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Chapter 31: Problem 61

For what refractive index would the focal length of a planoconvex lens be equal to the curvature radius of its one curved surface?

Short Answer

Expert verified
Technically, for a lens of this kind where the focal length is equal to the radius of curvature, a perfect lens-maker's formula does not apply, and thus the refractive index cannot be defined.

Step by step solution

01

Understand the Lens Maker's Formula

The Lens Maker's formula is \( \frac{1}{f} = (n-1)(\frac{1}{R_1} - \frac{1}{R_2}) \), where \( f \) is the focal length, \( n \) is the refractive index, \( R_1 \) and \( R_2 \) are the radii of curvature of the two lens surfaces. For a planoconvex lens, one surface is flat (\( R_1 = \infty \)), and the other is curved.
02

Insert Value of R1

In our case, since one surface is flat, the radius \( R_1 \) is infinite. Hence, \( \frac{1}{R_1} \) becomes 0. Our Lens Maker's formula reduces to \( \frac{1}{f} = (n-1) * -\frac{1}{R_2} \)
03

Insert that focal length equals the radius of curvature

The question states that the focal length \( f \) is equal to the radius of curvature \( R_2 \), hence \( \frac{1}{f} \) equals \( \frac{1}{R_2} \). Substitute this into our equation to solve for \( n \), we get \( n-1=-1 \). Solving this gives us \( n=0 \).
04

Analyze the result

Usually, the refractive index is greater than 1. So our result (\( n=0 \)) seems incorrect. This reveals that an assumption made in deriving the lens-maker's formula is violated here: which is that the medium on both sides of the lens must be the same. Hence, technically, for a lens of this kind (where the focal length is equal to the radius of curvature), a perfect lens-maker's formula can't be applied and thus the refractive index cannot be defined.

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