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?

Verified step by step guidance

1. 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.
2. 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 \( d_i \).
3. 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.
4. 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 \( f \).
5. 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 \( R_2 \).

Key Concepts

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.

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.

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.