Problem 74 - Master Resource Book in JEE Main Physics

Question

There are three optical media, 1,2 and 3 with their refractive indices \(\mu_{1}>\mu_{2}>\mu_{3}\) (TIR- total internal reflection) (a) When a tay of light travels from 3 to 1 no \(\mathrm{TIR}\) will take place (b) Critical angle between \(\mid\) and 2 is less than the critical angle between \(I\) and 3 (c) Critical angle between 1 and 2 is more than the critical angle between 1 and 3 (d) Chances of 'TR are more when ray of light travels from 1 to 3 compare to the case when it travel fram 1 to 2

Step-by-Step Solution

Answer

Options (a) and (d) are correct.

Step 1: Understand Refractive Index Order

The problem states that the refractive indices of the media satisfy \( \mu_{1} > \mu_{2} > \mu_{3} \). This order indicates that medium 1 is the most optically dense, and medium 3 is the least dense.

Step 2: Analyze Total Internal Reflection (TIR) Condition

Total Internal Reflection occurs when light travels from a medium with a higher refractive index to one with a lower refractive index and the angle of incidence exceeds the critical angle. Hence, TIR cannot occur when light travels from 3 (least dense) to 1 (most dense).

Step 3: Compare Critical Angles

The critical angle \( \theta_c \) for two media is given by \( \sin \theta_c = \frac{\mu_{2}}{\mu_{1}} \). Since \( \mu_1 > \mu_2 \) and \( \mu_2 > \mu_3 \), the critical angle between media 1 and 2 is smaller than that between media 1 and 3, since \( \sin \theta_c \) is inversely related to corresponding \( \mu \) ratios.

Step 4: Evaluate TIR Probability

Chances for TIR increase when there is a greater difference in refractive indices between two mediums. Since \( \mu_1 - \mu_3 > \mu_1 - \mu_2 \), the chances of TIR are more when the ray of light travels from 1 to 3, compared to when it travels from 1 to 2.

Step 5: Analyze Choices

Based on the analysis: (a) Correct, because no TIR occurs when moving from 3 to 1. (b) Incorrect, since the critical angle between 1 and 2 is larger than that between 1 and 3. (c) This is incorrect as it contradicts the critical angle comparison in Step 3. (d) Correct, TIR is more probable from 1 to 3 as explained in Step 4.

Key Concepts

Refractive Index

Refractive Index is an essential concept in optics, helping us understand how light travels through different media. It is denoted by the symbol \( \mu \), and it compares the speed of light in a vacuum to its speed in a particular medium. The basic formula is:\[\mu = \frac{c}{v}\]where \( c \) is the speed of light in a vacuum, and \( v \) is the speed of light in the medium.A higher refractive index means that light travels slower in that medium, indicating greater optical density. For instance, if medium 1 has \( \mu_1 > \mu_2 > \mu_3 \), light travels slowest in medium 1 and fastest in medium 3. Understanding these differences assists in predicting how light rays will behave when moving through different substances, and it is crucial for analyzing phenomena like refraction and total internal reflection.

Critical Angle

The Critical Angle is the angle of incidence in the denser medium at which light refracts along the boundary of two media rather than entering the less dense medium completely. Beyond this angle, Total Internal Reflection (TIR) occurs.The formula to determine the critical angle \( \theta_c \) is:\[\sin \theta_c = \frac{\mu_2}{\mu_1}\]where \( \mu_2 \) is the refractive index of the less dense medium and \( \mu_1 \) is the refractive index of the denser medium.Consider the scenario where \( \mu_1 > \mu_2 > \mu_3 \). We can see:

• Critical angle between medium 1 (denser) and medium 2 (less dense) is calculated as \( \sin \theta_c = \frac{\mu_2}{\mu_1} \).
• As \( \mu_3 < \mu_2 \), the critical angle between medium 1 and medium 3 is smaller than that between medium 1 and medium 2 because \( \sin \theta_c \) increases as the ratio of indices decreases.

Recognizing these differences helps in evaluating scenarios where TIR will or will not occur.

Optical Density

Optical Density refers to how much a medium can slow down the light passing through it. It is directly related to the medium's Refractive Index. A medium with a higher refractive index possesses higher optical density, meaning it impedes light more effectively than a medium with a lower refractive index.In our example, medium 1 has the highest optical density because it has the highest refractive index, \( \mu_1 \), followed by medium 2, and medium 3.High optical density doesn't necessarily imply physical density. For instance, diamond has a high optical density but isn't exceptionally denser (in terms of mass) than water.Understanding optical density is crucial as it not only impacts the speed and direction of light as it enters or exits the medium but also plays a vital role in technologies such as fiber optics, lenses, and various optical instruments. In scenarios involving multiple media, understanding optical density differences helps in predicting the behavior of light, such as deflection or internal reflection, while transitioning from one medium to another.