A cylindrical rod of glass with index of refraction 1.50 is...

A cylindrical rod of glass with index of refraction 1.50 is immersed in water with index 1.33 . The diameter of the rod is 9.00 $\mathrm{cm}$ . The outer part of each end of the rod has been ground away to form each end into a hemisphere of radius $4.50 \mathrm{cm} .$ The central portion of the rod with straight sides is 75.0 $\mathrm{cm}$ long. An object is situated in the water, on the axis of the rod, at a distance of 100 $\mathrm{cm}$ from the vertex of the nearer hemisphere. (a) Find the location of the final image formed by refraction at both surfaces. (b) Is the final image real or virtual? Upright or inverted? Enlarged or diminished?

Key Concepts

Spherical Surface Refraction

When light encounters a curved (spherical) interface between two media, its refraction is described using formulas that include the radius of curvature of the surface. This approach is similar to the mirror or lens equations but modified to account for the change in refractive index. Analyzing refraction at spherical surfaces involves applying the appropriate sign conventions and using formulae that help determine the relationships between object distance, image distance, and the radius of curvature.

Image Formation

Image formation involves the convergence or divergence of light rays to create an image of an object. In the context of refraction through curved surfaces, one can determine the location, orientation, and size of the image by tracking how light rays are bent at each interface. The principles used help distinguish whether an image is real or virtual, upright or inverted, and whether it is magnified or reduced in size.

Refraction

Refraction is the bending of light when it passes from one medium to another with a different optical density. This phenomenon is governed by Snell's law, which relates the angles of incidence and refraction to the indices of refraction of the two media. Understanding refraction is essential when analyzing how light paths are altered at the boundaries between substances such as water and glass.

Index of Refraction

The index of refraction is a measure of how much the speed of light is reduced inside a medium compared to its speed in a vacuum. It plays a crucial role in determining the degree to which light bends when entering or exiting a material. Differences in the indices of refraction between media are responsible for the change in direction of light rays during the refraction process.

Sign Conventions in Geometric Optics

Sign conventions provide a systematic way to assign positive or negative values to distances, angles, and radii of curvature in geometric optics. They are crucial when applying equations for refraction or reflection, especially with spherical surfaces and lens combinations, to maintain consistency and accurately predict the behavior of light rays as they form images.

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