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 find the size of the letter G when viewed directly from above, we need to consider the refraction of light as it passes through the transparent hemisphere.
Here's how we can approach this problem step by step:
1. The refractive index of the transparent hemisphere, n = 1.5. This means that light travels slower through the hemisphere than in air (refractive index of air is approximately 1).
2. The radius of curvature of the hemisphere, R = 20 cm. Since the hemisphere is flat side down, the radius of curvature is positive.
3. Place the transparent hemisphere on top of the paper with the letter G. The letter G is 4.0 mm tall (or 0.4 cm).
4. Let's consider the path of the light rays as they pass through the hemisphere. The light ray from the top of the letter G will bend toward the normal (perpendicular) of the hemisphere surface since it travels from a less dense medium (air) to a denser medium (hemisphere).
5. Using Snell's law of refraction, we can calculate the angle of refraction (θ2) given the angle of incidence (θ1) and the refractive indices:
n1*sin(θ1) = n2*sin(θ2)
Since air has a refractive index approximately equal to 1, we have:
1*sin(θ1) = 1.5*sin(θ2)
We can rearrange the equation to find sin(θ2):
sin(θ2) = (1*sin(θ1)) / 1.5
6. Now, we need to find the angle of incidence (θ1). Since θ2 is the angle of light inside the hemisphere, we can use Snell's law again to find θ1 using the following formula:
n2*sin(θ2) = n1*sin(θ1)
Substituting the known values:
1.5*sin(θ2) = 1*sin(θ1)
sin(θ1) = (1.5*sin(θ2)) / 1
7. Calculate the size of G when viewed directly from above. Since the angle of incidence (θ1) is very small (almost 0), we can use the small-angle approximation:
Size of G = 2 * (Radius of curvature) * tan(θ1)
Size of G = 2 * 20 cm * tan(θ1)
8. Plug in the value of sin(θ1) calculated from step 6 into the equation from step 7 to find the size of G.