For a thin lens, the thickness d is taken to be zero. If a convex lens, having focal length f, is cut along the principal axis, both the resulting pieces will have the same focal length f. Water droplets can be considered as convex lenses and the lens maker formula is applicable. Lens Maker's Formula : Using the positive optical sign convention, the lens maker's formula states where f is the focal length, n is the index of refraction, and and are the radii of curvature of the two sides of the lens. This is lens maker’s formula. Lenses of different focal lengths are used for various optical instruments. the thin lens approximation of the power is P = diopters which corresponds to focal length f = cm. 1. The focal length of a lens depends upon the refractive index of the material of the lens and the radii of curvatures of the two surfaces. The principal axis intersects the surfaces at points C and D. The optical center of the lens is at P. Since the lens is thin, the points C,D and P are considered to be overlapping. The formula is used to construct lenses with desired focal lengths. The focal length of a lens, made up of glass, is 5 cm in air. Consider two thin convex lenses L 1 and L 2 of focal length f 1 and F 2 placed coaxially in contact with each other. Solution: The radii of curvature of the lens are \[R_{1}\] and \[R_{2}\]. Applying lens maker equation for air, \[\frac{1}{f_{a}}\] = (\[\mu _{g}\] - 1)  \[\left ( \frac{1}{R_{1}} - \frac{1}{R_{2}}   \right )\], \[\frac{1}{5cm}\] = (1.51 - 1) \[\left ( \frac{1}{R_{1}} - \frac{1}{R_{2}}   \right )\], \[\left ( \frac{1}{R_{1}} - \frac{1}{R_{2}}   \right )\] =   \[\frac{1}{2.55cm}\]. The focal length, f, of a lens in air is given by the lensmaker's equation: = (−) [− + (−)], where n is the index of refraction of the lens material, and R 1 and R 2 are the radii of curvature of the two surfaces. The lensmaker's equation relates the focal length of a simple lens with the spherical curvature of its two faces: , where and represent the radii of curvature of the lens surfaces closest to the light source (on the left) and the object (on the right). . Basic assumptions in derivation of Lens-maker’s formula: (i) Aperture of lens should be small (ii) Lenses should be thin (iii)Object should be point sized and placed on principal axis. Lens is a refracting device, consisting of a transparent material. If we know the refractive index and the radius of the curvature of both the surface, then we can determine the focal length of the lens by using the given lens maker’s formula: 1 f = ( μ − 1) × ( 1 R 1 – 1 R 2) \frac {1} {f} = (\mu -1) \times (\frac {1} {R_1} – \frac {1} {R_2}) f 1. . An image I' is formed due to refraction at the first surface with radius of curvature \[R_{1}\] . Using lens formula the equation for magnification can also be obtained as . Firstly, lets define a reference system. Your email address will not be published. This gives, \[\frac{1}{f}\] = \[\left ( \frac{\mu_{1}}{\mu_{2}} - 1 \right )\]   \[\left ( \frac{1}{R_{1}} - \frac{1}{R_{2}} \right )\]. Applying the object-image relation due to refraction at the second surface, \[\frac{\mu_{2}}{v}\] - \[\frac{\mu_{1}}{{v}'}\] = \[\frac{\mu_{2} \mu_{1}}{R_{2}}\], \[\frac{1}{v}\] - \[\frac{1}{u}\] = \[\left ( \frac{\mu_{1}}{\mu_{2}} - 1 \right )\]   \[\left ( \frac{1}{R_{1}} - \frac{1}{R_{2}} \right )\]. Khan Academy is a 501(c)(3) nonprofit organization. The medium used on both sides of the lens should always be the same. \[R_{1}\] = -  \[R_{2}\] = R. Refractive index =1.5 and R=20 cm. The lens maker equation for a thin lens is given by, \[\frac{1}{f}\] = (\[\mu\] - 1)  \[\left ( \frac{1}{R_{1}} - \frac{1}{R_{2}} \right )\]. The lensmaker's equation relates the focal length of a simple lens with the spherical curvature of its two faces: , where and represent the radii of curvature of the lens surfaces closest to the light source (on the left) and the object (on the right). Distances, measured along the direction of incident light, are positive. For a thin lens, the power is approximately the sum of the surface powers.. Lens Maker's Formula Using the positive optical sign convention, the lens maker's formula states where f is the focal length, n is the index of refraction, and and are the radii of curvature of the two sides of the lens. If the ambient medium is taken to be air i.e. The distance between the optical center and the focus is called the focal length. Applying the object-image relation due to refraction at the second surface. 1 v - 1 u = (μ1 μ2 − 1) ( 1 R1 − 1 R2) If the object is at infinity, the image is formed at the focus i.e. The sign of is determined by the location of the center of curvature along the optic axis, with the origin at the center of the lens. u=∞ and v=f. The lens maker formula for a lens of thickness d and refractive index \[\mu\]  is given by, \[\frac{1}{f}\] = (\[\mu\] - 1) \[\left [ \frac{1}{R_{1}} - \frac{1}{R_{2}} + \left ( 1 - \frac{1}{\mu} \right ) \frac{d}{R_{1} R_{2}}\right ]\]. Check the limitations of the lens maker’s formula to understand the lens maker formula derivation is a better way. Radius of curvature is negative i.e. Then the object-image relation due to refraction at the first surface gives, The intermediate image serves as the object for the second surface. The magnification is negative for real image and positive for virtual image. Let the refractive indices of the surrounding medium and the lens material be n1 and n2 respectively. Where μ is the refractive index of the material. The sign of is determined by the location of the center of curvature along the optic axis, with the origin at the center of the lens. To avoid this double refraction, thin lenses are considered. This approximation is valid when the thickness is very small compared to the radii of curvature. Lens Maker Formula for Concave Lens and Convex Lens, Solution: The radii of curvature of the lens are, \[R_{1}\] and \[R_{2}\]. The lens equation essentially states that the magnification of the object = - distance of the image over distance of the object. This is the lens maker formula derivation. Using the Lens Maker’s Equation (3) and the appropriate sign for radii R1 and R2, determine the formulae for the focal distance of the hemisphere and the sphere in terms of R and n. Once you have these equations, you should be able to find n from the