Question

$$ \begin{array}{|l|l|} \hline \text { Column - I } & \text { Column - II } \\ \hline \text { (i) Minimum deviation } & \text { (a) }(\mathrm{n}-1) \mathrm{A}+\left(\mathrm{n}^{\prime}-1\right) \mathrm{A}^{\prime}=0 \\ \text { (ii) Angular dispersion } & \text { (b) }\left[\left(\mathrm{n}_{\mathrm{v}}-\mathrm{n}_{\mathrm{r}}\right) /(\mathrm{n}-1)\right] \\ \text { (iii) Dispersive power } & \text { (c) }\left(\mathrm{n}_{\mathrm{v}}-\mathrm{n}_{\mathrm{r}}\right) \mathrm{A} \\ \text { (iv) Condition for no deviation } & \text { (d) }(\mathrm{n}-1) \mathrm{A} \\ \hline \end{array} $$ (A) \(i-c\), ii \(-d\), ii \(-b\), iv-a (B) \(i-a, 1 i-b\), iii \(-c\), iv-d (C) \(i-c\), ii \(-b\), ii \(-a\), iv-d (D) \(\mathrm{i}-\mathrm{d}, \mathrm{i} i-\mathrm{c}, \mathrm{ii}-\mathrm{b}, \mathrm{iv}-\mathrm{a}\)

Short answer

The short answer is: (D) i-d, ii-b, iii-c, iv-a.

Step-by-step solution

(Step 1: Understanding the terms)

We need to understand the terms in Column-I and their meaning to be able to match them to their correct expressions in Column-II. - Minimum Deviation: In the context of optics and prisms, it is the smallest angle between the incident light ray and the refracted light ray in a prism. - Angular Dispersion: It refers to how the prism separates light of different colors (wavelengths) by angular spread. - Dispersive Power: It is the measure of how effectively a prism disperses light of various colors. - Condition for No Deviation: It is the condition for when light passes through a prism without any deviation from its original path.

(Step 2: Matching the terms to the expressions)

Now we will analyze each term and find the correct expressions in Column-II. (i) Minimum Deviation: The formula for minimum deviation is given by \( (\mathrm{n} - 1) \mathrm{A} \), corresponding to option (d) in Column-II. (ii) Angular Dispersion: The formula for angular dispersion is given by \(\dfrac{\mathrm{n}_v - \mathrm{n}_r}{\mathrm{n}-1}\), corresponding to option (b) in Column-II. (iii) Dispersive Power: The formula for dispersive power is given by \((\mathrm{n}_{\mathrm{v}} - \mathrm{n}_{\mathrm{r}}) \mathrm{A}\), corresponding to option (c) in Column-II. (iv) Condition for No Deviation: The formula for condition for no deviation is given by \((\mathrm{n}-1)\mathrm{A} + (\mathrm{n'}-1)\mathrm{A'} = 0\), corresponding to option (a) in Column-II.

(Step 3: Finding the correct option)

With the terms and expressions matched, we have: - i-d - ii-b - iii-c - iv-a Based on these matches, the correct option is (D). i-d, ii-b, iii-c, iv-a.

Key concepts

Minimum Deviation

In prism optics, minimum deviation refers to the smallest angle between the incident and emerging light rays as they pass through a prism. This phenomenon is significant because it allows for the determination of the refractive index of the prism material. At minimum deviation, the light travels symmetrically through the prism, which means the angle of incidence is equal to the angle of emergence. This symmetry results in the least amount of bending compared to other conditions. The formula associated with this is \((n - 1)A\), where \(n\) is the refractive index of the material and \(A\) is the angle of the prism. Understanding this concept is crucial for many applications in optical instruments and experiments.

When measuring anything involving refraction, using the minimum deviation angle ensures the highest accuracy. Opticians often rely on this principle to design lenses and prisms that have precise optical properties.

Angular Dispersion

Angular dispersion is a phenomenon that occurs when a prism separates light into its constituent colors. This separation happens because different colors of light have different wavelengths, and they bend to varying degrees when passing through the prism. The result is a spectrum or spread of colors emerging from the prism.

The degree of this separation is quantified by the expression \(\frac{n_v - n_r}{n - 1}\). Here, \(n_v\) and \(n_r\) represent the refractive indices of the violet and red light, respectively, and \(n\) is the refractive index for a standard wavelength. Angular dispersion is vital for understanding how light interacts with different materials and is the foundation for devices like spectrometers, which analyze the spectra of light sources.

Dispersive Power

Dispersive power measures how effectively a prism can separate light into its component colors. It is an intrinsic property of the prism material and provides insights into how much the colors will be spread out.

The formula for dispersive power is \((n_v - n_r)A\), where \(n_v\) and \(n_r\) denote the refractive indices of violet and red light, and \(A\) is the prism's angle. Higher dispersive power means the light colors are spread out more, creating a broader spectrum. Materials with high dispersive power are often used in creating prisms for optical devices that require precise color separation, such as in cameras and spectrographs.

Condition for No Deviation

The condition for no deviation is a unique scenario where light passes straight through a prism without bending. This occurs under specific circumstances involving the prism's angles and refractive indexes of the surrounding media.

The mathematical condition for no deviation is given by \((n - 1)A + (n'-1)A' = 0\), where \(n\) and \(n'\) are the refractive indices of the prism and the surrounding medium, respectively, while \(A\) and \(A'\) are angles related to the prism. When this condition is satisfied, the light exits the prism with its path unchanged. Instruments designed to align light with minimal alteration in its pathway, such as certain types of photolithography systems, utilize this principle to ensure high precision.