Problem 58 - University Physics with Modern Physics

Question

Given small samples of three liquids, you are asked to determine their refractive indexes. However, you do not have enough of each liquid to measure the angle of refraction for light refracting from air into the liquid. Instead, for each liquid, you take a rectangular block of glass (\(n\) = 1.52) and place a drop of the liquid on the top surface of the block. You shine a laser beam with wavelength 638 nm in vacuum at one side of the block and measure the largest angle of incidence \(\theta_a\) for which there is total internal reflection at the interface between the glass and the liquid (\(\textbf{Fig. P33.58}\)). Your results are given in the table: What is the refractive index of each liquid at this wavelength?

Step-by-Step Solution

Answer

Calculate each liquid's refractive index using the formula: \(n_{liquid} = 1.52 \cdot \sin(\theta_{critical})\).

Step 1: Understanding Total Internal Reflection

Total internal reflection occurs when a light ray traveling from a medium with a higher refractive index to a medium with a lower refractive index reaches a critical angle. Beyond this angle, the light is completely reflected back into the original medium (glass in this case).

Step 2: Identify the Known Values

The refractive index of glass, \(n_g = 1.52\), and the critical angle \(\theta_c\) is essentially the largest angle of incidence \(\theta_a\) that results in total internal reflection. These angles are provided in your results table.

Step 3: Applying Snell's Law

Snell's Law relates the refractive indices and angles of incidence and refraction: \( n_{glass} \cdot \sin(\theta_{critical}) = n_{liquid} \cdot \sin(90^\circ) \). Since \(\sin(90^\circ) = 1\), the refractive index of the liquid, \(n_{liquid}\), can be calculated as \(n_{liquid} = n_{glass} \cdot \sin(\theta_{critical})\).

Step 4: Calculation for Each Liquid

For each liquid, substitute the provided \(\theta_{critical}\) (i.e., the largest angle of incidence from your table) into the equation: \(n_{liquid} = 1.52 \cdot \sin(\theta_{critical})\). Calculate this for each angle to find the refractive index of each liquid.

Step 5: Verification and Conclusion

Ensure that your calculated refractive indices make sense in the context of the problem (i.e., they should be less than 1.52 since the liquid is supposed to have a lower refractive index than the glass).

Key Concepts

Snell's Law

Snell's Law is a fundamental principle in optics.It describes how light behaves when it passes through different media.This law relates the angles and refractive indices of two media through which light travels.
According to Snell's Law, the relationship is given by: \[n_1 \cdot \sin(\theta_1) = n_2 \cdot \sin(\theta_2)\]where

• \(n_1\) and \(n_2\) are the refractive indices of the first and second media, respectively
• \(\theta_1\) is the angle of incidence
• \(\theta_2\) is the angle of refraction

In the context of the given exercise, light is moving from glass (\(n = 1.52\)) to a liquid.The angle where light continues to refract or starts reflecting completely depends on the differences in refractive indices between these media.

Total Internal Reflection

Total internal reflection is an optical phenomenon that occurs when light tries to move from a denser medium to a less dense medium. When light hits the boundary at an angle larger than a specific critical angle, instead of refracting, it reflects completely within the denser medium. This is called total internal reflection.
For total internal reflection to happen, two conditions are necessary:

• The light must be traveling from a medium with a higher refractive index to one with a lower refractive index.
• The angle of incidence must be greater than the critical angle.

In the exercise provided, light is traveling from glass to the less dense liquid, and total internal reflection marks the threshold angle beyond which light ceases to refract.

Critical Angle

The critical angle is a crucial concept in understanding total internal reflection.It is the minimum angle of incidence at which total internal reflection occurs.Light exceeding this angle will be completely reflected back into the original medium, with none passing into the second medium.
This angle can be derived using Snell's Law.At the critical angle, the angle of refraction is \(90^\circ\), and since \(\sin(90^\circ) = 1\), the formula simplifies to:\[ \sin(\theta_{c}) = \frac{n_2}{n_1} \] where

• \(\theta_{c}\) is the critical angle
• \(n_1\) is the refractive index of the denser medium (glass in the exercise)
• \(n_2\) is the refractive index of the less dense medium (the liquid)

Understanding the critical angle allows us to determine at what point total internal reflection begins.

Refractive Index Calculation

Calculating the refractive index of a medium is essential in optics.The refractive index describes how much the light slows down when traveling through a medium compared to vacuum.
In the provided exercise, you calculate the refractive index of the liquid using the conditions of total internal reflection and the critical angle.According to Snell's Law, at the critical angle, the formula becomes:\[ n_{liquid} = n_{glass} \cdot \sin(\theta_{critical}) \]where

• \(n_{glass} = 1.52\)
• \(\theta_{critical}\) is the maximum angle of incidence given for total internal reflection with each liquid

By substituting the known values, you can solve for the refractive index for each liquid, helping to identify their optical properties.