Radius of curvature of partially cut glass surface

[desmond iking]

Homework Statement

when the glass is partially cut( as shown in the photo ) , the centre of curvature is inside the denser medium (glass), so the centre of curvature should be lower than point Q in the diagram . am i correct? by saying that the centre of curvature is inside the denser medium , the radius of curvature should be positive ( upload error, so i attached the image directly here ) , but the notes from my book give sample ans which the radius of curvature is negative because the object is concave relative to the refracting surface.


how can the notes spherical surface convex to the less dense medium or C in the denser medium correct?

Homework Equations

The Attempt at a Solution

 

[Simon Bridge]

Homework Statement

when the glass is partially cut( as shown in the photo ) , the centre of curvature is inside the denser medium (glass), so the centre of curvature should be lower than point Q in the diagram .

I don't see a point Q in any of the attachments.

 

[desmond iking]

I don't see a point Q in any of the attachments.

point Q is here. refer to the latest attachment please.

 

[Simon Bridge]

Oh OK - the top of the block is 3cm above Q, which is on the table top. It's radius of curvature is 10cm.
Therefore the center of curvature is below the table-top.

If the center of curvature is on the same side as where the light comes from, then it's sign is negative.
Please review: http://www.math.ubc.ca/~cass/courses/m309-01a/chu/MirrorsLenses/refraction-curved.htm

Let Side A of an optical component be the side from which light starts, and let Side B be the side to which light travels. With mirrors, Sides A and B are identical. If s is the source distance, i is the image distance, R is the radius of curvature, and f is the focal length, then, our sign conventions are as follows.

* The sign of s is determined by Side A. If the source is on Side A, s is positive; if it is on the side opposite to Side A, s is negative.
* The signs of i, R, f are determined by Side B. For the image and the focal point, their measurements are positive if they are on Side B, and negative if they are on the side opposite to Side B. R is positive provided the centre of curvature is on Side B; it will be negative if it is on the side opposite Side B.
* It cannot be stressed enough that for a mirror, Sides A and B are the same.​

 

[HalfLostAndSearching]

I would like to clarify that the radius of curvature of a partially cut glass surface can be positive or negative depending on the orientation of the surface. In general, the radius of curvature is defined as the distance between the center of curvature and the surface of the sphere that the surface would form if it were complete. Therefore, if the surface is concave, the radius of curvature would be positive, and if it is convex, the radius of curvature would be negative.

In the case of a partially cut glass surface, the center of curvature is indeed inside the denser medium, as stated in the question. This means that the surface would be concave relative to the refracting surface, and thus the radius of curvature would be positive. The notes from your book may have given a negative radius of curvature for a concave surface because they may have considered the orientation of the surface relative to the less dense medium. However, it is important to note that the radius of curvature is always measured relative to the center of curvature and not the medium in which it is located.

To answer your question about the notes stating that the spherical surface is convex to the less dense medium, it is possible for a concave surface to appear convex when viewed from the opposite side. This is because the curvature of the surface changes as it transitions from one medium to another, leading to a change in direction of light rays. This can create an illusion of the surface being convex when it is, in fact, concave.

In conclusion, it is important to consider the orientation of the surface and the medium in which it is located when determining the radius of curvature of a partially cut glass surface. The notes may have provided a different answer due to the different perspectives being considered. However, the radius of curvature should always be measured relative to the center of curvature and not the medium.