E2.6 Calculate light speed in a medium from its refractive index

Ontario Grade 10 Science

Physics: Light and Geometric Optics

Using refractive index to calculate how fast light travels

Light travels through empty space and through materials such as water and glass. Its speed is not the same in every setting. In this lesson, you will use a material’s refractive index to calculate the speed of light in that material. A medium is the material through which light travels. Air, water, and glass are examples of media. The calculation uses one relationship and a known value for the speed of light in empty space.

What you will learn

Explain in simple terms what a medium and refractive index are.
Use the refractive index of a medium to calculate the speed of light in it.
Choose suitable units and check whether a calculated speed is reasonable.

1. A Grade 9 bridge: speed and units

You have used speed to describe how far something travels in a certain time. For example, a cyclist moving at

5\ \mathrm{m/s}

travels

5

metres each second. The unit

\mathrm{m/s}

means metres per second.

Light also has a speed. In empty space, its speed is about

3.00 \times 10^8\ \mathrm{m/s}

. This is a very large number: light travels about 300 000 000 metres in one second. The symbol

c

is used for this speed.

A transparent material lets light pass through it. When light travels through a transparent material, its speed is less than its speed in empty space. The material’s refractive index tells us how the speed compares with the speed in empty space.

c = 3.00 \times 10^8\ \mathrm{m/s}

A medium is a material through which light travels.
The speed of light in empty space is represented by $c$ .
Keep the speed unit as metres per second unless a question asks for a different unit.

2. What the refractive index tells us

Imagine comparing how fast light travels in a clear material with how fast it travels in empty space. The refractive index, represented by

n

, is that comparison written as a number. For this calculation, a larger refractive index means light travels more slowly in the material.

For example, suppose a material has a refractive index of

1.50

. This does not mean that its light speed is

1.50

times the speed in empty space. The refractive index compares the speed in empty space with the speed in the material. So the material’s light speed is found by dividing the empty-space speed by

1.50

.

The refractive index has no unit. It is a ratio of two speeds, and the speed units cancel. This makes it straightforward to use with the value of

c

in metres per second.

The model is a calculation tool: take the speed of light in empty space and divide by the refractive index. When the index is greater than

1

, the calculated speed is lower than

c

. That is a useful reasonableness check.

n = \frac{c}{v}\quad\Longrightarrow\quad v = \frac{c}{n}

The refractive index is represented by $n$ and has no unit.
Divide by the refractive index; do not multiply by it.
For an index greater than $1$ , the calculated speed should be less than $c$ .

3. A reliable calculation method

Start by identifying the refractive index given in the question. Then write down the known empty-space speed,

c = 3.00 \times 10^8\ \mathrm{m/s}

. Use

v = c/n

to find the speed in the material.

Substitute the values carefully. Keep the power of ten and the units with the speed. Since

n

has no unit, the answer keeps the unit

\mathrm{m/s}

. Round only at the end, and use the level of precision requested in the question.

Finally, check the result. For a refractive index greater than

1

, the result should be less than

3.00 \times 10^8\ \mathrm{m/s}

. If it is larger, check whether you multiplied instead of divided or entered the index incorrectly.

v = \frac{3.00 \times 10^8\ \mathrm{m/s}}{n}

Identify $n$ and use $c = 3.00 \times 10^8\ \mathrm{m/s}$ .
Calculate $v$ by dividing $c$ by $n$ .
Include units and check that the answer is below $c$ when $n > 1$ .

4. Evidence, models, and safe use

The relationship connects a measurable property of a material, its refractive index, with the speed of light in it. In this lesson, the index is supplied, so no experiment is needed to answer the calculation questions. The examples use hypothetical values for practice; they are not reported laboratory measurements.

A model is a simplified way to describe something. Here, the model is the equation relating

n

,

c

, and

v

. It helps predict the speed from the supplied index. A calculated value depends on the value and units given in the question, so it should not be treated as a new experimental measurement.

No special equipment or home experiment is needed for these calculations. If you encounter demonstrations involving bright light sources, do not look directly into a laser or other intense beam. Use teacher-approved demonstrations and safety instructions.

The calculation uses a supplied refractive index; the practice values below are hypothetical.
A calculated result is a prediction from the model, not a laboratory measurement.
Never look directly into a laser or intense light beam.

Worked example

A hypothetical clear material

A hypothetical transparent material has a refractive index of

1.50

. Calculate the speed of light in the material.

Choose the relationship
Use the equation that gives the speed in a medium from its refractive index. The empty-space speed is $3.00 \times 10^8\ \mathrm{m/s}$ .
$v = \frac{c}{n}$
Substitute and calculate
Put $1.50$ in for the index. Since the index has no unit, the result keeps the speed unit $\mathrm{m/s}$ .
$v = \frac{3.00 \times 10^8}{1.50} = 2.00 \times 10^8\ \mathrm{m/s}$
Check the result
The result is below the speed in empty space, as expected for an index greater than $1$ .

Answer: The speed of light in the hypothetical material is

2.00 \times 10^8\ \mathrm{m/s}

.

