A points source of light is placed at the bottom of jar having a liquid of refractive index 5/3.An opaque disc of radius 1.0 cm is placed on the liquid surface with its centre vertically above the source. What is the maximum height of liquid for which the source is not visible from above
measuring angles from the normal
sinTheta=3/5
from Snells law.
then using proprotion
height/4=1cm/3
height= 1.333cm
To find the maximum height of the liquid for which the source is not visible from above, we can use the concept of critical angle.
The critical angle is the angle of incidence at which the refracted ray just grazes along the boundary between two mediums. In this case, the medium is changing from the liquid (with refractive index 5/3) to air (with refractive index 1).
Here's how we can calculate the critical angle:
1. Determine the refractive index of the interface between the liquid and air:
Refractive index of liquid (n₁) = 5/3
Refractive index of air (n₂) = 1
2. Use Snell's Law to calculate the critical angle (θc) using the formula:
n₁ * sin(θc) = n₂ * sin(90°)
sin(θc) = n₂ / n₁
θc = arcsin(n₂ / n₁)
3. Calculate the maximum height of liquid:
Since the disc is opaque, the source of light will not be visible from above when its rays are totally internally reflected within the liquid.
- Let R be the radius of the disc (1.0 cm),
- d be the depth of the disc (distance from the disc to the bottom of the jar), and
- h be the height of the liquid (distance from the top of the liquid to the bottom of the jar).
In order to find the maximum height h for which the source is not visible, the angle of incidence (θi) at the point where the light ray just grazes along the boundary will be equal to the critical angle (θc). The distance travelled by the ray in the liquid medium will be the diagonal of a right triangle with base R and height h+d:
sin(θi) = R / √(h² + (d+R)²)
Since θi = θc, we substitute θc as:
sin(θc) = R / √(h² + (d+R)²)
Rearranging the formula, we can solve for h:
h = √((R / sin(θc))² - (d+R)²) - d
Plug in the known values to calculate the maximum height h for which the source is not visible.
Please note that for accurate calculations, you'll need to convert the measurements to suitable units (e.g., meters) and ensure precision in calculations.