An object is placed 96.0 cm from a glass lens (n = 1.52) with one concave surface of radius 22.0 cm and one convex surface of radius 18.5 cm.
(a) Where is the final image?
(b) What is the magnification?

Verified step by step guidance

1. Determine the focal length of the lens using the Lensmaker's Equation: \( \frac{1}{f} = (n - 1) \left( \frac{1}{R_1} - \frac{1}{R_2} \right) \), where \( n \) is the refractive index of the lens material, \( R_1 \) is the radius of curvature of the first surface (positive for convex, negative for concave), and \( R_2 \) is the radius of curvature of the second surface.
2. Substitute the given values into the Lensmaker's Equation: \( n = 1.52 \), \( R_1 = 18.5 \ \text{cm} \) (convex, positive), and \( R_2 = -22.0 \ \text{cm} \) (concave, negative). Solve for \( f \), the focal length of the lens.
3. Use the thin lens equation to find the image distance \( d_i \): \( \frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i} \), where \( d_o \) is the object distance (96.0 cm) and \( f \) is the focal length calculated in the previous step. Rearrange to solve for \( d_i \): \( \frac{1}{d_i} = \frac{1}{f} - \frac{1}{d_o} \).
4. Calculate the magnification \( M \) using the formula \( M = -\frac{d_i}{d_o} \), where \( d_i \) is the image distance (calculated in the previous step) and \( d_o \) is the object distance (96.0 cm).
5. Interpret the results: The sign of \( d_i \) will indicate whether the image is real or virtual (positive for real, negative for virtual), and the sign of \( M \) will indicate whether the image is upright or inverted (positive for upright, negative for inverted). The magnitude of \( M \) gives the size ratio of the image to the object.

Key Concepts

Lens Maker's Equation

The Lens Maker's Equation relates the focal length of a lens to the radii of curvature of its surfaces and the refractive index of the lens material. It is given by the formula: 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 of the lens surfaces. This equation is essential for determining the focal length, which is crucial for locating the image formed by the lens.

Image Formation by Lenses

Image formation by lenses involves the interaction of light rays as they pass through the lens. Depending on the type of lens (convex or concave), the image can be real or virtual, upright or inverted, and magnified or reduced in size. The position of the image can be found using the lens formula: 1/f = 1/do + 1/di, where do is the object distance and di is the image distance. Understanding this concept is vital for solving the problem.

Magnification

Magnification is a measure of how much larger or smaller an image appears compared to the object. It is defined as the ratio of the height of the image to the height of the object and can also be calculated using the formula: M = -di/do, where M is the magnification, di is the image distance, and do is the object distance. This concept is important for determining the size and orientation of the image produced by the lens.