question_answer

Diameter of a plano?convex lens is 6 cm and thickness at the centre is 3 mm. If the speed of light in the material of the lens is 2 \[\times \] 108 m/sec, the focal length of the lens is [CPMT 1989]

A)            15 cm                                      

B)            20 cm

C)            30 cm                                      

D)            10 cm

Correct Answer: C

Solution :

                   According to lens formula \[\frac{1}{f}=(\mu -1)\left[ \frac{1}{{{R}_{1}}}-\frac{1}{{{R}_{2}}} \right]\]                    The lens is plano-convex i.e., \[{{R}_{1}}=R\] and \[{{R}_{2}}=\infty \]                    Hence \[\frac{1}{f}=\frac{\mu -1}{R}\Rightarrow f=\frac{R}{\mu -1}\]                    Speed of light in medium of lens \[v=2\times {{10}^{8}}\] \[m/s\]                    Þ \[\mu =\frac{c}{v}=\frac{3\times {{10}^{8}}}{2\times {{10}^{8}}}=\frac{3}{2}=1.5\]                    If r is the radius and y is the thickness of lens (at the centre), the radius of curvature R of its curved surface in accordance with the figure is given by            \[{{R}^{2}}={{r}^{2}}+{{(R-y)}^{2}}\Rightarrow {{r}^{2}}+{{y}^{2}}-2Ry=0\]            Neglecting \[{{y}^{2}};\] we get \[R=\frac{{{r}^{2}}}{2y}=\frac{{{(6/2)}^{2}}}{2\times 0.3}=15\] cm            Hence \[f=\frac{15}{1.5-1}=30\]