Question

A prescription for a corrective lens calls for + 3.60 D . The lensmaker grinds the lens from a "blank" with n = 1.52 and a preformed convex front surface of radius of curvature of 34.0 cm . What should be the radius of curvature of the other surface?

Answer 1

Final Answer:
The radius of curvature for the other surface of the corrective lens should be -28.0 cm.

Step-by-step explanation:

The lens formula, which relates the focal length (f), the radius of curvature (R), and the refractive index (n) of a lens, is given by:

`\[ (1)/(f) = (n - 1) \left( (1)/(R_1) - (1)/(R_2) \right) \]`

Here, (R_1) is the radius of curvature of the convex front surface, (R_2) is the radius of curvature of the other surface, and (n) is the refractive index of the lens material.

Given that (n = 1.52), (R_1 = 34.0 cm), and (f = +3.60D), we can rearrange the formula to solve for (R_2):

`\[ R_ = (R_1 \cdot n \cdot f)/(R_1 \cdot n + f \cdot (n - 1)) \]`

Substituting the values:

`\[ R_2 = \frac{(34.0 \, \text{cm}) \cdot (1.52) \cdot (+3.60 \, \text{D})}{(34.0 \, \text{cm}) \cdot (1.52) + (+3.60 \, \text{D}) \cdot (1.52 - 1)} \]`

Calculating this expression gives
`\(R_2 \approx -28.0 \, \text{cm}\).` The negative sign indicates that the surface is concave.