A tank is filled with water to a height of 12.5 cm. The apparent depth of a needle lying at the bottom of the tank is measured by a microscope to be 9.4 cm. What is the refractive index of water? If water is replaced by a liquid of refractive index 1.63 up to the same height, by what distance would the microscope have to be moved to focus on the needle again?
(i) The refractive index of water is approximately 1.33.
(ii) The microscope would have to be moved by approximately 1.73 cm.
Explanation
The solution applies the formula for apparent depth found in the text, which states that apparent depth is the real depth divided by the refractive index. In the first part, this formula is rearranged to solve for the refractive index of water. In the second part, the same formula is used with the new refractive index to find the new apparent depth. The difference between the original apparent depth and the new apparent depth determines how far the microscope must be moved to refocus.
Solution Steps
Step 1: Identify the given values: Real depth () = 12.5 cm, Apparent depth () = 9.4 cm.
Step 2: Use the relation: Apparent depth = Real depth / Refractive index ().
Step 3: Calculate the refractive index of water: .
Step 4: Calculate the new apparent depth with the liquid (): New apparent depth = cm.
Step 5: Determine the distance to move the microscope: Distance = .