Question

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 the water? If water is replaced by a liquid of refractive index 1.63 up to the same height, by what distance would a microscope have to be moved to focus on the needle again?

Answer 1

Hint: Use the formula for apparent depth of the medium of refractive index \[\mu \]. We have given both apparent depth and real depth. Therefore, determine the refractive index from it. For the second part, calculate the apparent depth of the needle using the same formula. Subtract it from the former apparent depth to calculate the distance to which the microscope should be moved upward.

Formula used:
The apparent depth of the medium of refractive index \[\mu \] is,
\[{\text{Apparent depth}} = \dfrac{{{\text{Real depth}}}}{\mu }\]

Complete step by step answer:
We know that when the light ray travels from denser medium to rare medium, the light ray bends away from the normal. Therefore, the depth of the medium appears to be shortened.

We know the apparent depth of the medium of refractive index \[\mu \] is,
\[{\text{Apparent depth}} = \dfrac{{{\text{Real depth}}}}{\mu }\]
\[ \Rightarrow \mu = \dfrac{{{\text{Real depth}}}}{{{\text{Apparent depth}}}}\]

We have given that the real depth of the tank is 12.5 cm and the apparent depth of the needle at the bottom of the tank is 9.4 cm. Therefore, the refractive index of the water is,
\[\mu = \dfrac{{{\text{12}}{\text{.5}}\,{\text{cm}}}}{{{\text{9}}{\text{.4}}\,{\text{cm}}}}\]
\[ \Rightarrow \mu = 1.329\]

Therefore, the refractive index of the water is 1.329.


We know the apparent depth of the medium of refractive index \[\mu \] is,
\[{\text{Apparent depth}} = \dfrac{{{\text{Real depth}}}}{\mu }\]

We have to substitute 12.5 cm for real depth and 1.63 for \[\mu \] in the above equation.
\[{\text{Apparent depth}} = \dfrac{{{\text{12}}{\text{.5}}\,{\text{cm}}}}{{1.63}}\]
\[ \Rightarrow {\text{Apparent depth}} = 7.669\,{\text{cm}}\]

Therefore, we see the depth of the needle appears to be 7.669 cm as seen from the above. Since the apparent depth of the needle in water is 9.4 cm, the image of the needle is shifted,
\[9.4\,cm - 7.669\,cm = 1.731\,cm\] upwards.

Therefore, we have to move the microscope 1.731 cm upward to focus on the needle again.

Note:
In questions where students are asked to determine how much the microscope can be moved to focus on the object again, once you determine the apparent depth of the image of the object, you should always subtract it from the real depth. This subtraction is the height of the image of the object from the bottom. Therefore, this is the distance you should move the microscope upward to focus again.