Q.(a) The refractive index of glass is $1.5$. What is the speed of light in glass? (Speed of light in vacuum is $3.0 \times 10^{8}\ \text{m s}^{-1}$)

Concept: Refractive Index Calculation

1. The refractive index $n$ relates the speed of light in vacuum $c$ to its speed in the medium $v$ by $n = \frac{c}{v}$. Given $n = 1.5$ and $c = 3.0 \times 10^{8}\ \text{m s}^{-1}$, we solve for $v$:

   $v = \frac{c}{n} = \frac{3.0 \times 10^{8}}{1.5} = 2.0 \times 10^{8}\ \text{m s}^{-1}$

2. No, the speed of light in glass depends on colour (wavelength) due to dispersion. The refractive index is higher for shorter wavelengths. Violet has a shorter wavelength than red, so it slows down more. Hence, violet travels slower in a glass prism.
   ✓Final answer

   1. The speed of light in glass is $\boxed{2.0 \times 10^{8}\ \text{m s}^{-1}}$.
   2. No, it is not independent; violet travels slower than red in a glass prism.

Why This Works

The refractive index $n$ of a material is defined as the ratio of the speed of light in vacuum $c$ to the speed of light in that material $v$:

This is not just a number — it tells you how much the material "slows down" light. A higher $n$ means slower light. For part (b), the key is that $n$ is not constant for all colours; it varies slightly with wavelength, a phenomenon called dispersion. This is why a prism splits white light into a rainbow.

Step-by-Step Solution

Part (a) — Speed of light in glass

1. Write down what you know.

   Refractive index of glass, $n = 1.5$

   Speed of light in vacuum, $c = 3.0 \times 10^{8}\ \text{m/s}$

2. Use the definition of refractive index.

3. Substitute the values.

Part (b) — Dependence on colour

4. Does speed depend on colour?

   Yes. The refractive index of glass is slightly different for different wavelengths (colours) of light. This is why a prism disperses white light.

5. Which colour travels slower?

   Violet light has a shorter wavelength and is bent more than red light when entering a prism. A larger bend means a higher refractive index for violet.

   Since $v = c/n$, a higher $n$ gives a lower speed.

   Therefore, violet travels slower than red in glass.

Method: Direct Formula Substitution (Refractive Index Relation)

This method uses the fundamental definition of absolute refractive index to find the speed of light in a medium.

Steps:

Step 1: Recall the formula

The absolute refractive index $n$ of a medium is defined as:

Step 2: Identify given values

• $n = 1.5$ (for glass)
• $c = 3.0 \times 10^{8}\ \text{m s}^{-1}$

Step 3: Rearrange and solve for $v$

Step 4: Calculate

Answer (a): The speed of light in glass is $2.0 \times 10^{8}\ \text{m s}^{-1}$.

Answer (b):

No, the speed of light in glass is not independent of the colour of light. This phenomenon is called dispersion.

• Violet light travels slower in a glass prism than red light.
• Reason: The refractive index of glass is higher for violet light than for red light. Since $v = c/n$, a higher $n$ means a lower speed.

Key result: In a glass prism, violet travels slower than red.

Here are the common mistakes students make on this question, along with how to avoid each.

(a) Calculating the speed of light in glass

Mistake 1: Using the wrong formula (multiplying instead of dividing)

• The error: Students often write $v = n \times c$ (multiplying refractive index by the speed in vacuum), which gives a speed greater than light in vacuum — physically impossible.
• Why it happens: Confusing the definition: $n = \frac{c}{v}$ means $v = \frac{c}{n}$, not $c \times n$.
• How to avoid: Always recall that refractive index tells you how much slower light moves in the medium. Since $n > 1$, the speed $v$ must be less than $c$. So you must divide $c$ by $n$.
  • Correct: $v = \frac{3.0 \times 10^8}{1.5} = 2.0 \times 10^8\ \text{m/s}$

Mistake 2: Forgetting units or writing them incorrectly

• The error: Writing the answer as just $2.0 \times 10^8$ without units, or using wrong units like $\text{cm/s}$.
• Why it happens: Rushing through the final step.
• How to avoid: Always write the unit m/s (or $\text{m s}^{-1}$) after the numerical value. In exams, missing units costs marks.

Mistake 3: Incorrect scientific notation or rounding

• The error: Writing $2 \times 10^8$ instead of $2.0 \times 10^8$, or miscalculating $3.0/1.5$ as $2.5$.
• How to avoid: Do the division carefully: $3.0 \div 1.5 = 2.0$. Keep the same number of significant figures as the given data (here, two significant figures).

(b) Dependence of speed on colour of light

Mistake 4: Saying the speed is independent of colour

• The error: Answering "Yes, speed is the same for all colours" — this is wrong.
• Why it happens: Assuming that refractive index is a fixed number for a material, forgetting that it varies slightly with wavelength (dispersion).
• How to avoid: Remember that refractive index depends on the colour (wavelength) of light. This is why a prism splits white light into a spectrum.

Mistake 5: Confusing which colour travels slower

• The error: Saying "red travels slower than violet" or "violet is faster".
• Why it happens: Mixing up the relationship between refractive index and speed.
• How to avoid: Use the logic:
  • Violet light has a higher refractive index in glass than red light.
  • Since $v = c/n$, a higher $n$ means a lower $v$.
  • Therefore, violet travels slower in a glass prism.

Quick memory aid: In a prism, violet bends the most (higher $n$), so it slows down the most.

Final Answer Summary