Question

Find the length of the optical path two medium in contact of length \[{d_1}\] and \[{d_2}\]of refractive indices \[{\mu _1}\] and \[{\mu _2}\] respectively:
(A) \[{\mu _1}{d_1} + {\mu _2}{d_2}\]
(B) \[{\mu _1}{d_2} + {\mu _2}{d_1}\]
(C) \[\dfrac{{{d_1}{d_2}}}{{{\mu _1}{\mu _2}}}\]
(D) \[\dfrac{{{d_1} + {d_2}}}{{{\mu _1}{\mu _2}}}\]

Answer 1

Hint:In this question use the concept of the optical path as the optical path length is the product length of the path which is followed by the light through a given system, the refractive index through which the light passes.

Complete step by step answer:
As we know that the optical path length is the product length of the path which is followed by the light through a given system, the refractive index through which the light passes. The optical path length allows us to find out the phase of the light will be at any point. And it is also called the optical distance.
The index of refraction is known as the refractive index, and which is the measure of the bending of a ray of light when passing through from one medium into the other medium. The refractive index is equal to the velocity of light of the wavelength in empty space divided by its velocity in a substance.

Considering the given values, we are given the medium of length \[{d_1}\] and \[{d_2}\] and the refractive indices of the medium are \[{\mu _1}\] and \[{\mu _2}\] respectively,
Calculate the optical path length,

\[OPL = \mu t\]

Calculate the optical path length of the \[{1^{st}}\] medium and Substitute the values in the formula

\[{1^{st}}_{OPL} = {\mu _1}.{d_1}\]

The optical path length of \[{1^{st}}\] medium is\[{1^{st}}_{OPL} = {\mu _1}.{d_1}\].
Calculate the optical path length of the \[{2^{nd}}\] medium and Substitute the values in the formula

\[{2^{nd}}_{OPL} = {\mu _2}.{d_2}\]

The optical path length of \[{2^{nd}}\]medium is\[{2^{nd}}_{OPL} = {\mu _2}.{d_2}\].
So, the total path length of the system is the sum of the optical path length of \[{1^{st}}\] medium and the optical path length of \[{2^{nd}}\].
Let us denote the total path length by \[x\].
Calculate the total path length,
\[\therefore x = {\mu _1}.{d_1} + {\mu _2}.{d_2}\]

Therefore, the total path length of the system of two mediums is \[{\mu _1}.{d_1} + {\mu _2}.{d_2}\].

Thus, the correct option is (A).

Note:Do not be confused between actual path length and the optical path length. The actual path length is the actual distance travelled by the light and the optical path is dependent on the refractive index of the material.