

*q*+

_{o}*p*. Show that this is

_{e}*f*, where again the subscripts

_{o}+f_{e}*o*and

*e*refer to the objective and the eyepiece. Also, use the principal ray through the center of each lens to derive the angular magnification of the telescope:

*M= - f*. (The minus sign is a sign convention like that for the image formed by a single lens, having to do with whether the image is inverted or right-side-up.)

_{o}/f_{e}A Galilean telescope is one in which the eyepiece is a negative lens.
You can drag a focal point of the eyepiece through the lens in the applet above
to bring it to the wrong side, thus making a Galilean telescope. You will have
to position the eyepiece in the right place, analogous to focusing a real
telescope.

The most important characteristic of a lens is its principal focal length or its inverse which is called the lens strength or lens "power". Optometrists usually prescribe corrective lenses in terms of the lens power in diopters. The lens power is the inverse of the focal length in meters: the physical unit for lens power is 1/meter which is called diopter.

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