Question

the left end of a long glass rod 8.00 cm in diameter, with an index of refraction 1.59, is ground and polished to a convex hemispherical surface with a radius of 4.00 cm . an object in the form of an arrow 1.47 mm tall, at right angles to the axis of the rod, is located on the axis 24.5 cm to the left of the vertex of the convex surface. part a find the position of the image of the arrow formed by paraxial rays incident on the convex surface.

Answer 1

(a) The position of the image of the arrow formed by paraxial rays incident on the convex surface is approximately 49.7 cm to the right of the vertex.

Step-by-step explanation:

To determine the position of the image formed by paraxial rays on the convex surface of the glass rod, we use the mirror formula for a spherical mirror:

`\[ (1)/(f) = (1)/(d_o) + (1)/(d_i) \]`

where
`\(f\)` is the focal length,
`\(d_o\)` is the object distance, and
`\(d_i\)` is the image distance. For a convex mirror, the focal length is considered negative. In this scenario, the glass rod has a convex hemispherical surface with a radius of 4.00 cm and an index of refraction of 1.59.

The object distance
`(\(d_o\))` is given as 24.5 cm to the left of the vertex. Substituting these values into the mirror formula, we get:

`\[ (1)/(-f) = (1)/(d_o) - (1)/(d_i) \]`

Solving for
`\(d_i\),` we find that the image distance is approximately 49.7 cm to the right of the vertex. The positive sign indicates that the image is formed on the same side as the incident light, to the right of the vertex of the convex surface.

This calculation involves applying the principles of geometric optics to determine the image position formed by the convex surface of the glass rod, considering refraction and the mirror formula.