A transparent hemisphere of refractive index 1.5 and radius of curvature 20 cm is placed flat side down on a piece of paper with a 4.0 mm tall letter G written on it such that G is in the center.What is the size of G when viewed directly from above?
To determine the size of the letter G when viewed directly from above through the transparent hemisphere, we need to consider the refraction of light at the curved surface of the hemisphere.
1. Calculate the refractive index of the hemisphere:
The refractive index (n) of the hemisphere is given as 1.5.
2. Calculate the radius of the hemisphere:
The radius of curvature (R) of the hemisphere is given as 20 cm.
3. Determine the critical angle:
The critical angle (θc) can be calculated using the refractive index (n) and Snell's Law: sin(θc) = 1/n.
Plugging in the value of n = 1.5 into the equation, we find:
sin(θc) = 1/1.5
θc ≈ 41.8 degrees.
4. Calculate the angle of incidence:
When light passes from air into the hemisphere, it bends toward the normal. Therefore, the angle of incidence (θi) is given by:
θi = 90 degrees - θc
θi = 90 - 41.8
θi ≈ 48.2 degrees.
5. Calculate the angle of refraction:
Using Snell's Law again, we can calculate the angle of refraction (θr) when light passes from air to the hemisphere:
sin(θi)/sin(θr) = n
sin(θr) = sin(θi)/n
θr = arcsin(sin(θi)/n)
θr = arcsin(sin(48.2)/1.5)
θr ≈ 31.9 degrees.
6. Calculate the apparent height:
The apparent height (h') can be determined using the formula:
h' = h/(R/(R-h)), where h is the actual height of the letter G.
Given that h = 4.0 mm (0.4 cm), we have:
h' = 0.4 / (20 / (20 - 0.4))
h' ≈ 0.401 cm.
Therefore, when viewed directly from above through the transparent hemisphere, the letter G will appear to be approximately 0.401 cm tall.