Ch 34: Ray Optics

Knight Calc - Physics for Scientists and Engineers 5th Edition

Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796당신이 사용하는 게 아니라요?교과서 변경

Knight Calc 5th Edition

Ch 34: Ray Optics

문제 69

34장, 문제 69

CALC A converging lens with focal length f creates a real image of an object. What is the minimum possible distance between the object and its image? Your answer will be a multiple of f.

검증된 단계별 안내

Start by recalling the lens equation:

\frac{1}{f} = \frac{1}{d}

_o + 1d_i, where

f

is the focal length,

d

_o is the object distance, and

d

_i is the image distance.

To find the minimum possible distance between the object and its image, define the total distance as

D = d

_o + d_i. Substitute

d

_i from the lens equation into this expression.

Rearrange the lens equation to express

d

_i in terms of

d

_o:

d

_i = d_of / (d_o - f). Substitute this into

D

Simplify the expression for

D

to find a single equation in terms of

d

_o and

f

. Then, differentiate

D

with respect to

d

_o to find the critical points.

Solve the derivative equation to find the value of

d

_o that minimizes

D

. Use this value to calculate the minimum distance

D

in terms of

f

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Converging Lens

A converging lens, or convex lens, is a transparent optical device that bends light rays inward to a focal point. It has a positive focal length and is used to form real images when the object is placed outside its focal length. The behavior of light through a converging lens is governed by the lens formula, which relates the object distance, image distance, and focal length.

Lens Formula

The lens formula is a mathematical relationship that describes the relationship between the object distance (u), image distance (v), and focal length (f) of a lens. It is expressed as 1/f = 1/v + 1/u. This formula is essential for determining the positions of the object and image in relation to the lens, particularly when calculating distances in optical systems.

Real Image

A real image is formed when light rays converge and can be projected onto a screen. In the case of a converging lens, a real image occurs when the object is placed outside the focal length. The characteristics of a real image include being inverted and having a distance from the lens that can be calculated using the lens formula, which is crucial for determining the minimum distance between the object and its image.