Ch 34: Geometric Optics

Young & Freedman Calc - University Physics 15th Edition

Young & Freedman Calc15th EditionUniversity PhysicsISBN: 9780135159552Non è quello che usi tu?Cambia libro di testo

Young & Freedman Calc 15th Edition

Ch 34: Geometric Optics

Problema 38

Capitolo 33, Problema 38

A lensmaker wants to make a magnifying glass from glass that has an index of refraction n = 1.55 and a focal length of 20.0 cm. If the two surfaces of the lens are to have equal radii, what should that radius be?

Guida verificata passo dopo passo

Start with the lensmaker's equation:

\frac{1}{f} = (n - 1) * (\frac{1}{R} - \frac{1}{R'})

, where

f

is the focal length,

n

is the index of refraction, and

R

and

R'

are the radii of curvature of the two surfaces.

Since the two surfaces of the lens are to have equal radii, set

R = - R'

(the negative sign accounts for the opposite curvature directions). Substitute this into the lensmaker's equation:

\frac{1}{f} = (n - 1) * (\frac{1}{R} - \frac{1}{-}).

Simplify the equation:

\frac{1}{f} = (n - 1) * (\frac{1}{R} + \frac{1}{R}) = (n - 1) * \frac{2}{R} .

Rearrange the equation to solve for

R

R = \frac{2}{f * (n - 1)} .

Substitute the given values:

f = 20.0

cm and

n = 1.55

. This will allow you to calculate the radius

R

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Lensmaker's Equation

The Lensmaker's Equation relates the focal length of a lens to the radii of curvature of its two surfaces and the refractive index of the material. It is expressed as 1/f = (n - 1) * (1/R1 - 1/R2), where f is the focal length, n is the refractive index, and R1 and R2 are the radii of curvature. For a lens with equal radii, R1 = R2 = R, simplifying the equation significantly.

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Lens Maker Equation

Index of Refraction

The index of refraction (n) is a dimensionless number that describes how light propagates through a medium. It is defined as the ratio of the speed of light in a vacuum to the speed of light in the medium. A higher index indicates that light travels slower in that medium, affecting how light bends when entering or exiting the lens, which is crucial for determining the lens's focal length.

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Index of Refraction

Focal Length

The focal length of a lens is the distance from the lens to the focal point, where parallel rays of light converge after passing through the lens. It is a key parameter in lens design, influencing the magnification and image formation. In this scenario, the focal length is given as 20.0 cm, which is essential for calculating the required radius of curvature for the lens surfaces.

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Spinning on a string of variable length

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