Suppose Fig. 32–37 shows a cylindrical rod whose end has a radius of curvature R = 2.0 cm, and the rod is immersed in water with index of refraction of 1.33. The rod has index of refraction 1.49. Find the location and height of the image of an object 2.0 mm high located 23 cm away from the rod.

Verified step by step guidance

1. Step 1: Understand the problem setup. The cylindrical rod acts as a lens due to the difference in refractive indices between the rod (n₁ = 1.49) and water (n₂ = 1.33). The radius of curvature of the rod's end is R = 2.0 cm, and the object is located at a distance of 23 cm from the rod. The height of the object is 2.0 mm. We need to find the location and height of the image formed.
2. Step 2: Use the lensmaker's equation for spherical surfaces to calculate the focal length of the rod's end. The equation is: $\frac{1}{f}=\left(\frac{1}{R}\right)\left(\frac{n}{-}\right)$, where n is the ratio of refractive indices (n₁/n₂). Substitute the values: n₁ = 1.49, n₂ = 1.33, and R = 2.0 cm.
3. Step 3: Once the focal length is determined, use the thin lens equation to find the image distance (s'). The thin lens equation is: $\frac{1}{f}=\frac{1}{s}+\frac{1}{\mathrm{s\text{'}}}$, where s is the object distance (23 cm) and f is the focal length calculated in Step 2. Rearrange the equation to solve for s'.
4. Step 4: To find the height of the image, use the magnification formula: $M=\frac{-}{\mathrm{s\text{'}}}$. The magnification (M) is the ratio of the image height (h') to the object height (h). Substitute the values of s' (from Step 3) and s (23 cm) to calculate M, and then use $\mathrm{h\text{'}}=M\times h$ to find the image height.
5. Step 5: Interpret the results. The sign of s' will indicate whether the image is real or virtual, and the sign of h' will indicate whether the image is upright or inverted. Combine the calculated values of s' and h' to describe the location and height of the image.

Key Concepts

Refraction

Refraction is the bending of light as it passes from one medium to another with a different index of refraction. This phenomenon occurs because light travels at different speeds in different materials. The degree of bending can be described by Snell's Law, which relates the angles of incidence and refraction to the indices of refraction of the two media.

Lens Maker's Equation

The Lens Maker's Equation is used to determine the focal length of a lens based on its curvature and the indices of refraction of the lens material and the surrounding medium. For a curved surface, the equation takes into account the radius of curvature and the refractive indices, allowing for the calculation of how light will converge or diverge after passing through the lens.

Image Formation

Image formation refers to the process by which light rays converge or diverge to create an image of an object. In optics, the characteristics of the image, such as its location, height, and orientation, can be determined using ray diagrams or mathematical formulas derived from the principles of refraction and lens behavior. The position and size of the image depend on the object's distance from the lens and the lens's focal length.