A ray of light passes through a prism in a principle plane the deviation being equal to angle of incidence which is equal to `2alpha`. It is given that `alpha` is the angle fo prism and `mu` is the refractive index of the material of prism, then

Options

A. `"cos"alpha = sqrt((mu^(2)-1)/(2))`
B. `"cos"alpha = sqrt((mu^(2)-1)/(8))`
C. `"sin"alpha = sqrt(mu^(2)-1)`
D. `"sin"alpha = sqrt((mu^(2)-1)/(8))`

Answer

B

Explanation

`2alpha = (2alpha + e) - alpha`
e= `alpha, r_(1) + r_(2) = alpha`
`"sin"2alpha = mu "sin"r_(1)`,
`mu "sin"(alpha-r_(1)) = "sin"alpham("sin"alpha."cos"r_(1) - "cos"alpha"sin"r) = "sin"alpha`
`mu["sin"alphasqrt(1-(sin^(2)2alpha)/(mu^(2)))-("cos"alpha"sin"2alpha)/(mu)] = "sin"alpha`
`"sin"alphasqrt(mu^(2) - sin^(2)2alpha)-"cos"alpha"sin"2alpha= "sin"alpha`
`sqrt(mu^(2) -sin^(2)2alpha)- 2cos^(2)alpha=1`
`mu^(2) - sin^(2)2alpha = 1 + 4 cos^(4)alpha " " 4 cos^(2)alpha`
`mu^(2) = 1 + 4cos^(4)alpha + 4cos^(2)alpha + 4 sin^(2)alpha cos^(2)alpha`
`mu^(2) = 1 + 4cos^(2)alpha(sin^(2)alpha + cos^(2)alpha)`
`"cos"alpha= sqrt((mu^(2)-1)/(8))`