Understanding Refractive Index and Critical Angle

The question asks for the refractive index of water given its critical angle. The refractive index of a medium tells us how much the speed of light is reduced in that medium compared to the speed of light in a vacuum. The critical angle is related to total internal reflection and occurs when light travels from a denser medium (like water) to a less dense medium (like air or vacuum).

Calculating Refractive Index from Critical Angle

When light travels from a denser medium (with refractive index $n_1$) to a rarer medium (with refractive index $n_2$, where $n_1 > n_2$), the critical angle $C$ is the angle of incidence in the denser medium for which the angle of refraction in the rarer medium is 90 degrees. The relationship between the refractive indices and the critical angle is given by Snell's Law at the critical angle:

$\frac{\sin C}{\sin 90^{\circ}} = \frac{n_2}{n_1}$

Since $\sin 90^{\circ} = 1$, the formula simplifies to:

$\sin C = \frac{n_2}{n_1}$

In the case where the rarer medium is air or vacuum, its refractive index $n_2$ is approximately 1. Let the refractive index of water be $n_1 = n$. Then the formula becomes:

$\sin C = \frac{1}{n}$

Rearranging the formula to find the refractive index $n$:

$n = \frac{1}{\sin C}$

Applying the Critical Angle Value for Water

The question states that the critical angle for water is $48.2^{\circ}$. Using the formula derived above, we can calculate the refractive index of water:

Given critical angle, $C = 48.2^{\circ}$

Refractive index of water, $n = \frac{1}{\sin(48.2^{\circ})}$

Step-by-Step Calculation

Let's calculate the value of $\sin(48.2^{\circ})$ and then find its reciprocal.

• Find the sine of the critical angle: $\sin(48.2^{\circ})$
• Using a calculator, $\sin(48.2^{\circ}) \approx 0.7455$
• Calculate the refractive index: $n = \frac{1}{0.7455}$
• $n \approx 1.3415$

Rounding to two decimal places, the refractive index of water is approximately 1.34.

Result

Based on the calculation using the given critical angle of $48.2^{\circ}$, the refractive index of water is approximately 1.34.

Refractive Index and Critical Angle Revision

Additional Information on Refractive Index and Critical Angle

• The refractive index is a dimensionless quantity.
• A higher refractive index means light travels slower in the medium.
• Total internal reflection occurs when the angle of incidence in the denser medium is greater than the critical angle.
• The critical angle depends on the pair of media involved. For light going from medium 1 to medium 2 ($n_1 > n_2$), $\sin C = n_2/n_1$. If medium 2 is air ($n_{air} \approx 1$), $\sin C = 1/n_1$.
• Applications of total internal reflection and critical angle include optical fibers, prisms in binoculars, and the sparkling of diamonds.