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Ch 34: Ray Optics
Knight Calc - Physics for Scientists and Engineers 5th Edition
Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoKnight Calc 5th EditionCh 34: Ray OpticsProblema 70
Capitolo 34, Problema 70

A plano-concave glass lens (flat on one side, concave on the other) creates an with magnification +0.40 of an object 75 cm from the lens. What is the radius of curvature of the lens's curved surface?

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Identify the given values: The magnification \( M = +0.40 \), the object distance \( d_o = 75 \ \text{cm} \), and the lens is plano-concave (one side flat, the other concave). The goal is to find the radius of curvature \( R \) of the curved surface.
Use the magnification formula \( M = -\frac{d_i}{d_o} \), where \( d_i \) is the image distance. Rearrange to solve for \( d_i \): \( d_i = -M \cdot d_o \). Substitute \( M = +0.40 \) and \( d_o = 75 \ \text{cm} \) to find \( d_i \).
Apply the lens maker's equation for a plano-concave lens: \( \frac{1}{f} = (n - 1) \left( \frac{1}{R_1} - \frac{1}{R_2} \right) \). Since one side is flat, \( R_1 = \infty \), so the equation simplifies to \( \frac{1}{f} = (n - 1) \left( -\frac{1}{R_2} \right) \), where \( R_2 \) is the radius of curvature of the concave surface.
Relate the focal length \( f \) to the object and image distances using the lens formula: \( \frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i} \). Substitute the values of \( d_o \) and \( d_i \) to calculate \( f \).
Combine the simplified lens maker's equation \( \frac{1}{f} = -(n - 1) \frac{1}{R_2} \) with the value of \( f \) obtained in the previous step. Rearrange to solve for \( R_2 \): \( R_2 = -(n - 1) \cdot f \). Use the refractive index of glass (\( n \)) to find \( R_2 \).

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Lens Formula

The lens formula relates the object distance (u), image distance (v), and focal length (f) of a lens. It is given by the equation 1/f = 1/v - 1/u. Understanding this formula is crucial for determining the focal length of the lens, which is necessary for further calculations regarding the radius of curvature.
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Lens Maker Equation

Magnification

Magnification (M) is the ratio of the height of the image to the height of the object, and it can also be expressed as M = -v/u for lenses. In this case, a magnification of +0.40 indicates that the image is upright and smaller than the object. This relationship helps in finding the image distance, which is essential for applying the lens formula.
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Mirror Equation

Radius of Curvature

The radius of curvature (R) of a lens is the radius of the sphere from which the lens surface is derived. For a plano-concave lens, the relationship between the focal length (f) and the radius of curvature is given by the formula f = R/2. Knowing the focal length allows us to calculate the radius of curvature, which is a key aspect of understanding the lens's optical properties.
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