The `xz` plane separates two media `A` and `B` with refractive indices `mu_(1)` & `mu_(2)` respectively. A ray of light travels from `A` to `B`. Its directions in the two media are given by the unut vectors, `vec(r)_(A)=a hat i+ b hat j` & `vec(r)_(B) alpha hat i + beta hat j` respectively where `hat i` & `hat j` are unit vectors in the `x` & `y` directions. Then :

Options

A. `mu_(1)a=mu_(2)alpha`
B. `mu_(1)alpha=mu_(2)a`
C. `mu_(1)b=mu_(2)beta`
D. None of the above

Answer

A

Explanation

(a) From the Snell's law,`mu_(1)sini_(1)=mu_(2)sini_(2)`

`therefore(mu_(1)a)/(sqrt(a^(2)+b^(2)))=(mu_(2)alpha)/(sqrt(alpha^(2)+beta^(2)))(becausesintheta=("Height")/"Hypotenuse")`
`({:(,becauser_(A)=ahati+bhatjand),(,r_(B)=alphahati+betahatj" are unit vectors"):})`
`sqrt(a^(2)+b^(2))=sqrt(alpha^(2)+beta^(2))=1`
`therefore mu_(1)a=mu_(2)alpha`