Q.A ray of light incident at an angle $\theta$ on a refracting face of a prism emerges from the other face normally. If the angle of the prism is $5^{\circ}$ and the prism is made of a material of refractive index 1.5, the angle of incidence is

The exit condition ("emerges normally") means the emergence angle is $0^\circ$, so by Snell's law at the second face, the internal angle there is also $r_2=0^\circ$.

1. Since $r_1+r_2=A=5^\circ$ for any prism, $r_1=5^\circ$.
2. Snell's law at the first face: $\sin\theta = n\sin r_1 = 1.5\sin5^\circ \approx 1.5\times0.0872=0.1308$.
3. $\theta = \sin^{-1}(0.1308) \approx 7.5^\circ$.

(No total internal reflection is involved anywhere in this problem - it's a straightforward Snell's-law application at both faces.)

Setting up the prism geometry

A ray enters the first face of the prism at angle of incidence $\theta$, refracts to angle $r_1$ inside, travels to the second face, hits it at internal angle $r_2$, and emerges there at angle $e$. For any prism, the two internal angles are related to the prism (apex) angle $A$ by

Here $A = 5^\circ$ and the refractive index is $n = 1.5$.

Using the "emerges normally" condition

"Emerges normally" means the ray leaves the second face along the normal, so the emergence angle $e = 0^\circ$. By Snell's law at the second face,

(This makes sense physically too: a ray travelling exactly along the normal inside the glass hits the exit face perpendicularly and passes straight through with no bending.)

Finding $r_1$

From $r_1 + r_2 = A$:

Applying Snell's law at the first face

Using $\sin 5^\circ \approx 0.0872$:

Method: Prism Refraction Using the Angle-Sum Relation $r_1+r_2=A$

This method solves any prism problem where you're given (or must find) the angle of incidence, the exit condition, or the prism's apex angle.

Steps

Step 1: Set up Snell's law at both faces

Let $i$ be the angle of incidence at the first face, $r_1$ the internal refraction angle there, $r_2$ the internal angle of incidence at the second face, and $e$ the emergence angle. Write:

Step 2: Bring in the prism's own geometric relation

For any prism of apex angle $A$, the two internal angles are always related by

regardless of what $i$ or $e$ happen to be.

Step 3: Translate the problem's stated exit condition into a value for $r_2$

A condition like "emerges normally" means $e=0^\circ$, which by Snell's law at the second face forces $r_2=0^\circ$. Other conditions (e.g. minimum deviation, or a given emergence angle) translate into $r_2$ differently — always convert the given condition into $r_2$ first, since that's the variable the angle-sum relation needs.

Step 4 (Applying to this problem): Solve for $r_1$, then for the unknown incidence angle

Use $r_1=A-r_2$ from Step 2, then substitute into Snell's law at the first face ($\sin i = n\sin r_1$) and solve for $i$. Never substitute the minimum-deviation shortcut ($r_1=A/2$) unless the problem actually states symmetric passage — here the exit condition already fixes $r_2$ directly.