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 calculate the size of the letter G when viewed directly from above, we need to consider the effect of refraction at the curved surface of the hemisphere. To simplify the calculation, let's break down the problem into steps:
Step 1: Determine the height of the letter G as viewed from the flat side of the hemisphere.
Since the letter G is 4.0 mm tall when viewed from the flat side of the hemisphere, this will remain unchanged.
Step 2: Determine the height of the letter G as it undergoes refraction at the curved surface.
To find this, we need to use the formula for refraction at a curved surface, which relates the height of the object before and after refraction, and the refractive index of the medium.
The formula for refraction at a curved surface is given by:
n1 / h1 = n2 / h2
Where:
- n1 is the refractive index of the medium before refraction (in this case, air which has a refractive index of approximately 1),
- h1 is the height of the object before refraction (4.0 mm),
- n2 is the refractive index of the medium after refraction (1.5), and
- h2 is the height of the object after refraction (which we want to find).
Plugging in the values, we have:
1 / 4.0 mm = 1.5 / h2
Cross-multiply to solve for h2:
h2 = (4.0 mm) * (1.5 / 1)
= 6.0 mm
Step 3: Calculate the size of the letter G when viewed directly from above.
To find the size of the letter G as viewed directly from above, we need to consider the geometry of the situation. Since we are viewing the hemisphere from directly above, we can consider it as a circle on a plane.
The size of the letter G can be calculated as the ratio of the height of the letter to the radius of curvature of the hemisphere:
Size of G = (h2 / 2) / R
Where:
- h2 is the height of the letter G after refraction (6.0 mm), and
- R is the radius of curvature of the hemisphere (20 cm = 200 mm).
Plugging in the values, we have:
Size of G = (6.0 mm / 2) / 200 mm
= 0.015
So, the size of the letter G when viewed directly from above is approximately 0.015.