A ray of light moving along the unit vector (`-i-2j`)undergoes refraction at an interface two media,which is the x-zplane. The refracive index for `ygt0`,it is`sqrt(5)//2`.the unit vector along which the refracted ray moves is:

Options

A. `((3hat(i)-5hat(j))/(sqrt(34)))`
B. `((-4hat(i)-3hat(j))/(5))`
C. `((-3hat(i)-4hat(j))/(5))`
D. None of these

Answer

B

Explanation

From snell's law
`mu_(1)[hat(r)_(i)xxhat(N)]=mu_(2)[hat(r)_(r)xxhat(N)] rArr2[(-hat(i)-2hat(j))/(sqrt(5))xx(-hat(j))]=sqrt(5)/(2)[((-hat(i)-yhat(j)))/(sqrt(1+y^(2)))xx(-hat(j))]`
`(2)/(sqrt(5))=(sqrt(5)/(2))((1)/(sqrt(1+y^(2)))) rArr sqrt(1+y^(2))=(5)/(4) rArr y=(3)/(4)`
`hat(r)=((hat(i)-yhat(j)))/(sqrt(1+y^(2))) rArr hat(r)=((-hat(i)-(3)/(4)hat(j)))/(sqrt(1+9//16))`
`hat(r)=(-4hat(i)-3hat(j))/(5)`
Method II

`vec(r)_(i).(-vec(j))=r_(i)cosi`
`(-hat(i)-2hat(j)).(-hat(j))=sqrt(5)cosi`
`cosi=(2)/(sqrt(5)) or sini=(1)/(sqrt(5))`
Applying Snell's law
`mu_(1) sini=mu_(2)sin r`
`2xx(1)/(sqrt(5))=(sqrt(5))/(2) sin r`
`sin r=(4)/(5) ,cosr=(3)/(5)`
`vec(r).(-hat(j))=r_(r)cosr`
`(-hati-yhatj)(-hat(j))=sqrt(1+y^(2))(3)/(5)`
`5y=3sqrt(1+y^(2))rArr y=(3)/(4)`
`hat(r)_(r)=((-hat(i)-yhat(j))/r_(B))rArr hat(r)_(r)=((-hati-(3)/(4)hat(j)))/(sqrt(1+((3)/(4))^(2)))`
`hatr =(-4hat(i)-3hat(j))/(5)`