The left end of a long glass rod 8.00 cm in diameter and with an index of refraction of 1.60 is

The left end of a long glass rod 8.00 cm in diameter and...

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

Key Concepts

Lateral Magnification

Lateral magnification in optical systems quantifies how the size of the image compares to the size of the object. It is defined as the ratio of the image height to the object height and is related to the distances of the object and image from the optical surface. In the context of a refracting spherical surface, calculating the magnification involves the derived image distance and helps determine whether the image is upright or inverted relative to the object.

Spherical Refracting Surface

This concept refers to the treatment of light refraction at a curved (spherical) boundary between two media of different refractive indices. For a spherical surface, there is a specific formula, (n?/s) + (n?/s') = (n? - n?)/R, that relates the object distance, image distance, and the radius of curvature while taking into account the refractive indices of the media. This formula is fundamental in determining where an image is formed when light passes through a curved surface under the paraxial approximation.

Paraxial Approximation

The paraxial approximation is an assumption used in optics where rays make only small angles with the optical axis. This simplification allows the use of linear approximations (small-angle approximations) in mathematical formulas, making it easier to analyze and calculate image positions and magnifications in optical systems, especially when using the spherical refracting surface equation.

Sign Conventions in Geometrical Optics

Understanding and applying the correct sign conventions is critical when solving problems involving curved surfaces in optics. The sign conventions determine the positive and negative directions for object distances, image distances, and radii of curvature, ensuring that the formulas for refraction and reflection yield physically meaningful results. For instance, the radius of curvature is taken as positive or negative depending on the direction the center of curvature lies with respect to the incoming light.

Transcript

00:01 In this problem, i drew a picture to indicate all the pertinent bits of information that are given.

00:07 So let's start with the object.

00:12 We're told that the object is at right angles to the optic axis and it's an arrow.

00:18 And we're given the height of 1 .5 millimeters.

00:22 And we're given s, which is the distance from the object to the vertex of this glass rod, that at 24 centimeters.

00:29 And we're told that we have paraxial rays that are basically illuminating this object and it's going to get refracted into this glass.

00:41 And we're asked to find the image distance, s -prime, and the image height, y -prime.

00:52 So i drew a figure here just to indicate how the image will look, and we can see podality from these two rays that are being refracted.

01:04 Well, this one is going straight to the center here, through the center of this glass rod where the radius of curvature is.

01:15 So it will go straight through.

01:18 However, this beam that's coming to the vertex will get diffracted.

01:22 So then where these two intersect is where my image will form.

01:28 And we have the annex of refraction of glass.

01:30 We have the radius of curvature.

01:35 Here we have an a, which is assumed to be here.

01:39 And now we're going to use equations 34 .11 and 1 -2 to find.

01:51 So from this one, we'll find s -primed...

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