Question

A double convex lens of glass has both faces of the same radius of curvature 17 cm. Find its focal length if it is immersed in water. The refractive indices of glass and water are 1.5 and 1.33 respectively.

Hint:

The focal length of a lens changes when immersed in a medium other than air due to the change in the relative refractive index.

Answer 1

The lens maker's formula relates the focal length \( f \) of a lens in a medium to its refractive index and the radii of curvature of its surfaces:\[ \frac{1}{f} = (n_{\text{lens}} - n_{\text{medium}}) \left( \frac{1}{R_1} - \frac{1}{R_2} \right) \] For a double convex lens submerged in water, the radii of curvature are \( R_1 = 17 \, \text{cm} \) and \( R_2 = -17 \, \text{cm} \). The refractive index of the glass lens is \( n_{\text{lens}} = 1.5 \), and that of water is \( n_{\text{medium}} = 1.33 \). Substituting these values into the formula yields:\[ \frac{1}{f} = (1.5 - 1.33) \left( \frac{1}{17} - \frac{1}{-17} \right) \]\[ \frac{1}{f} = 0.17 \left( \frac{2}{17} \right) = 0.17 \times \frac{2}{17} = 0.02 \, \text{cm}^{-1} \]Therefore, the focal length \( f \) is calculated as:\[ f = \frac{1}{0.02} = 50 \, \text{cm} \]The focal length of the lens in water is \( 50 \, \text{cm} \).