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Question: The speed of light in vacuum is \( 3 \times {10^8}m/s \) . If the refractive index of glass is 1.5, ...

The speed of light in vacuum is 3×108m/s3 \times {10^8}m/s . If the refractive index of glass is 1.5, the time taken by the light to travel 400m in glass in microsecond is
(A) 4.3
(B) 8
(C) 2
(D) 3

Explanation

Solution

Hint
From the formula of the refractive index of a medium we can find the velocity of light in that medium. So we can find the velocity of glass as the refractive index of glass and the velocity of light in vacuum is given. Then we use that velocity to calculate the time taken for the light to travel 400m in the glass.
In the solution of this question we use the following formulas,
⇒R.I.=speed of light in vacuumspeed of light in that medium\Rightarrow R.I. = \dfrac{{{\text{speed of light in vacuum}}}}{{{\text{speed of light in that medium}}}}
where R.I.R.I. is the refractive index of that medium.

Complete step by step answer
The refractive index of a medium is the ratio of the speed of light in a vacuum to the speed of light in that medium.
Therefore we can write,
⇒R.I.=speed of light in vacuumspeed of light in that medium\Rightarrow R.I. = \dfrac{{{\text{speed of light in vacuum}}}}{{{\text{speed of light in that medium}}}}
Now in the question, we are provided with the refractive index of glass and the speed of light in vacuum. So by substituting these values, we get
⇒1.5=3×108v\Rightarrow 1.5 = \dfrac{{3 \times {{10}^8}}}{v}
where vv is the speed of light in that medium.
Therefore, we can find vv as,
⇒v=3×1081.5\Rightarrow v = \dfrac{{3 \times {{10}^8}}}{{1.5}}
On doing the calculation we get,
⇒v=2×108m/s\Rightarrow v = 2 \times {10^8}m/s
This is the speed of light in the glass. Now the light has to travel a distance of 400m. So the time taken by light is found by dividing the distance by the speed. Hence we get,
⇒t=dv\Rightarrow t = \dfrac{d}{v}
So substituting the values,
⇒t=4002×108\Rightarrow t = \dfrac{{400}}{{2 \times {{10}^8}}}
So we get the time as,
⇒t=2×10−6s\Rightarrow t = 2 \times {10^{ - 6}}s
It is asked in the question to find the answer in the units of microseconds. So,
⇒t=2×10−6s=2μs\Rightarrow t = 2 \times {10^{ - 6}}s = 2\mu s
So the time taken is 2μs2\mu s.
Therefore, the correct answer is option (C); 2.

Note
The time taken by light to travel in vacuum is the highest speed of light. The speed of light in any medium cannot exceed this value. So from the formula of the refractive index, the numerator is always greater than the denominator. Therefore the value of the refractive index can never be less than 1.