Question

A layer of oil 3cm thick is floating on a layer of colored water 5cm thick. The refractive index of the colorless water is \[\dfrac{5}{3}\] and the apparent depth of the two liquids is \[\dfrac{36}{7}\]cm, then refractive index of the oil is-

Answer 1

Hint: In order to solve this question, we will use the relationship between refractive index, real depth and apparent depth where apparent depth is distance of virtual image from the surface of liquid and real depth is distance of real object from surface of liquid.

Complete step-by-step solution:
In order to solve the above question, let us first discuss what is a refractive index?
The ratio between the speed of light in a medium to the speed of light in a vacuum is the refractive index.
When we use Snell’s law with the formula of refractive index, we get a relationship between refractive index and real depths and apparent depths of the given liquids.
The given question is solved by using the relationship between refractive index, real depth and apparent depth
In the above question we are given that,
The real depth of coloured liquid is 5cm
And the refractive index of the coloured liquid is \[\dfrac{5}{3}\]
Now according to the relationship between refractive index, apparent depth and real depth which is as follows-
Refractive index =\[\dfrac{\text{real depth}}{\text{apparent depth}}\]
We can say that,
\[\Rightarrow \dfrac{5}{3}=\dfrac{5}{\text{apparent depth}}\]
Apparent depth of the coloured liquid comes out to be = 3 cm
In the above question we are also given the apparent depth of two liquids that is \[\dfrac{36}{7}\]cm
So, now let’s find the apparent depth of oil,
Which we can find by using the equation,
Apparent depth of two liquids – apparent depth of coloured liquid = apparent depth of oil
So, put the values in above equation, we get
Apparent depth of oil =\[\dfrac{36}{7}-3=\dfrac{15}{7}\]
So, we can say that apparent depth of oil is \[\dfrac{15}{7}\]cm
Now, we will find the refractive index of oil,
Refractive index \[\Rightarrow \dfrac{\text{real depth of oil}}{\text{apparent depth of oil}}\]
So, we can clearly say that refractive index of oil is-
\[\Rightarrow \dfrac{3}{{}^{15}/{}_{7}}\]
Refractive Index\[=1.4\]
Hence, we found the refractive index of oil that is 1.4

Note: Don’t get confused about what will be the unit of refractive index as refractive index is the ratio of the real depth and the apparent depth so it does not have any unit. It is a unitless quantity. Also keep in mind the unit of other quantities in the question they may be different and can result in wrong answers.