Question

For an isosceles prism of angle $A$ and refractive index $\mu$, it is found that the angle of minimum deviation ${\delta }_m=A$. Which of the following option(s) is/are correct?

• A. At minimum deviation, the incident angle $i_1$ and the refracting angle $r_1$ at the first refracting surface are related by $r_1=\left(i_1/2\right)$
• B. For this prism the refractive index $\mu$ and the angle of prism A are related as $A=\frac{1}{2}{\cos }^{-1}\left(\frac{\mu }{2}\right)$
• C. For this prism, the emergent ray at the second surface will be tangential to the surface when the angle of incidence at the first surface is $i_1={\sin }^{-1}[\sin A\sqrt{4{\cos }^2\frac{A}{2}-1}-\cos A]$
• D. For the angle of incidence $i_1=A$, the ray inside the prism is parallel to the base of the prism

Answer 1

Answer: (A), (C), (D)

For the angle of incidence $i_1=A$, the ray inside the prism is parallel to the base of the prism

Option (A)

We know for minimum deviation, $i_1=\frac{A+{\delta }_m}{2}$( ${\delta }_m$ : angle of minimum deviation)

Given ${\delta }_m=A$

Hence $i_1=A$

Now, for minimum deviation condition $r_1=A/2$

Hence $r_1=i_1/2$

Correct

Option (B)

$\mu =\frac{sin\left(i_1\right)}{sin\left(r_1\right)}$

$\mu =\frac{sin\left(\frac{A+{\delta }_m}{2}\right)}{Sin\left(A/2\right)}$

$\mu =\frac{sinA}{sinA/2}$ = $2cosA/2$
Incorrect

Option (C)

For tangential emergence, $i_2={90}^o$

$\sin r_2=\frac{1}{\mu }$

Also $r_1=A-r_2$

$\sin i_1=\mu sin{r}_1$

$\sin i_1=\mu sin(A-r_2)$

$\begin{array}{}\sin i_1=\mu (\sin A\cos r_2-\cos A\sin r_2)& \text{(1)}\end{array}$

But, $\mu \sin r_2=1$
and
$A=i_1$ and $r_1=\frac{A}{2}$
$\begin{array}{}\mu =\frac{sin{i}_1}{sin{r}_1}=\frac{sinA}{sin\frac{A}{2}}=2cos\frac{A}{2}& \text{(2)}\end{array}$
From eq. (1) and eq. (2)
$\sin i_1=\mu [\sin A\frac{\sqrt{{\mu }^2-1}}{\mu }-\cos A\frac{1}{\mu }]$
$\Rightarrow \sin i_1=\sin A\sqrt{4{\cos }^2\frac{A}{2}-1}-\cos A$

Correct

Option (D)
At minimum deviation the angle of incidence is equal to the angle of prism and hence the ray inside the prism is parallel to the base of the prism.

Correct