(II) A planoconvex lens (Fig. 33–2a) has one flat surface and the other has R = 15.3 cm. This lens is used to view a red and yellow object which is 62.0 cm away from the lens. The index of refraction of the glass is 1.5106 for red light and 1.5226 for yellow light. What are the locations of the red and yellow images formed by the lens?

Verified step by step guidance

1. Step 1: Understand the problem. The planoconvex lens has one flat surface and one curved surface with radius of curvature R = 15.3 cm. The object is located 62.0 cm away from the lens, and the refractive indices for red and yellow light are given as 1.5106 and 1.5226, respectively. We need to find the image locations for both red and yellow light using the lens maker's formula and the thin lens equation.
2. Step 2: Use the lens maker's formula to calculate the focal length of the lens for each wavelength of light. The formula is: $\frac{1}{f}=(n-1)\left[\frac{1}{R}\right]$, where n is the refractive index, and R is the radius of curvature. For the flat surface, the curvature is zero, so only the curved surface contributes to the calculation.
3. Step 3: Substitute the refractive indices for red and yellow light into the lens maker's formula to find the focal lengths for each color. For red light, use n = 1.5106, and for yellow light, use n = 1.5226. Perform the calculations separately for each wavelength.
4. Step 4: Apply the thin lens equation to find the image distances for both red and yellow light. The thin lens equation is: $\frac{1}{f}=\frac{1}{d}+\frac{1}{\mathrm{d\text{'}}}$, where f is the focal length, d is the object distance (62.0 cm), and d' is the image distance. Rearrange the equation to solve for d'.
5. Step 5: Solve for d' (image distance) for both red and yellow light using their respective focal lengths calculated in Step 3. This will give the locations of the images formed by the lens for each wavelength of light.

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. For a planoconvex lens, the focal length can be determined using the radius of curvature (R) with the formula f = R/2. Understanding this formula is essential for calculating the positions of the images formed by the lens.

Refraction and Snell's Law

Refraction is the bending of light as it passes from one medium to another, which is governed by Snell's Law. This law states that n1 * sin(θ1) = n2 * sin(θ2), where n is the refractive index and θ is the angle of incidence or refraction. The different refractive indices for red and yellow light in the lens material will affect how each color is focused, leading to different image positions.

Chromatic Aberration

Chromatic aberration occurs when a lens fails to focus all colors to the same convergence point due to varying refractive indices for different wavelengths of light. In this case, the planoconvex lens has different indices for red and yellow light, causing the images of these colors to form at slightly different locations. Understanding this phenomenon is crucial for predicting the positions of the red and yellow images.