Calculating the Power of a Convex Lens in a Liquid Medium: A Comprehensive Guide

Introduction

The calculation of the power of an optical system, such as a convex lens surrounded by a liquid of different refractive index, involves several fundamental principles from optics. This article aims to guide you through the process of determining the power of such a system using the lensmakers equation and the effective power formula.

Understanding the System

Consider a thin-walled transparent cell containing a glass convex lens of refractive index 1.5 and a radius of curvature of 10 cm at each surface. The surrounding spaces are filled with a liquid with a refractive index of 4/3.

Step 1: Calculating the Focal Length of the Lens

The focal length of the lens can be calculated using the lensmakers equation:

[ frac{1}{f} (n - 1) left( frac{1}{R_1} - frac{1}{R_2} right) ]

For a convex lens, the refractive index (n 1.5), the radius of curvature of the first surface (R_1 10 , text{cm}), and the radius of curvature of the second surface (R_2 -10 , text{cm}).

Substituting these values into the lensmakers equation:

[ frac{1}{f} 1.5 - 1 left( frac{1}{10} - frac{1}{-10} right) ] [ frac{1}{f} 0.5 left( frac{1}{10} - frac{1}{10} right) 0.5 times 0.2 0.1 , text{s}^{-1} ] [ f frac{1}{0.1} 10 , text{cm} 0.1 , text{m} ]

Step 2: Calculating the Power of the Lens

The power (P) of a lens is given by:

[ P frac{1}{f} , text{in meters} ]

Therefore,

[ P frac{1}{0.1} 10 , text{D} ]

Step 3: Considering the Effect of the Surrounding Liquid

To find the effective focal length of the system, we use the formula for the effective focal length of a lens in a medium:

[ frac{1}{f_{text{eff}}} (n_{text{lens}} - n_{text{medium}}) left( frac{1}{R_1} - frac{1}{R_2} right) ]

Here, the refractive index of the liquid (n_{text{medium}} frac{4}{3}).

Substituting the values:

[ frac{1}{f_{text{eff}}} (1.5 - frac{4}{3}) left( frac{1}{10} - frac{1}{-10} right) ]

Calculating (1.5 - frac{4}{3}):

[ 1.5 frac{3}{2} frac{9}{6} quad, quad frac{4}{3} frac{8}{6} quad Rightarrow quad 1.5 - frac{4}{3} frac{9}{6} - frac{8}{6} frac{1}{6} ]

Substituting back into the equation:

[ frac{1}{f_{text{eff}}} frac{1}{6} left( frac{2}{10} right) frac{1}{6} times frac{1}{5} frac{1}{30} , text{s}^{-1} ] [ f_{text{eff}} 30 , text{cm} 0.3 , text{m} ]

Step 4: Calculating the Effective Power of the System

Finally, the effective power of the system can be calculated as:

[ P_{text{eff}} frac{1}{f_{text{eff}}} frac{1}{0.3} approx 3.33 , text{D} ]

Conclusion

The power of the optical system, consisting of the convex lens and the surrounding liquid, is approximately 3.33 diopters (D). This calculation demonstrates the importance of considering the refractive index of surrounding media in determining the overall performance of an optical system.