The medium used on both sides of the lens should always be the same. To avoid this double refraction, thin lenses are considered. This point is called the focus of the lens. Lens, Thin Lens Formula Lens manufacturers use this relation to construct a lens of a particular power. m = h 2 /h 1 = v//u = (f-v)/f = f/(f+u) This equation is valid for both convex and concave lenses and for real and virtual images. The magnification is negative for real image and positive for virtual image. The radii of these spheres are called the radii of curvature of the lens. Using the formula for refraction at a single spherical surface we can say that, For the first surface, For the second surface, Now adding equation (1) and (2), When u = ∞ and v = f. But also, Therefore, we can say that, Where μ is the refractive index of the material. The principal axis intersects the surfaces at points, If an object (at a medium with refractive index \[mu_{1}\]) is placed at a distance, , in front of a spherical surface of refractive index \[mu_{2}\], having a radius of curvature, For refraction at the first surface, object distance OC \[\approx\] OP = - u and image distance is. Sample Problem #1 A 4.00-cm tall light bulb is placed a distance of 45.7 cm from a double convex lens having a focal length of 15.2 cm. Lens is a refracting device, consisting of a transparent material. 3. Determine the image distance and the image size. The sign convention should be followed in the application of the lens maker’s equation. If … The lens equation essentially states that the magnification of the object = - distance of the image over distance of the object. is the refractive index of water. \[R_{1}\] = -  \[R_{2}\] = R. Refractive index =1.5 and R=20 cm. The lens maker formula takes the following form. In the case of a concave lens, it is always positive. Substituting the value of \[\left ( \frac{1}{R_{1}} - \frac{1}{R_{2}}   \right )\], \[\frac{1}{f_{w}}\] =  \[\left ( \frac{1.51}{1.33} - 1 \right )\] \[\frac{1}{2.55cm}\]. If an object (at a medium with refractive index \[mu_{1}\]) is placed at a distance u, in front of a spherical surface of refractive index \[mu_{2}\], having a radius of curvature R, an image is formed at a distance v from that surface such that, \[\frac{\mu_{2}}{v}\] - \[\frac{\mu_{1}}{u}\]  = \[\frac{\mu_{2} \mu_{1}}{R}\]. This is the lens maker formula derivation. Solved example: Lens makers formula Our mission is to provide a free, world-class education to anyone, anywhere. Using lens formula the equation for magnification can also be obtained as . Your email address will not be published. When parallel light rays are incident on a lens, the refracted rays converge to a point (for a converging lens) or appear to diverge from a point (for a diverging lens). μ1 ≈ 1 and μ2 = μ is considered, the lens maker formula can be given in the usual form. \[R_{1}\] = -  \[R_{2}\] = R. Refractive index, , is cut along the principal axis, both the resulting pieces will have the same focal length, NCERT Solutions for Class 9 Maths Chapter 12 Heron's Formula, NCERT Solutions for Class 9 Maths Chapter 12 Heron's Formula In Hindi, NCERT Solutions for Class 9 Maths Chapter 12 Heron's Formula (Ex 12.2) Exercise 12.2, NCERT Solutions for Class 9 Maths Chapter 12 - Heron s Formula Exercise 12.1, NCERT Solutions for Class 11 Physics Chapter 2, NCERT Solutions for Class 12 Physics Chapter 5, NCERT Solutions For Class 12 Physics Chapter 4 Moving Charges and Magnetism, NCERT Solutions for Class 11 Physics Chapter 6, NCERT Solutions for Class 12 Physics Chapter 2, Vedantu Lens, Thin Lens Formula Focal Length and Radius of Curvature Definition, The lens maker formula for a lens of thickness. The object and the images lie on the principal axis. For a convex lens, \[R_{1}\] is positive and \[R_{2}\]is negative. 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. Using the formula for refraction at a single spherical surface we can say that. 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) \[\left ( \frac{1}{R_{1}} - \frac{1}{R_{2}}   \right )\], \[\left ( \frac{1}{R_{1}} - \frac{1}{R_{2}}   \right )\] =   \[\frac{1}{2.55cm}\]. Applying the object-image relation due to refraction at the second surface. Radius of curvature is negative i.e. The lens equation allows us to understand geometric optic in a quantitative way where 1/d0 + 1/di = 1/f. An image I' is formed due to refraction at the first surface with radius of curvature \[R_{1}\] . Lens-Maker's Formula. Your email address will not be published. The refractive index of the lens is \[mu_{2}\]   and it is kept in a medium of refractive index \[mu_{1}\]. Lenses of different focal lengths are used for various optical instruments. Combination of Thin Lenses. 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. The complete derivation of lens maker formula is described below. A light ray gets refracted two times (at two surfaces) while passing through a lens. Solution: The radii of curvature of the two surfaces are equal i.e. Let us consider the thin lens shown in the image above with 2 refracting surfaces having the radii of curvatures R1 and R2 respectively. The reciprocal of the focal length is called the refractive power that has units dioptre (inverse meter). The distance between the optical center and the focus is called the focal length. The Lensmaker's formula works with this convention, although you can modify the formula and define your reference system accordingly. The refractive power (inverse of focal length) can be computed from this formula. If the lens is in another medium, such as water, its lens strength will be diminished. and refractive index \[\mu\]  is given by. 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 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. A convex lens of negligible thickness is considered in the above figure. 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. Here µ is refractive index of lens material to the medium outside. As a demonstration of the effectiveness of the lens equation and magnification equation, consider the following sample problem and its solution. . Check the limitations of the lens maker’s formula to understand the lens maker formula derivation is a better way. If the focal length is \[f_{w}\]in water, the lens maker equation gives, \[\frac{1}{f_{w}}\] = \[\left ( \frac{\mu_{g}}{\mu_{w}} - 1 \right )\]   \[\left (     \frac{1}{R_{1}} - \frac{1}{R_{2}} \right )\]. According to the sign convention for refraction at a curved surface. u=∞ and v=f. 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. Therefore, object distance is I'DI'P=v' and final image distance is IDIP=v. What is the lens maker formula and why is it called so? the thin lens approximation of the power is P = diopters which corresponds to focal length f = cm. Distances, measured along the direction of incident light, are positive. 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. Depending on the shape of the lens, the radii change. The refractive index of glass is  \[\mu _{g}\]  = 1.51. This gives, \[\frac{1}{f}\] = \[\left ( \frac{\mu_{1}}{\mu_{2}} - 1 \right )\]   \[\left ( \frac{1}{R_{1}} - \frac{1}{R_{2}} \right )\]. This is lens maker’s formula. The lens should not be thick so that the space between the 2 refracting surfaces can be small. Applying the object-image relation due to refraction at the second surface, \[\frac{\mu_{2}}{v}\] - \[\frac{\mu_{1}}{{v}'}\] = \[\frac{\mu_{2} \mu_{1}}{R_{2}}\], \[\frac{1}{v}\] - \[\frac{1}{u}\] = \[\left ( \frac{\mu_{1}}{\mu_{2}} - 1 \right )\]   \[\left ( \frac{1}{R_{1}} - \frac{1}{R_{2}} \right )\].