Lenses Physics 202 Professor Lee Carkner Lecture 23

Refraction  Mirrors can be used to magnify an object or to gather light  Lenses can be used for the same purposes  Lenses gather light and magnify objects through refraction instead of reflection  Lenses have focal lengths and real and virtual images, but their properties also depend on the index of refraction  Unlike a mirror, you look through a lens  It has two sides we have to account for

Lenses  Light incident on a lens is refracted twice, once when entering and once when leaving  We will consider only thin lenses, i.e. thickness much smaller than i, p or f  Our thin lenses are composed of two refracting surfaces placed back to back  If the two surfaces are the same, the lens is symmetric

Lenses and Mirrors  Mirrors produce virtual images on the opposite side from the object  Lenses produce virtual images on the same side as the object  i is negative in both cases  Mirrors produce real images on the same side as the object  Lenses produce real images on the opposite side as the object  i is positive in both cases  If a mirror curves towards the object, f and r are positive (real focus)  If a lens curves towards the object, f and r are negative (virtual focus) Real is positive, virtual is negative

Converging and Diverging

Converging Lens  A lens consisting of two convex lenses back to back is called a converging lens  Rays initially parallel to the central axis are focused to the focal point after refraction  The focal point is on the opposite side from the incoming rays  f is real and positive  Converging lenses produce images larger than the object  Magnification is same as for mirrors m = -i/p

Diverging Lens  A lens consisting of two concave lenses back to back is called a diverging lens  Rays initially parallel to the central axis diverge after refraction, but can be traced back to a virtual focus  f is virtual and negative  Th focal point is on the same side as the incoming rays  Diverging lenses produce images smaller than the object

Converging and Diverging

Lens Equations  A thin lens follows the same equation as a mirror, namely: 1/f = 1/p + 1/i  We can also relate f to the index of refraction of the lens n 1/f = (n-1) (1/r 1 -1/r 2 )  Where r 1 and r 2 are the radii of curvature of each side of the lens (r 1 is the side nearest the object)  If the lens curves towards the object, r is negative, if the lens curves away from the object, r is positive  For symmetric lenses r 1 and r 2 have opposite sign

Three Types of Images

Converging Lenses and Images  The type of image produced by a converging lens depends on the distance of the object from the focal point  Objects in front of the focal point (nearer to the lens) produce virtual images on the same side as the object  The image is not inverted  Image is virtual so i is negative  Objects behind the focal point (further from the lens) produce real images on the opposite side of the lens  The image is inverted  Image is real so i is positive

Diverging Lenses and Images  No matter where the object is, a diverging lens produces an upright, virtual image on the same side as the object  For either lens the location of images is the reverse of that for mirrors:  Virtual images form on the same side as the object, real images form on the opposite side  Real images have positive i, virtual images have negative i

Three Types of Images

1) Rays that are initially parallel to the central axis will pass through the focal point after refraction and vice versa

2) Rays that pass through the center of the lens will not be refracted

Two Lenses  Many common optical instruments are formed from more than one lens (microscope, telescope)  To find the final image we find the image produced by the first lens and use that as the object for the second lens  For a two lens system the magnification is: M = m 1 m 2

Dual Lenses

Optical Instruments  We can approximate several common optical instruments as being composed of a simple arrangement of thin lenses  In reality the lenses are not thin and may be arranged in a complex fashion

Near Point  You can increase an objects angular size by moving it closer to your eye  The largest clear (unlensed) image of an object is obtained when it is at the near point (about 25 cm for most people)  If you move the object any closer it will not be in focus  A converging lens will increase the angular diameter of an object m  =  ’/ 

Magnifying Lens  You can use a magnifying lens to overcome the limitation of your eye’s near point  If the object is inside the near point you can view it through a lens which will produce a virtual image outside of the near point  The magnification is: m  = 25 cm /f  This is the size of the object seen through the lens compared to its size at the near point

Magnifying Glass

Compound Microscope  A simple compound microscope consists of an objective and eyepiece  The objective creates a real image focused at the focal point of the eyepiece  The eyepiece acts as a magnifying glass  The magnification of the objective is m = -i/p  i is very close to the distance between the lenses, s  p is very close to the focal length of the objective, f ob  The total magnification is the product of the magnification of each M = (-s/f ob )(25 cm/f ey )  where s is the distance between the focal point of the lenses (the tube length) and f is the focal length

Microscope

Refracting Telescope  In a telescope the two lenses are placed so that the two inner focal points are in the same place  The rays coming in from infinity are refracted by the objective to create a real image at the common focal point  The eyepiece then magnifies the real image  The total angular magnification of the telescope depends on the ratio of the eyepieces m  = -f ob /f ey

Refracting Telescope

Telescopes  The magnification of the telescope can be altered by changing eyepieces  Short focal length means more magnification  Magnification is not the most important property of a telescope  Limited by blurring effects of atmosphere  The true purpose of the objective lens is to gather more light than your eye can and focus it so that it can be viewed  The largest practical refracting telescope has an objective with a diameter of about 1m  The objective becomes so large it is hard to build and support  Most large telescopes are reflectors

Giant 40 inch Refractor at Yerkes Observatory, Williams Bay Wisconsin

Newtonian Telescope