A right angled prism is placed on a printed page as shown in fig. The observer $\boldsymbol{A}$ look

Question

A right angled prism is placed on a printed page as shown in fig. The observer $\boldsymbol{A}$ looks through the vertical face while the observer $\boldsymbol{B}$ looks through the inclined face. Identify the correct statement.

Professor PC Thomas Classes · Answer

Observer $A$ cannot see the object if $\mu>\sin \frac{\pi}{4}$ or $\mu>\sqrt{2}$. Observer $B$ cannot see the object if $\mu>\sin \frac{\pi}{6}$ or $\mu>3.86$. Observer A looks through the vertical face. The normal to this face is horizontal. The light rays from the page are centered around the vertical (normal to the base). Inside the prism, the light rays make an angle $r$ with the vertical, where $0 \leq r \leq heta_c$. The angle of incidence on the vertical face $\left(i_A ight)$ is $90^{\circ}-r$. To see the letters, the smallest possible angle of incidence must be less than the critical angle: $$ 90^{\circ}- heta_c< heta_c \Longrightarrow 2 heta_c>90^{\circ} \Longrightarrow heta_c>45^{\circ} $$Since $\sin heta_c=\frac{1}{\mu}$, this means $\frac{1}{\mu}>\sin 45^{\circ}$, or $\mu<\sqrt{2} \approx 1.414$. Observer B looks through the inclined face, which is at $60^\circ$ to the base. The normal to this face makes an angle of $30^{\circ}$ with the vertical. The light rays inside the prism are still within the cone $r \leq heta_c$ relative to the vertical. The angle of incidence on this face ( $i_B$ ) for a ray at angle $r$ is $\left|30^{\circ}-r ight|$. To see the letters, we need the smallest angle of incidence to be less than the critical angle. - If $ heta_c \geq 30^{\circ}$, a ray can hit the face at $0^{\circ}$ (the normal), so visibility is guaranteed. - If $ heta_c<30^{\circ}$, the ray closest to the normal hits at $30^{\circ}- heta_c$. The limit for B to see anything is $30^{\circ}- heta_c< heta_c \Longrightarrow 2 heta_c>30^{\circ} \Longrightarrow heta_c>15^{\circ}$. This corresponds to $\mu<\frac{1}{\sin 15^{\circ}} \approx 3.86$.