A point source of light is placed at the centre of the bottom of a 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 the liquid for which the source is not visible from above?
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In order to find the maximum height of the liquid for which the source is not visible from above, we need to consider the concept of critical angle and total internal reflection.
To understand this, let's break down the problem step by step:
Step 1: Identify the relevant formulas and concepts.
- Snell's Law: n₁sinθ₁ = n₂sinθ₂
- Critical angle (θc) and total internal reflection.
Step 2: Determine the critical angle.
- The critical angle (θc) is the angle of incidence that produces an angle of refraction of 90°, resulting in light being completely reflected back into the original medium.
- For a medium with a refractive index of n₁ and another medium with a refractive index of n₂, the critical angle can be found using the equation: sinθc = n₂/n₁.
Step 3: Calculate the critical angle using the given refractive indices.
- In this case, the refractive index of the liquid is given as 5/3.
- So, we can calculate the critical angle using the formula: sinθc = (5/3) / 1 = 5/3.
- Therefore, the critical angle (θc) is sin⁻¹(5/3).
Step 4: Determine the height of the liquid that would cause total internal reflection.
- When the angle of incidence (θ₁) is greater than the critical angle (θc), total internal reflection occurs.
- In this case, the angle of incidence is the angle between the normal and the ray of light just before it enters the liquid.
- The angle of incidence (θ₁) can be determined using trigonometry, considering the radius of the opaque disc and the maximum height of the liquid.
- The height of the liquid where total internal reflection occurs can be found using the formula: max_height = radius / tan(θ₁).
Step 5: Calculate the maximum height of the liquid.
- To find the value of θ₁, we consider a triangle formed by the radius of the disc, the height of the liquid (max_height), and the hypotenuse created by the refracted light ray within the liquid.
- Using trigonometry, we can determine that tan(θ₁) = max_height / radius.
- Rearranging the equation, we find: max_height = radius * tan(θ₁).
- In this case, the radius of the disc is given as 1.0 cm, and θ₁ is the angle of incidence (which is equal to the critical angle θc).
- So, the maximum height of the liquid for which the source is not visible from above is: max_height = 1.0 cm * tan(sin⁻¹(5/3)).
Step 6: Calculate the final result.
- Perform the calculation using the formula above to find the maximum height of the liquid.
By following these steps, you will be able to calculate the maximum height of the liquid for which the source is not visible from above based on the given parameters.