question_answer

Derive the lens formula giving the relation between \[u,\upsilon \]and \[f\] for a thin convex lens Define the term 'linear magnification' and draw a graph showing the variation of linear magnification with image distance for a thin convex lens. How can this graph be used for finding the focal length of the lens?

Answer:

                Assumptions used in the derivation of lens formula:                 (i) The lens used in thin.                 (ii) The aperture of the lens is small.                 (iii) The incident and refracted rays make small angles with the principal axis.                 (iv) The object is a small object placed on the principal axis.                 Derivation of thin lens formula for a convex lens when it forms a real image. As shown in Fig. consider an object AB placed perpendicular to the principal axis of thin convex lens between its F? and C?. A real, inverted and magnified image A?B? is formed beyond C on the other side of the lens.                                 \[\Delta A'B'O\,and\,\Delta ABO\,are\,similar,\]                 \[\therefore \] \[\frac{A'B'}{AB}=\frac{OB'}{BO}\]    ?(1)                 \[Also\,\Delta A'B'F\,and\,\Delta MOF\,are\,similar,\]                 \[\therefore \] \[\frac{A'B'}{MO}=\frac{FB'}{OF}\]                 \[But\,MO=AB,\]                 \[\therefore \] \[\frac{A'B'}{AB}=\frac{FB'}{OF}\]    ?(2)                 From (1) and (2), we get                 \[\frac{OB'}{BO}=\frac{FB'}{OF}=\frac{OB'-OF}{OF}\]                 Using new Cartesian sign convention, we get                 Object distance, \[BO=-u\]                 Image distance, \[OB'=+\upsilon \]                 Focal length, \[OF=+f\]                 \[\therefore \] \[\frac{\upsilon }{-u}=\frac{\upsilon -f}{f}\]                 or \[\upsilon f=-u\upsilon +uf\,\,or\,\,u\upsilon =uf-\upsilon f\]                 Dividing both sides by \[u\upsilon f,\] we get                 \[\frac{1}{f}=\frac{1}{\upsilon }-\frac{1}{u}\]                 This proves the lens formula for a convex lens when it forms a real image. The linear magnification produced by a lens is define as the ratio of the size of the image (h\[({{h}_{2}})\] formed by the lens to the size of the object \[({{h}_{1}})\]. \[m=\frac{{{h}_{2}}}{{{h}_{1}}}=\frac{\upsilon }{u}\] The graph drawn between linear magnification \[m\] an the image distance \[\upsilon \] is a straight line as shown in Fig. By reading the values of \[\text{m}\] and \[\upsilon \] from this graph, we can find the focal length \[f\] of the lens from the following relation: \[m=\frac{f-\upsilon }{f}\]