Chapter 33: Q59P (page 1005)

In the figure, a ray of light is perpendicular to the face $a b$ of a glass prism $(n = 1 .52)$
.(a) Find the largest value for the angle $ϕ$ so that the ray is totally reflected at face ac if the prism is immersed in air, and
(b) Find the largest value for the angle $ϕ$ so that the ray is totally reflected at face ac if the prism is immersed in water.

Short Answer

Expert verified

Largest value for $ϕ$ so that the ray is totally reflected at face ac if the prism is immersed in air will be $48.9 °$ .
Largest value for $ϕ$ so that the ray is totally reflected at face ac if the prism is immersed in water will be $29 °$ .

Step by step solution

Given

$n_{1} = 1.52$

Understanding the concept

Using Snell’s law, we can determine the critical angle. Using this, we can find the largest value for $ϕ$ if the prism is immersed in air and in water.

Formula:

$n_{1} \sin θ_{1} = n_{2} \sin θ_{2}$

(a) Calculate the largest value for so that the ray is totally reflected at face ac if the prism is immersed in air

As the ray is travelling perpendicular to the surface, it does not refract on the initial face of bc.

Due to this,thetotal internal reflection occurs at the critical angle.

Snell’s law gives that

$n_{1} \sin θ_{1} = n_{2} \sin θ_{2}$

Sincetheprism is immersed in air,

$n_{2} = 1$ And $θ_{2} = 90 °$

$\begin{matrix} 1.52 \sin θ_{1} = 1 \sin 90 ° \\ \sin θ_{1} = \frac{1}{1.52} \\ θ_{1} = \sin^{- 1} 0.65 \\ θ_{1} = 41.1 ° \end{matrix}$

Since it is a prism,

$\begin{matrix} θ_{1} + ϕ = 90 ° \\ ϕ = 90 - θ_{1} \\ ϕ = 90 - 41.1 ° \\ ϕ = 48.9 ° \end{matrix}$

(b) Calculate the Largest value for ϕ so that the ray is totally reflected at face ac if the prism is immersed in water

Similarly,

Ifthe prism is immersed in water,

$n_{2} = 1.33$ and $θ_{2} = 90 °$

$\begin{matrix} 1.52 \sin θ_{1} = 1 \sin 90 ° \\ \sin θ_{1} = \frac{1.33}{1.52} \\ θ_{1} = \sin^{- 1} 0.875 \\ θ_{1} = 61.0 ° \end{matrix}$

Since it is a prism,

$\begin{matrix} θ_{1} + ϕ = 90 ° \\ ϕ = 90 - θ_{1} \\ ϕ = 90 - 61 \\ ϕ = 29 ° \end{matrix}$

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Short Answer

Step by step solution

Given

Understanding the concept

(a) Calculate the largest value for so that the ray is totally reflected at face ac if the prism is immersed in air

(b) Calculate the Largest value for ϕ so that the ray is totally reflected at face ac if the prism is immersed in water

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