Question

Two biconcave lenses of glass (\( n_g = \frac{3}{2} \)) of radius of curvature 10 cm are placed in contact. Water (\( n_w = \frac{4}{3} \)) is filled in between the lenses. Find the power and nature of the combined lens.

Hint:

The power of a combined lens is the sum of the powers of the individual lenses. Biconcave lenses always produce divergent rays.

Answer 1

Step 1: Lens Formula.
The power \( P \) of a lens is related to its focal length \( f \) by the formula: \[ P = \frac{1}{f} \] For a lens with two surfaces, the focal length is given by the lensmaker’s formula: \[ \frac{1}{f} = (n - 1) \left[ \frac{1}{R_1} - \frac{1}{R_2} \right] \] where \( n \) is the refractive index of the material of the lens, and \( R_1 \) and \( R_2 \) are the radii of curvature of the two surfaces.
Step 2: Formula for the Power of a Combined Lens.
For the combined power of two lenses in contact, the total power is the sum of the individual powers: \[ P_{\text{total}} = P_1 + P_2 \]
Step 3: Refractive Indices of the Lenses.
Let the refractive index of the glass be \( n_g = \frac{3}{2} \), and the refractive index of water be \( n_w = \frac{4}{3} \).
Step 4: Calculate the Focal Length of Each Lens.
Each lens is biconcave, meaning both radii of curvature are negative. The radius of curvature \( R \) of each lens is given as \( 10 \, \text{cm} \), or \( 0.1 \, \text{m} \). For the first lens (glass lens): \[ \frac{1}{f_1} = \left( \frac{3}{2} - 1 \right) \left[ \frac{1}{-R} - \frac{1}{R} \right] = \frac{1}{2} \times \left( \frac{-2}{R} \right) = \frac{-1}{R} = \frac{-1}{0.1} = -10 \, \text{diopters} \] For the second lens (water lens): \[ \frac{1}{f_2} = \left( \frac{4}{3} - 1 \right) \left[ \frac{1}{-R} - \frac{1}{R} \right] = \frac{1}{3} \times \left( \frac{-2}{R} \right) = \frac{-2}{3R} = \frac{-2}{3 \times 0.1} = -6.67 \, \text{diopters} \]
Step 5: Calculate the Total Power.
The total power of the combined lens is the sum of the individual powers: \[ P_{\text{total}} = P_1 + P_2 = -10 + (-6.67) = -16.67 \, \text{diopters} \]
Step 6: Conclusion.
The combined power of the lenses is \( -16.67 \, \text{diopters} \), indicating that the combined lens is a divergent lens.