Question

A layer of methyl alcohol, with index of refraction \(1.329,\) rests on a block of ice, with index of refraction \(1.310 .\) A ray of light passes through the methyl alcohol at an angle of \(\varphi_{1}=61.07^{\circ}\) relative to the alcohol-ice boundary. What is the angle \(\varphi_{2}\) relative to the boundary at which the ray passes through the ice?

Short answer

Answer: The angle of refraction in ice is approximately 62.13 degrees.

Step-by-step solution

Identify the formula to use

We will use Snell's Law for this problem, which is given by: \(n_1 \cdot \sin(\varphi_1) = n_2 \cdot \sin(\varphi_2)\) where \(n_1\) and \(n_2\) are the indices of refraction of the two substances, and \(\varphi_1\) and \(\varphi_2\) are the angles of incidence and refraction, respectively.

Plug in the given values

We are given the indices of refraction and the angle of incidence, so we can plug these into the formula: \((1.329) \cdot \sin(61.07^{\circ}) = (1.310) \cdot \sin(\varphi_2)\)

Solve for the angle of refraction \(\varphi_2\)

Now, we need to isolate \(\varphi_2\) by dividing both sides of the equation by 1.310 and then taking the inverse sine (arcsin) of the result: \(\varphi_2 = \arcsin \left(\frac{1.329 \cdot \sin(61.07^{\circ})}{1.310}\right)\)

Calculate the angle of refraction

Using a calculator, we can find the angle: \(\varphi_2 = \arcsin \left(\frac{1.329 \cdot \sin(61.07^{\circ})}{1.310}\right) \approx 62.13^{\circ}\)

State the final answer

The angle at which the ray passes through the ice relative to the boundary is \(\varphi_2 \approx 62.13^{\circ}\).

Key concepts

Index of Refraction

The index of refraction, often symbolized as n, is a fundamental property of materials that measures how much they can bend (refract) a ray of light. It is defined as the ratio of the speed of light in a vacuum (which is approximately 299,792 kilometers per second) to the speed of light in the material. The higher the index of refraction, the more the light will be bent as it enters the material from a vacuum or air.

The index of refraction is not only about light slowing down but also provides insight into the optical density of a material. Optical density is indicative of how tightly packed the particles within a material are, which affects how the light is bent. For instance, the index of refraction for methyl alcohol is 1.329, meaning light travels through methyl alcohol at a slower pace compared to a vacuum. Similarly, ice has an index of refraction of 1.310. Although these values are relatively close, the different indices signify that light will change direction slightly when passing from methyl alcohol to ice due to the difference in optical density.

Angle of Incidence

The angle of incidence is defined as the angle between a ray of light approaching a surface and the line perpendicular to that surface at the point of contact, known as the normal. This angle is crucial because it determines how much the light will bend when it passes from one medium into another with a different index of refraction, according to Snell's Law. It is always measured from the normal, which means a light ray hitting a surface straight-on has an angle of incidence of 0 degrees.

The value of this angle affects the bending of the light. For instance, in the given exercise, the light ray travels through methyl alcohol with an angle of incidence of 61.07 degrees. This information, combined with the indices of refraction for the substances involved, allows us to use Snell's Law to determine the angle at which the light will continue to travel as it passes into the ice, known as the angle of refraction.

Angle of Refraction

The angle of refraction is the angle between the refracted ray and the normal to the surface at the point of refraction. In other words, it tells us the direction in which light continues to travel after it has passed from one material into another. According to Snell's Law, this angle is dependent on both the angle of incidence and the relative indices of refraction of the two media involved.

Due to the change in speed as light enters a different medium, the path of the light bends. If the light enters a more optically dense medium, it bends towards the normal, and if it enters a less optically dense medium, it bends away from the normal. In our problem, we're trying to find the angle of refraction as light passes from methyl alcohol to ice. The solution process involves mathematically manipulating Snell's Law to find this angle, given the known angle of incidence and the indices of refraction. The resulting angle of refraction tells us how the path of light will alter as it emerges into the ice from the alcohol layer.