a room which is h = 9.0 feet tall and
r = 12.0 feet wide. Attached to the ceiling is a piece of glass (with unknown index of refraction) which is 2.5 feet thick. A laser pointer in the bottom left corner is aimed so that the ray which reflects off the glass hits the room's bottom right corner.
(a) What is the ray's angle of incidence at the glass?
°
(b) If the ray which refracts into the glass hits the ceiling 7.7 feet away from the left wall, what is the refracted ray's angle of refraction at the glass?
°
(c) What is the glass's index of refraction?
Draw the figure, use geometry to calculate the angles.
To answer these questions, we need to use Snell's law, which states that the ratio of the sine of the angle of incidence to the sine of the angle of refraction is equal to the ratio of the velocities of light in the two mediums.
(a) The first step is to calculate the height of the room above the glass. Since the glass is 2.5 feet thick and the room's height is 9.0 feet, the height above the glass is 9.0 - 2.5 = 6.5 feet.
Next, we need to calculate the distance from the bottom left corner to the bottom right corner of the room along the base. The width of the room is given as 12.0 feet.
Using these values, we can form a right-angled triangle with the height and base as the two sides. We can calculate the angle of incidence at the glass using the formula: angle = atan(height/base).
Here, height = 6.5 feet and base = 12.0 feet.
Plugging in the values, we get: angle = atan(6.5/12.0) ≈ 30.96°
Therefore, the ray's angle of incidence at the glass is approximately 30.96°.
(b) To find the angle of refraction at the glass, we need to use the information that the refracted ray hits the ceiling 7.7 feet away from the left wall. Using this distance as the base and the height above the glass (6.5 feet) as the side, we can again form a right-angled triangle.
Using the formula: angle = atan(height/base), with height = 6.5 feet and base = 7.7 feet, we can calculate the angle of refraction at the glass.
Plugging in the values, we get: angle = atan(6.5/7.7) ≈ 41.52°
Therefore, the refracted ray's angle of refraction at the glass is approximately 41.52°.
(c) To find the glass's index of refraction, we need to use the relationship between the angles of incidence and refraction and the refractive indices of the mediums.
According to Snell's law, the ratio of the sines of the angles of incidence and refraction is equal to the ratio of the refractive indices:
sin(angle of incidence) / sin(angle of refraction) = n2 / n1,
where n1 is the refractive index of the medium where the incident ray comes from (air in this case) and n2 is the refractive index of the medium where the refracted ray enters (the glass in this case).
Rearranging the equation, we have:
n2 = (sin(angle of incidence) * n1) / sin(angle of refraction).
The angle of incidence is 30.96° (from part a) and the angle of refraction is 41.52° (from part b). Assuming that light is passing from air into glass, the refractive index of air can be considered to be approximately 1.
Substituting the values into the formula:
n2 = (sin(30.96°) * 1) / sin(41.52°).
Using a calculator, we can find:
n2 ≈ 1.633.
Therefore, the glass's index of refraction is approximately 1.633.