Question

Draw (\(i - \delta\)) curve for a prism, and show angle of minimum deviation in the curve.

Hint:

The condition for minimum deviation is crucial for many prism-related problems. At \(\delta = \delta_m\), we have \(i = e\) and \(r_1 = r_2\), where \(r_1\) and \(r_2\) are the angles of refraction inside the prism. This symmetry simplifies the derivation of the prism formula.

Answer 1

Step 1: Understanding the Concept:
When a ray of light passes through a prism, it deviates from its original path. The angle between the incident ray and the emergent ray is called the angle of deviation (\(\delta\)). This angle depends on the angle of incidence (\(i\)). The (\(i - \delta\)) curve is a graphical representation of this relationship.

Step 2: Detailed Explanation and Diagram:
The graph of the angle of deviation (\(\delta\)) as a function of the angle of incidence (\(i\)) has the following characteristics:
\begin{itemize} \item Axes: The angle of incidence (\(i\)) is plotted on the X-axis, and the angle of deviation (\(\delta\)) is plotted on the Y-axis.
\item Shape of the Curve: As the angle of incidence \(i\) is increased from a small value, the angle of deviation \(\delta\) first decreases, reaches a minimum value, and then starts to increase again. This results in a U-shaped curve that is not symmetric.
\item Angle of Minimum Deviation (\(\delta_m\)): The lowest point on the curve corresponds to the minimum possible value of the angle of deviation, denoted by \(\delta_m\). At this specific point, the angle of incidence is equal to the angle of emergence (\(i = e\)). For every other value of \(\delta\), there are two corresponding values of the angle of incidence.
\end{itemize} Graphical Representation:
\begin{center} \begin{tikzpicture} \begin{axis}[ xlabel={Angle of Incidence ($i$)}, ylabel={Angle of Deviation ($\delta$)}, xmin=20, xmax=100, ymin=30, ymax=60, axis lines=left, ticks=none, clip=false, width=0.7\textwidth, height=0.5\textwidth, ] \addplot[smooth, thick, blue, domain=30:90] {37 + 0.01*(x-50)^2}; \node[label={below:$i_m$}, circle, fill, inner sep=1.5pt] at (axis cs:50,37) {}; \draw[dashed, gray] (axis cs:20,37) -- (axis cs:50,37) node[left, black] {$\delta_m$}; \draw[dashed, gray] (axis cs:50,30) -- (axis cs:50,37); \end{axis} \end{tikzpicture} \end{center}

Step 3: Final Answer:
The (\(i - \delta\)) curve for a prism shows that the angle of deviation initially decreases with an increase in the angle of incidence, reaches a minimum value (\(\delta_m\)), and then increases. The point where the deviation is minimum is called the angle of minimum deviation.