Question

A layer of water, with index of refraction 1.333 , rests on a block of an unknown transparent material. A ray of light passes through the water at an angle of \(\varphi_{1}=68.77^{\circ}\) relative to the boundary between the materials and then passes through the unknown material at an angle of \(\varphi_{2}=72.98^{\circ}\) relative to the boundary. What is the speed of light in the unknown material?

Short answer

Answer: The speed of light in the unknown material is approximately \(2.31 \times 10^8 m/s\).

Step-by-step solution

Write down Snell's law of refraction

Snell's law states that the ratio of the sines of the angles of incidence and refraction is equal to the ratio of the indices of refraction of the two materials. The formula for Snell's law is: \(n_1 \sin{\varphi_1} = n_2 \sin{\varphi_2}\) Where \(n_1\) and \(n_2\) are the indices of refraction of the first and second materials, and \(\varphi_1\) and \(\varphi_2\) are the angles of incidence and refraction, respectively.

Insert the given values into Snell's law formula

We are given \(n_1 = 1.333\), \(\varphi_1 = 68.77^{\circ}\), and \(\varphi_2 = 72.98^{\circ}\). We need to solve for the unknown index of refraction, \(n_2\). Plug in the given values into the Snell's law formula: \((1.333) \sin{68.77^{\circ}} = n_2 \sin{72.98^{\circ}}\)

Solve for \(n_2\)

Now, let's solve for the unknown material's index of refraction, \(n_2\): \(n_2 = \frac{(1.333) \sin{68.77^{\circ}}}{\sin{72.98^{\circ}}}\) Calculating the value, we get: \(n_2 \approx 1.299\)

Find the speed of light in the unknown material

Finally, to find the speed of light in the unknown material, use the relationship between the index of refraction and the speed of light: \(n = \frac{c}{v}\) Where \(n\) is the index of refraction, \(c\) is the speed of light in a vacuum (\(3 \times 10^8 m/s\)), and \(v\) is the speed of light in the unknown material. Now we can solve for \(v\): \(v = \frac{c}{n}\) Plugging in the values, we get: \(v = \frac{3 \times 10^8 m/s}{1.299}\) Calculating the speed of light in the unknown material, we find: \(v \approx 2.31 \times 10^8 m/s\) So, the speed of light in the unknown material is approximately \(2.31 \times 10^8 m/s\).

Key concepts

Understanding Snell's Law

Snell's Law is a foundational principle in optics that governs the behavior of light as it moves from one medium to another. When a light ray encounters a boundary between two different transparent materials, it generally bends, or refracts, at an angle. This behavior can be precisely described by Snell's Law, which is mathematically represented as:
\[\begin{equation} n_1 \sin(\varphi_1) = n_2 \sin(\varphi_2)\end{equation}\]
Here, \(\varphi_1\) is the angle of incidence—this is the angle between the incident ray and an imaginary line called the normal, which is perpendicular to the boundary. Similarly, \(\varphi_2\) is the angle of refraction, which is the angle between the refracted ray and the normal. The variables \(n_1\) and \(n_2\) represent the indices of refraction for the first and second material, respectively.

• Understand the term 'normal': It is an imaginary line perpendicular to the boundary between two materials.
• Angle of incidence and refraction: These are measured from the normal, not the boundary.
• Implications of Snell's Law: It explains phenomena like bending of light in a pool and the formation of rainbows.

Snell's Law not only helps us understand why light bends, but it is also instrumental in calculating the extent of this bending, which is essential for designing optical devices such as lenses and telescopes.

Index of Refraction

The index of refraction, denoted as \(n\), is a dimensionless number that describes how light propagates through a particular medium. It is defined as the ratio of the speed of light in a vacuum, \(c\), to the speed of light in the material, \(v\):
\[\begin{equation} n = \frac{c}{v}\end{equation}\]
For any material, the index of refraction is always equal to or greater than 1, since light travels fastest in a vacuum. The higher the index of refraction, the more the light slows down and the more it bends or refracts when entering the material from a vacuum or air.

• Low index of refraction: Materials with a low \(n\) value, such as air, cause less bending of light.
• High index of refraction: Materials with high \(n\) values, like diamonds, significantly bend light, which is why they sparkle.

Knowing the index of refraction is crucial for various practical applications, including the manufacturing of eyeglass lenses, fiber optic cables for telecommunications, and even for understanding the visual distortions under water.

Speed of Light in Materials

The speed of light is regarded as one of the fundamental constants of nature when it travels through a vacuum, approximately \(3 \times 10^8 \) meters per second. However, this speed reduces when light travels through different materials. This reduction is due to the interaction of light with the atoms within the material, which absorb and re-emit the light waves, effectively causing it to travel slower than it would in a vacuum.

• Measuring light's speed: Typically, the speed of light within a material is calculated using the material's index of refraction.
• Relevance to everyday life: The speed of light in materials affects how we see things. For example, when wearing glasses, the light bends to correct vision based on the speed at which it travels through the lens.

The slower speed of light in materials compared to a vacuum is not simply of academic interest; it has real-world implications. In optometry, optical fibers, and even in the functioning of natural phenomena, understanding how light speed changes in materials enables us to craft technologies and gain insight into the world around us.