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Chapter 31: Problem 74

Derive an expression for the thickness \(t\) of a plano-convex lens with diameter \(d\), focal length \(f\), and refractive index \(n\)

Short Answer

Expert verified
The thickness \(t\) of the plano-convex lens is approximately given by \( t = f \cdot (n-1)-d/2 \)

Step by step solution

01

Define Parameters

The parameters provided are the diameter of the lens \(d\), the focal length of the lens \(f\), and the refractive index \(n\).
02

Lens Maker's formula

The lens maker's formula is given as \(1/f= (n-1) \cdot (1/R1 - 1/R2)\), where \(R1\) and \(R2\) are the radii of curvature of the two surfaces of the lens.
03

Properties of a Plano-Convex Lens

Understanding a Plano-Convex lens: this is a lens where one surface is flat (plane) while the other is convex. Hence in this case \(R1 = \infty \) (for the flat surface) and \(R2 = R\) (for the convex surface).
04

Substitute into Lens Maker's formula

Substitute \(R1 = \infty\) and \(R2 = R\) into the lens maker's formula to get \(1/f = (n-1) \cdot (1/R - 0)= (n-1)/R\).
05

Derive the thickness

To express \(R\) in terms of the diameter \(d\), we know that in a plano-convex lens, the thickness at the center \(t\) is approximately \(R-d/2\) when the curvature is small compared to the diameter of the lens. So, the lens maker’s formula can be re-arranged as:\( t = R - d/2 = f \cdot (n-1)-d/2 \)

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Most popular questions from this chapter

A contact lens is in the shape of a convex meniscus (see Fig. 31.25). The inner surface is curved to fit the eye, with curvature radius \(7.80 \mathrm{mm} .\) The lens is made from plastic with refractive in\(\operatorname{dex} n=1.56 .\) If it has a \(44.4-\mathrm{cm}\) focal length, what's the curvature radius of its outer surface?

How can you see a virtual image, when it's not "really there"?

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

Is there any limit to the temperature you can achieve by focusing sunlight?

A lens with 50 -cm focal length produces a real image the same size as the object. How far from the lens are image and object?
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