A LARGE BOWL CONTAINS WATER 10cm DEEP AND HAS A SMALL ELLECTRIC LAMP AT THE BOTTOM.A DISC OF CARDBOARD IS FLOATING ON THE SURFACE WITH ITS CENTRE VERTICALLY ABOVE THE LAMP, AND IS OF SUCH DIAMETER THAT THE LAMP CAN JUST NOT BE SEEN BY A PERSON LOOKING INTO THE WATER. CALCULATE THE DIAMETER OF THE DISC.REFRATIVE INDEX OF WATER IS 1,33.
To calculate the diameter of the disc, we need to use the concept of refraction and the properties of light when it passes from one medium (air) to another (water). Here are the steps to solve the problem:
1. Determine the critical angle: The critical angle is the angle at which light passing from a more dense medium (water) to a less dense medium (air) undergoes total internal reflection. To find the critical angle, we use the formula:
critical angle = inverse sine (1 / refractive index)
In this case, the refractive index of water is given as 1.33. So,
critical angle = inverse sine (1 / 1.33) = inverse sine (0.7518) = 48.8 degrees (approximately)
2. Draw a diagram: Visualize the setup described in the problem. Draw a side view of the water-filled bowl with the disc floating on the surface and the viewer's line of sight looking down into the water.
3. Apply Snell's Law at the surface of the water: At the surface of the water, we need to consider the path of light from the lamp to the person's eye. Snell's Law relates the angle of incidence (θ1) and the angle of refraction (θ2) with the refractive indices of the two media:
refractive index of air x sin(θ1) = refractive index of water x sin(θ2)
Since the angle of incidence (θ1) is 90 degrees (perpendicular to the surface), the equation simplifies to:
sin(θ2) = refractive index of air / refractive index of water
sin(θ2) = 1 / 1.33
4. Find the angle of refraction: Calculate the angle of refraction (θ2) using the inverse sine function:
θ2 = inverse sine (1 / 1.33) = inverse sine (0.7518) = 48.8 degrees (approximately)
5. Determine the minimum angle for the viewer: To find the minimum angle at which the viewer can observe the lamp through the water, we add the critical angle and the angle of refraction:
minimum angle = critical angle + angle of refraction
minimum angle = 48.8 + 48.8 = 97.6 degrees (approximately)
6. Calculate the height of the disc above the lamp: Since the person looking into the water cannot see the lamp, the disc must be blocking the direct line of sight. When the person is looking along the minimum angle, the height of the disc will equal the depth of the water:
height of the disc = depth of water = 10 cm
7. Find the diameter of the disc: The diameter of the disc is twice the distance from the center of the disc to the edge of the disc. To calculate this distance, we can use trigonometry. Using the right triangle formed by the diameter of the disc, the height of the disc, and half of the resulting triangle:
sin(minimum angle) = height of the disc / (0.5 x diameter of the disc)
Rearranging the equation:
diameter of the disc = 2 x (height of the disc / sin(minimum angle))
diameter of the disc = 2 x (10 cm / sin(97.6 degrees))
diameter of the disc = 28.1 cm (approximately)
Therefore, the diameter of the disc is approximately 28.1 cm.