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Chapter 24: Problem 45

The plane face of a plano-convex lens is silvered. If \(\mu\) be the refractive index and \(R\), the radius of curvature of curved surface, then the system will behave like a concave mirror of radius of curvature (a) \(\mu \mathrm{R}\) (b) \(R /(\mu-1)\) (c) \(R^{2} \mu\) (d) \([(\mu+1) /(\mu-1)] R\)

Short Answer

Expert verified
The system behaves like a concave mirror of radius of curvature \( R/ (\mu - 1) \).

Step by step solution

01

Understand the System

A plano-convex lens with its plane side silvered acts as a combination of a lens and a mirror. The light first refracts through the lens, then reflects off the silvered plane face, and finally refracts out again.
02

Identify Equivalent Focal Length

The focal length of the plano-convex lens when the plane face is silvered forms its equivalent to that of a mirror. The mirror's focal length is defined according to the lens-maker's formula and mirror formula.
03

Apply Lens-Maker's Formula

For a plano-convex lens, the lens formula is \( \frac{1}{f_l} = (\mu - 1) \left(\frac{1}{R} - \frac{1}{\infty}\right) \), simplifying to \( \frac{1}{f_l} = \frac{\mu - 1}{R} \), where \( f_l \) is the focal length of the lens.
04

Consider Reflective Action

Since the plane face is silvered, the combination behaves like a mirror. The effective focal length of the system \( f_{mirror} \) is connected to the focal length through the relation \( f_{mirror} = \frac{f_l}{2} \).
05

Derive the Radius of Curvature for the Mirror

The relationship between focal length \( f_{mirror} \) and radius of curvature \( R_{mirror} \) of a mirror is \( R_{mirror} = 2f_{mirror} \). Substituting from Step 4, \( R_{mirror} = 2 \times \frac{\mu R}{2(\mu - 1)} = \frac{\mu R}{\mu - 1} \).
06

Match with Given Options

Compare the derived expression \( \frac{R}{\mu - 1} \) with the given options. It matches option (b) \( R / (\mu - 1) \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Refractive Index
In the world of optics, the refractive index is a fundamental concept. It is a measure of how much a ray of light bends, or refracts, when it enters a different medium from the air. The refractive index, often represented by the Greek letter \( \mu \), is calculated as the ratio of the speed of light in a vacuum to the speed of light in the medium. This concept is crucial for lenses and mirrors, as it determines how much bending or focusing of light occurs. In a plano-convex lens, the refractive index impacts how the light bends when passing through the convex face, affecting the lens's focal properties.
Lens-Maker's Formula
For lens designers, the lens-maker's formula is an invaluable tool. It relates the focal length of a lens to its refractive index and curvature. For a simple lens like a plano-convex lens, the formula is \( \frac{1}{f_l} = (\mu - 1) \left(\frac{1}{R_1} - \frac{1}{R_2}\right) \). However, for a plano-convex lens with one flat face, \( R_2 \) is infinite, simplifying the formula to \( \frac{1}{f_l} = \frac{\mu - 1}{R} \). This formula helps understand that the focal length depends not just on the shape of the lens but also on the material from which the lens is made.
Radius of Curvature
The radius of curvature is a measure of the "arc" or segment of a circle that forms the lens surface. This parameter, represented by \( R \), is crucial because it determines how sharply the lens can bend light. Essentially, it鈥檚 the radius of an imaginary sphere of which the lens forms a part. When discussing mirrors and lenses, a smaller radius means a lens or mirror that is more curved, affecting its focusing power. In a plano-convex lens, only the curved side has a radius of curvature, and this plays a role in determining the lens's bending effect on light.
Plano-Convex Lens
A plano-convex lens is a type of lens with one flat side and one convex or outward-curving side. Its design gives it unique reflective and refractive properties that are useful in a variety of optical devices. When the flat side of a plano-convex lens is silvered, it behaves like a concave mirror. Such lenses are often used in situations where light needs to be focused or diverged, such as in shorthand projectors. The presence of the flat side simplifies the calculations because the radius of curvature is only relevant to the convex side.
Focal Length
The focal length of a lens or mirror is the distance from the lens or mirror where parallel rays of light converge or diverge. It's a key factor in optics, affecting how images are focused or spread out. In a plano-convex lens, the focal length is determined using the lens-maker's formula and is impacted by the refractive index and the radius of curvature. When the plane face of a plano-convex lens is silvered to act as a mirror, the effective focal length is altered. The relationship between the focal length and the radius of curvature becomes crucial in determining how the lens-mirror system behaves.

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Most popular questions from this chapter

Concave and convex lenses are placed touching each other. The ratio of magnitudes of their power is \(2: 3\). The focal length of the system is \(30 \mathrm{~cm}\). Then the focal lengths of individual lenses are (a) \(-75,50\) (b) \(-15,10\) (c) 75,50 (d) \(75,-50\)

A cubical room is formed with 6 plane mirrors. An insect moves along diagonal of the floor with uniform speed. The velocity of its image in two adjacent walls are \(20 \sqrt{2} \mathrm{~cm} / \mathrm{s}\). Then the velocity of image formed by the roof is (a) \(20 \mathrm{~cm} / \mathrm{sec}\) (b) \(40 \mathrm{~cm} / \mathrm{sec}\) (c) \(20 \sqrt{2} \mathrm{~cm} / \mathrm{sec}\) (d) \(10 \sqrt{2} \mathrm{~cm} / \mathrm{sec}\)

\(N\) plane mirrors are arranged parallel to one another each moving with a speed \(v\). The linear velocity of the Nth image of a point object placed in front of the first mirror is (a) \(N v\) (b) \(N v^{2}\) (c) \(N v^{3}\) (d) \(2 N v\)

A thin plano-convex lens acts like a concave mirror of focal length \(0.2 \mathrm{~m}\), when silvered on its plane surface. The refractive index of the material of lens is \(1.5 .\) The radius of curvature of the convex surface of the lens will be (a) \(0.1 \mathrm{~m}\) (b) \(0.2 \mathrm{~m}\) (c) \(0.4 \mathrm{~m}\) (d) 08

A convex driving mirror of focal length \(20 \mathrm{~cm}\), is fitted in a motor car. If the second car \(2 \mathrm{~m}\) broad and \(1.6 \mathrm{~m}\) high is \(6 \mathrm{~m}\) away from the first car and overtakes the first car at a relative speed of \(15 \mathrm{~m} / \mathrm{s}\), then how fast will the image be moving? (a) \(0.016 \mathrm{~m} / \mathrm{s}\) (b) \(0.0257 \mathrm{~m} / \mathrm{s}\) (c) \(0.162 \mathrm{~m} / \mathrm{s}\) (d) \(0.0073 \mathrm{~m} / \mathrm{s}\)
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