Question

To what height should water be filled in a container of height 21cm, so that it appears as half filled when viewed from the top?(Take ${}_a\mu {}_w = \dfrac{4}{3}$)
(A) 12cm
(B) 15cm
(C) 10.5cm
(D) 7cm

Answer 1

Hint:After filling the tank, take a look to study about the image. The image is formed due to the transmission of the light.

Formula used:To solve this type of question the following formula is used.
$\mu = \dfrac{h}{{h'}}$; here $\mu $ is a refractive index of medium, h is real height/width and h’ is apparent height/width.

Complete step-by-step answer:
When we fill a tank with water and look from above, the image we see will be an apparent image because light travels slower in water than air.
Now, let us assume that h cm is the height of the water filled in the tank. Then the apparent height is (21-h) cm. Now, let us introduce the concept of refractive index which is given by the formula$\mu = \dfrac{h}{{h'}}$.
Let us substitute the values in this equation and we get the following.
$\mu = \dfrac{h}{{21 - h}}$ (1)
We have given the refractive index of water with respect to air${}_a\mu {}_w = \dfrac{4}{3}$.
Let us substitute the value of the refractive index in equation (1).
$\dfrac{h}{{21 - h}} = \dfrac{4}{3}$
Let us further simplify it.
$3h = 84 - 4h$
$ \Rightarrow h = 12cm$
Hence, the apparent height is given below.
${h'} = 21 - 12$
$ = 9cm$
Hence, it is clear that the container should be filled up to 12cm so that it appears to be half filled.
Hence, option (A) “12cm” is the correct option.

Additional information:The phenomenon that describes how fast the light travels through the medium is known as refractive index. The refractive index of the material is a dimensionless number.

Note:
*Light travels slower in water than in air/vacuum, to measure this we use the concept of refractive index.
*Refractive index of water with respect to air is ${}_a\mu {}_w = \dfrac{4}{3}$. It means light travels 4/3 times faster in air than in water, vice-versa.