sign convention for radius of curvature of lens

Optical surfaces with non-spherical profiles, such as the surfaces of aspheric lenses, also have a radius of curvature. 1 r As the wave leaves the lens the outer portions move into the air first and so speed up first. {\displaystyle \alpha _{2}} . Therefore: Effect of a lens A lens adds curvature to a wavefront. This page was last edited on 25 September 2020, at 01:36. The sign convention for the optical radius of curvature is as follows: Thus when viewing a biconvex lens from the side, the left surface radius of curvature is positive, and the right radius of curvature is negative. The vertex of the lens surface is located on the local optical axis. The radius of curvature of a convex mirror used for rearview on a car is 4.00 m. If the location of the bus is 6 meters from this mirror, find the position of the image formed. converges the waves and negative if it decreases the curvature of the ways – i.e. If the vertex lies to the left of the center of curvature, the radius of curvature is positive. {\displaystyle K} Note however that in areas of optics other than design, other sign conventions are sometimes used. a b ğ ñ H I Ì Í Î Ù p ú û w x z ¿ À í î ï ğ ñ ı ı ı ı ı ı ø ø ø ø ø ı ı ö ø ø ı ô ò ı ı ı ı ı ı ı ı ı $a$ ñ ú û ü F ” – ¤ Ê * , ’ ” ˜ š �   ¤ ¦ ¸ º ö ğ î ö ğ î î î î ì ì î ì ì ì ì ì î î î î î î î ç î $a$ „h]„h „øÿ„ &`#$ º Æ È ò ô $ & P R V X \ ^ b d h j n p t v z | � � ¦ ¨ ¼ ¾ ú ø ú ø ú ø ú ø ø ø ø ø ø ø ø ø ø ø ø ø ø ø ú ø ú ø ú ø $a$ ¦ ¨ ¼ ¾ Ú Ü ö ø S U W X Z [ ] ^ ` a É Ë ñ ó ü ı ı ı ı ı û û ÷ í ÷ ê ı ı ı ı CJ 5�OJ QJ \�^J 5�\� 5�CJ ¾ Ú Ü ö ø [3] Care should be taken when using formulas taken from different sources. The shape of the incident waves have no effect on how much the lens changes their curvature and so we can say that a lens increases the curvature of waves that pass through it by a constant amount 1/f. a convex lens A surface that diverges a wavefront is taken as negative e.g. from the axis. {\displaystyle r=0} The lens adds a constant curvature (1/f) to the wave and so the equation relating the object distance (u) image distance (v) and the focal length (f) is: using the above sign convention called the Cartesian sign convention This can be shown in the following diagram. Curvature and the sign convention. As the waves enter the lens their curvature is changed. {\displaystyle \alpha _{1}} Sometimes this image is real (you can form it on a surface such as a piece of paper) and sometimes virtual (it cannot be formed on a surface) Curvature and the sign convention The radius of curvature (R) of a surface is taken as a positive number if it increases the curvature of the waves – i.e. For the general mathematical concept, see, https://en.wikipedia.org/w/index.php?title=Radius_of_curvature_(optics)&oldid=980175178, Creative Commons Attribution-ShareAlike License. i is the sag—the z-component of the displacement of the surface from the vertex, at distance α converges the waves and negative if it decreases the curvature of the ways – i.e. {\displaystyle K} ). If Waves and curved surfaces When light waves fall on a curved surface that surface changes the curvature of the wave. Looking at the diagram you can see that waves with no curvature have been converged to a point a distance f (the focal length) from the lens. F ” Ê ” ˜ š �   ¤ ¦ ¸ È ê î ğ ô & H L N R V X \ ^ b d h j n p t v z | � � ¦ ñ ç ñ ñ ñ İ ñ ñ ñ ÖÓÖÓ ÖÓÖËÖÓ È Å Å Â Â È Å ¿ º ¿ º ¿ º È È È È Â Â Â Å Å OJ QJ CJ CJ CJ CJ$ 0J mH nH u0J j 0J U5�B*CJ \�ph ÿ 5�B* CJ \�ph ÿj CJ UmH nH sH uI ) U V ¤ ¥ H I d N O [ ] ^ _ ` q c e f g à á ı ø ı ö ñ ñ ï ï ï ñ í ñ ö ï ï ï ï ï ï ï ï í ï ï ï ï ï $a$ $a$ ñ şşş á Lenses are made with at least one curved surface (convex or concave) and so lenses simply change the curvature of the waves that fall on them. Solution: Given: The radius of curvature (R)= +4.00 m. Object distance(u) = -6.00 m. Image distance(v) = ? α 2 diverges the waves. α {\displaystyle R} ( r A surface with a small radius of curvature therefore has a large curvature. This means that the outer portions of the wave ‘catch up’ so increasing the curvature to form a converging beam. For thin lenses we can use the lens maker's equation: 1 f = ( n − 1) ( 1 R 1 − 1 R 2) Where n is the index of refraction of the material, R 1 is the radius of curvature of the side the light hits first, and R 2 is the radius of curvature of the side the light hits last. The distance from the vertex to the center of curvature is the radius of curvature of the surface.[1][2]. 0 The overall action of the convex lens is therefore to converge light waves to a focus. A surface that converges a wavefront is taken a positive e.g. This means that the outer portions move off more rapidly first and so the curvature of the wave is further increased so converging the light more strongly. The coefficients (Remember that 1/f will be positive for a convex lens and negative for a concave lens) The example has been done with a plane wave but the same would occur with a spherical wave spreading outwards from a point object. * , à â [2], This article is about optical applications. {\displaystyle z(r)} This means that they have been given a curvature 1/f. ĞÏࡱá > şÿ y { şÿÿÿ x ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿì¥Á 7 ğ¿ bjbjUU .. 7| 7| ñ ÿÿ ÿÿ ÿÿ l H H H H , ¼ ¼ ¼ 8 ô 4 , *› ì H H " j j j -� -� -� …š ‡š ‡š ‡š ‡š ‡š ‡š $ œ 6� h «š 9 -� W� Ö -� -� -� «š K’ H H j j íy äš K’ K’ K’ -� L H x j j …š K’ -� …š K’ : K’ …š À X …š j °ıŞ“ñNÊ, � ¼ y� Ğ …š …š úš 0 *› …š �� I� �� …š K’ , , H H H H Ù Lenses Lenses affect the light that passes through them making an image of the object from which the light waves come.