Question

A paperweight is made of a solid glass hemisphere of index of refraction 1.59. The radius of the circular cross section is 4.0 cm. The hemisphere is placed on its flat surface, with the center directly over a 2.0 mm long line drawn on a sheet of paper. What length of line is seen by someone looking vertically down on the hemisphere? mm

Answer 1

Here the relevant equation to be used is the Snell's law for spherical surfaces to find the image distance, and then use the formula for magnification to find the length of the line observed. The Snell's law for spherical surface is:

where n1 is the index of refraction of the glass (given 1.59), n2 is the index of refraction of air (equal to 1)

s is the source distance. Since the light ray from the 2 mm line starts from inside the glass, s is positive (according to sign convention, if source is on the same side from which the ray of light starts, s is positive ).

Thus, s = 4.0 cm

i is the image distance and R is the radius of the hemisphere.

Given, R = -4.0 cm (negative because the outgoing ray is going opposite to the center of curvature)

Plugging in all the known values in the above equation, we can calculate the image distance.

or

i = - 4.0 cm. This is the image distance.

We can use this to find the magnification. The magnification equation is

, where h' is the length of the image observed and h is the original length of the object

Given h = 2 mm

or h' = 3.18 mm