Question

The refractive index of water is 4/3. If the ray travels 24cm in the air in a given time \[t\], the distance traveled by light waves in the same time is:
A. 32 cm
B. 24 cm
C. 18 cm
D. 15 cm

Answer 1

Hint: The refractive index of a can be defined as the ratio of the speed of light in a vacuum to the speed of light in the medium. Hence we can use the definition of speed (as distance over time) to find the refractive index in terms of distance traveled.

Formula used: In this solution we will be using the following formulae;
\[n = \dfrac{c}{v}\] where \[n\] is the refractive index of a particular material, \[c\] is the speed of light in vacuum, and \[v\] is the speed of light in the medium.
\[v = \dfrac{d}{t}\] where \[v\] is the speed of anything (whether object, wave or even an arbitrary point), \[d\] is the distance travelled by the thing, and \[t\] is the time taken to travel the distance.

Complete step by step answer:
To solve the above, we define the refractive index of a medium as the ratio of the speed of light in a vacuum to the speed of light in the medium. Mathematically,
\[n = \dfrac{c}{v}\] where \[n\] is the refractive index of a particular material, \[c\] is the speed of light in vacuum, and \[v\] is the speed of light in the medium.
Now, in the formula, above we can multiply both the numerator and the denominator by time \[t\]. Then we’ll have
\[n = \dfrac{{ct}}{{vt}}\]
From the definition of velocity, which is
\[v = \dfrac{d}{t}\] where \[d\] is the distance travelled by the thing, and \[t\] is the time taken to travel the distance.
Hence,
\[d = vt\]
This means that \[ct\] and \[vt\] above are the distance travelled by light in vacuum and medium respectively. Hence, we write
\[n = \dfrac{{{d_v}}}{{{d_m}}}\]
By inserting known values, we have that
\[\dfrac{4}{3} = \dfrac{{24}}{{{d_m}}}\]
\[ \Rightarrow {d_m} = \dfrac{{24 \times 3}}{4} = 18cm\]
Hence, the correct option is D.

Note: Alternatively, one could fairly make a good guess without any calculation being done. Since we know that the velocity in a medium other than vacuum is always less than that of vacuum, hence, the distance traveled, at the same time, would always be less. This leaves us with option C and D only. Although one cannot be exactly sure which is correct without calculation but it acts as a pointer to know that if one gets an answer not one of these, there’s an error somewhere.