I need 3 & 4... free free to answer 1 & 2 if youwant!
Lenses may be simple ones with two spherical curved surfaces ona piece of transparent material like glass or plastic, or much morecomplex and compounded of different elements each with sometimes adifferent material. The surfaces do not have to be spherical, andmanufacturing techniques today allow combining these \"aspheric\"lenses in designs that produce exquisite detail in an image. Yourcell phone camera lens is an example, as are the lenses of a largerdigital or photographic camera. This week's problem is chosen toget to the basics of lenses and how they work because they are themost common essential component of optical instruments. Startingwith Snell's law, you can show that a lens has a property called a\"focal length\" such that 1/f = 1/p + 1/q where p is the distance tothe object in front of the lens, and \(q\) is the distance from thelens to the image it forms. This applies to a lens so thin that thethickness of the glass is small compared to these distances. Lightfrom infinity must form an image at q = f Written this way, thereis a convention to measure the distance to the object as positiveto the left of the lens, and the distance to the image as positiveto the right of the lens.
1. Where does light coming from a distance f in front of thelens form an image? Explain.
2. If I want a lens to be halfway between an object and a screenwhere the image forms, what is the focal length of the lens? Youmay answer generally, or if you prefer a specific case let theobject and the screen be 10 cm apart.
3. The focal length of a thin lens is given by the \"lens maker'sequation\" 1 divided by f space equals space left parenthesis nminus 1 right parenthesis space left parenthesis 1 divided by Rsubscript 1 space minus space 1 divided by R subscript 2 rightparenthesis This works when you can neglect the spacing between thesurfaces, that is, when the radii are much bigger than thethickness of the lens. It is simple enough, but perilous forproblems because of how the signs have to be interpreted. A lenssurface that curves outward so that it is thicker at the center onthat surface is \"convex\". One that curves inward, making it thinnerat the center on that side, is concave. By convention, the sign of\(R\) is positive if the lens is convex to the incoming light, andnegative if it is concave. Here n is the index of refraction of theglass relative to the medium it is in (say air), and the \(R\)'sare the radii of the surfaces of the lens. Thinking of light ascoming from the left, the radius is positive if it is convex to theleft, concave to the right. For example, a lens that has convexsurfaces on both sides with radii 10 cm, an index of 1.5, wouldhave a focal length of 1/f = (1.5 - 1) (1/10 - (-1/10)) = 0.1 f =10 cm The second radius is negative because it is concave to theleft and convex to the right. The shape of the surface and theindex on both sides determine whether the lens converges the light,or diverges it.
--> 3. What would be the radius of curvature of the surfacesof a double convex lens with the same shape on both sides and afocal length of 1 meter? Assume an index of 1.5.
--> 4. Suppose you made a lens in which the first surface wasconvex to the left with a radius of 50 cm. Immediately after it theback surface is exactly the same, also convex to the left, with thesame radius of curvature. Now take this lens outside and letsunlight fall on it. What happens to the light that goes throughthe lens? Explain it with these equations for a thin lens, and alsowith the wave theory of light.