Check: The answer has speed units and is less than

3.00 \times 10^8\ \mathrm{m/s}

.

Worked example

A hypothetical medium with a higher index

A hypothetical medium has a refractive index of

2.00

. Calculate the speed of light in it.

Set up the calculation
The index is the divisor because it compares the empty-space speed with the speed in the medium.
$v = \frac{c}{n}$
Substitute the values
Use the given index and the known speed in empty space.
$v = \frac{3.00 \times 10^8\ \mathrm{m/s}}{2.00} = 1.50 \times 10^8\ \mathrm{m/s}$
Compare with the first example
This index is larger than $1.50$ , so the calculated speed is lower than the speed in the first example. That matches the relationship: a larger index gives a lower speed.

Answer: The speed of light in the hypothetical medium is

1.50 \times 10^8\ \mathrm{m/s}

.

Check: The result is less than the empty-space speed and less than the result for an index of

1.50

.

Worked example

Checking a proposed answer

A hypothetical material has a refractive index of

1.25

. A student proposes that light travels through it at

3.75 \times 10^8\ \mathrm{m/s}

. Calculate the speed and decide whether the proposed answer is reasonable.

Calculate from the index
Use division by the refractive index. This gives the speed predicted by the relationship, rather than relying on the proposed value.
$v = \frac{3.00 \times 10^8\ \mathrm{m/s}}{1.25} = 2.40 \times 10^8\ \mathrm{m/s}$
Judge the proposal
The proposed value is greater than the speed of light in empty space. That does not fit this calculation for a material with an index greater than $1$ . The calculated value is the reasonable one.
$3.75 \times 10^8\ \mathrm{m/s} > 3.00 \times 10^8\ \mathrm{m/s}$

Answer: The calculated speed is

2.40 \times 10^8\ \mathrm{m/s}

. The proposed answer is not reasonable.

Check: The correct result is below

c

, while the proposed speed is above

c

.

Common mistakes and how to avoid them

Multiplying the empty-space speed by the refractive index.

Correction: Divide the empty-space speed by the index:

v = c/n

. For an index greater than

1

, this makes the speed smaller than

c

.

Giving an answer with no units.

Correction: The refractive index has no unit, but the calculated speed does. Include

\mathrm{m/s}

.

Assuming the refractive index is the fraction of the speed in the material compared with the speed in empty space.

Correction: The index is the empty-space speed divided by the speed in the material:

n = c/v

.

Lesson summary

The refractive index $n$ compares the speed of light in empty space with its speed in a medium.
Use $v = c/n$ , where $c = 3.00 \times 10^8\ \mathrm{m/s}$ .
For an index greater than $1$ , the speed in the medium is less than $c$ .
Give the speed in $\mathrm{m/s}$ and check that the result is reasonable.

Check your understanding

Question 1

A hypothetical material has a refractive index of

1.20

. What is the speed of light in it?

$2.50 \times 10^8\ \mathrm{m/s}$
$3.60 \times 10^8\ \mathrm{m/s}$
$1.20 \times 10^8\ \mathrm{m/s}$
correctIndex ‏

Show answer and explanation

2.50 \times 10^8\ \mathrm{m/s}

Divide the empty-space speed by the index:

3.00 \times 10^8 / 1.20 = 2.50 \times 10^8\ \mathrm{m/s}

.

Question 2

A hypothetical material has an index of

1.60

. Which statement is correct?

Light travels faster in it than in empty space.
Light travels at $3.00 \times 10^8\ \mathrm{m/s}$ in it.
Light travels more slowly in it than in empty space.
correctIndex: 2,

Show answer and explanation

Light travels more slowly in it than in empty space.

Since the index is greater than

1

, dividing

c

by the index gives a speed below

c

.

Question 3

Which operation is used to find speed in a medium from its refractive index?

Multiply $c$ by $n$ .
Divide $c$ by $n$ .
Subtract $n$ from $c$ .
correctIndex

Show answer and explanation

Divide

c

by

n

.

The relationship is

v = c/n

, so divide the empty-space speed by the refractive index.

Key terms

Medium: A material through which light travels, such as water or glass.
Refractive index: A number that compares the speed of light in empty space with its speed in a medium.
Speed: The distance travelled in a certain amount of time.

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About this lesson and its review

Published by DoAssignment. This reviewed lesson follows Ontario Grade 10 Science (SNC2D), expectation E2.6. It is a study resource, not an official curriculum publication.

Before publication, content is checked for structure, mathematical or chemical notation, calculations, course boundaries, and readability. Errors can still occur, so corrections are welcomed.

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E2.6 · Calculate light speed in a medium from its refractive index

Ontario Grade 10 Science

Using refractive index to calculate how fast light travels

What you will learn

1. A Grade 9 bridge: speed and units

2. What the refractive index tells us

3. A reliable calculation method

4. Evidence, models, and safe use

A hypothetical clear material

A hypothetical medium with a higher index

Checking a proposed answer

Common mistakes and how to avoid them

Lesson summary

Check your understanding

Question 1

Question 2

Question 3

Key terms

Continue through SNC2D

About this lesson and its review