The left end of a long glass rod 8.00 cm in diameter, with an index of refraction of 1.60, is ground and polished to a convex hemispherical surface with a radius of 4.00 cm. An object in the form of an arrow 1.50 mm tall, at right angles to the axis of the rod, is located on the axis 24.0 cm to the left of the vertex of the convex surface. Find the position and height of the of the arrow formed by paraxial rays incident on the convex surface. Is the erect or inverted?

Verified step by step guidance

1. Identify the given values: radius of curvature (R) of the convex surface is 4.00 cm, index of refraction of glass (n2) is 1.60, index of refraction of air (n1) is 1.00, object height (h_o) is 1.50 mm, and object distance (s_o) is -24.0 cm (negative as the object is on the same side as the incoming light).
2. Use the lensmaker's formula for a spherical surface to find the image distance (s_i). The formula is \(\frac{n_2}{s_i} - \frac{n_1}{s_o} = \frac{n_2 - n_1}{R}\). Solve this equation for \(s_i\) by substituting the known values.
3. Calculate the magnification (m) using the formula \(m = -\frac{n_1}{n_2} \times \frac{s_i}{s_o}\). This will help determine the height of the image (h_i) using the relation \(h_i = m \times h_o\).
4. Determine the nature of the image (erect or inverted) based on the sign of the magnification. If magnification (m) is positive, the image is erect; if negative, the image is inverted.
5. Summarize the findings: the position of the image from the vertex of the convex surface, the height of the image, and whether the image is erect or inverted.

Key Concepts

Refraction and the Lens Maker's Equation

Refraction is the bending of light as it passes from one medium to another, which is quantified by the index of refraction. The Lens Maker's Equation relates the focal length of a lens to the radii of curvature of its surfaces and the refractive indices of the materials involved. Understanding this concept is crucial for determining how light behaves when it encounters the convex surface of the glass rod.

Ray Diagrams and Paraxial Rays

Ray diagrams are visual representations used to trace the path of light rays as they interact with optical elements. Paraxial rays are those that make small angles with the optical axis and can be approximated using linear equations. This concept is essential for predicting the position and characteristics of the image formed by the convex surface of the rod.

Image Formation and Magnification

Image formation refers to the process by which light rays converge to create an image of an object. The magnification of an image is the ratio of the height of the image to the height of the object, which can indicate whether the image is erect or inverted. This concept is vital for determining the final position, height, and orientation of the arrow image produced by the convex surface.