Problem 7 - An Introduction to Physical Science

Question

What percentage of the speed of light in vacuum is the speed of light in water?

Step-by-Step Solution

Answer

The speed of light in water is approximately 75.33% of the speed of light in vacuum.

Step 1: Understand the Refractive Index Concept

The speed of light in a medium is related to the speed of light in a vacuum by the refractive index of the medium. The refractive index, denoted by \( n \), is a dimensionless number that describes how much the speed of light is reduced inside the medium compared to a vacuum.

Step 2: Identify the Speed of Light in Vacuum

The speed of light in a vacuum is a constant, \( c \), which approximately equals \( 3 \times 10^8 \) meters per second.

Step 3: Determine the Refractive Index of Water

For water, the refractive index \( n \) is approximately 1.33. This means that light travels 1.33 times slower in water than in a vacuum.

Step 4: Calculate the Speed of Light in Water

The speed of light in water \( v \) can be calculated using the formula:\[v = \frac{c}{n}\]Substituting the known values:\[v = \frac{3 \times 10^8}{1.33}\]This gives a speed of approximately \( 2.26 \times 10^8 \) meters per second.

Step 5: Calculate the Percentage of the Speed of Light in Water

To find what percentage the speed of light in water is compared to the speed of light in a vacuum, use the formula:\[\text{Percentage} = \left( \frac{v}{c} \right) \times 100\%\]Substitute the values:\[\text{Percentage} = \left( \frac{2.26 \times 10^8}{3 \times 10^8} \right) \times 100\% = 75.33\%\]

Key Concepts

Refractive Index

Light behaves differently when it travels through various media, and this is due in part to the refractive index of those media. The refractive index involves a comparison of the speed of light in a vacuum to its speed in a given medium. If you consider a vacuum like outer space, light travels its fastest there at about 300,000 kilometers per second. The refractive index, represented by the symbol \( n \), tells us how much slower light travels in a medium compared to a vacuum. For example, when light enters water, it slows down because of the water’s refractive index, which is approximately 1.33. This means that light in water travels about 1.33 times slower than it would in a vacuum. Here's how it works: if you know the speed of light in a vacuum and the refractive index of a medium, you can find out how fast light travels in that medium using the formula:

• Speed of light in medium \( v = \frac{c}{n} \)

This fundamental concept helps us understand not just physics, but also how lenses and glasses work to bend and focus light.

Light in Different Media

Light doesn't always zoom through at the same speed everywhere; it varies depending on the medium it is traveling through. For instance:

• In a vacuum, like outer space, light moves fast at a speed of about 300,000 kilometers per second.
• Through air, which has a refractive index very close to 1, light speed slows down just a tiny bit.
• When it travels through water, its speed decreases significantly because of water's higher refractive index of 1.33.

The decrease in speed happens because light waves interact with the particles making up the medium, which slows them down. When light enters a denser medium, like glass or diamond, it has an even higher refractive index, which means the light slows down even more. Understanding these variations is key in fields like optics and photonics, where the control and manipulation of light interaction is crucial.

Light Speed Calculation

To calculate the speed of light in any medium, you need to consider two crucial factors: the speed of light in a vacuum and the medium's refractive index. The formula used to find the speed of light in a medium is:

• \( v = \frac{c}{n} \)

where:

• \( v \) is the speed of light in the medium,
• \( c \) is the speed of light in a vacuum, approximately \( 3 \times 10^8 \) meters per second,
• \( n \) is the refractive index of the medium.

For example, to find the speed of light in water, given its refractive index \( n = 1.33 \), plug the known values into the formula:\[ v = \frac{3 \times 10^8}{1.33} \]Upon solving, you'll find that the speed of light in water is about \( 2.26 \times 10^8 \) meters per second. To express this as a percentage of the speed of light in a vacuum, use the calculation:\[ \text{Percentage} = \left( \frac{v}{c} \right) \times 100\% \]to get approximately 75.33%. This means that light travels at only about 75.33% of its speed in a vacuum when moving through water. This type of calculation is vital for designing systems that depend on precise light measurements and interactions.