Animated Solution for Physics - Optics: The graph between angle of deviation ($\delta$) and angle of incidence ($i$) for a triangular prism is represented by

Have you ever looked at the iconic album cover of Pink Floyd's The Dark Side of the Moon? A single beam of white light hits a triangular glass prism and bursts into a brilliant spectrum of colors. While the dispersion of colors is mesmerizing, there is an equally fascinating physics story happening with just a single color of light (monochromatic light) as it bends through the glass.

When a ray of light enters a prism, it bends towards the normal. When it exits, it bends away from the normal. The net effect is that the ray's final path is deviated from its original path. We call this the angle of deviation, denoted by $\delta$. But how does this deviation change if we change the angle at which the light hits the prism (the angle of incidence, $i$)? Let's dive into the mathematics and geometry to find out.

To understand the relationship between the angle of incidence $i$ and the angle of deviation $\delta$, we first need to look at the geometry of the prism. Let the prism angle be $A$. When a light ray enters the prism at an angle $i$, it refracts at an angle $r_1$. It travels through the glass, hits the second surface at an angle $r_2$, and emerges into the air at an angle of emergence $e$.

By applying simple geometry to the quadrilateral and triangles formed inside the prism, we arrive at two fundamental equations:

Rearranging the second equation gives us our master equation for deviation:

This equation tells us that the deviation depends on the angle of incidence $i$, the angle of emergence $e$, and the constant prism angle $A$. However, $e$ is not independent; it is strictly determined by $i$ through Snell's Law at both interfaces. Because Snell's Law involves sine functions ($\sin i=\mu \sin r_1$), the relationship between $i$ and $e$ is highly non-linear.

Imagine a light ray entering the prism at $i=30^{\circ }$ and emerging at $e=60^{\circ }$. The deviation $\delta$ will be $30^{\circ }+60^{\circ }-A$.

Now, what if we reverse the direction of the light ray? According to the Principle of Reversibility of Light, if we shine the light back along its exit path, it will retrace its exact journey. So, if we make the light enter at $i=60^{\circ }$, it will emerge at $e=30^{\circ }$.

Notice what happens to the deviation: $\delta =60^{\circ }+30^{\circ }-A$. It's exactly the same!

This profound realization means that for almost any given value of deviation $\delta$, there are two possible angles of incidence: $i_1$ and $i_2$. On a graph, if you draw a horizontal line at a specific $\delta$, it must intersect the curve at two distinct points. This immediately rules out a straight line or a monotonic curve.

Minimum Deviation

If there are two angles of incidence for every deviation, is there ever a case where there is only one? Yes!

As we gradually increase the angle of incidence from a very small value, the angle of deviation initially decreases. The two values $i_1$ and $i_2$ get closer and closer together. Eventually, they merge into a single point. At this unique point, the light ray passes perfectly symmetrically through the prism.

This means the angle of incidence exactly equals the angle of emergence:

When this happens, the deviation reaches its absolute lowest possible value, known as the angle of minimum deviation, denoted by ${\delta }_m$. At this state, the refracted ray inside the prism travels perfectly parallel to the base of the prism (assuming it's an isosceles prism).

Let's put it all together.

1. We start with a small angle of incidence $i$. The deviation $\delta$ is high. 2. As we increase $i$, the deviation $\delta$ drops non-linearly. 3. It hits a rock-bottom minimum ${\delta }_m$ where $i=e$. 4. If we continue to increase $i$ beyond this point, the deviation $\delta$ starts to climb back up.

Because the underlying physics relies on the non-linear sine function from Snell's Law, the resulting graph cannot be made of sharp, straight lines (like a V-shape). Instead, it forms a smooth, continuous, U-shaped curve (often resembling a skewed parabola).

Therefore, when you are asked to identify the correct graph representing the variation of the angle of deviation with the angle of incidence for a triangular prism, you must always look for the smooth U-shaped curve. This makes option (c) the undeniably correct answer.