The refractive index of fused quartz is 1.46 and that of sapphire is 1.77. If V q is the speed of light in quartz and v s is the speed of light in sapphire, then which one of the following relations is correct ?
v q>v s
Understanding Refractive Index and Speed of Light
The question asks us to compare the speed of light in two different materials, fused quartz and sapphire, given their refractive indices. The refractive index is a property of a material that tells us how much the speed of light is reduced when it travels through that material compared to its speed in a vacuum.
Relationship between Refractive Index and Speed of Light
The refractive index (\(n\)) of a medium is defined as the ratio of the speed of light in vacuum (\(c\)) to the speed of light in the medium (\(v\)). This relationship is given by the formula:
\[n = \frac{c}{v}\]
From this formula, we can also express the speed of light in the medium (\(v\)) in terms of the speed of light in vacuum (\(c\)) and the refractive index (\(n\)):
\[v = \frac{c}{n}\]
Since the speed of light in vacuum (\(c\)) is a constant, this formula shows that the speed of light in a medium is inversely proportional to its refractive index. This means:
- If the refractive index (\(n\)) is high, the speed of light (\(v\)) in that medium is low.
- If the refractive index (\(n\)) is low, the speed of light (\(v\)) in that medium is high.
Comparing Speeds in Quartz and Sapphire
We are given the following information:
- Refractive index of fused quartz, \(n_q = 1.46\)
- Refractive index of sapphire, \(n_s = 1.77\)
Using the relationship \(v = \frac{c}{n}\), we can write the speeds of light in quartz (\(v_q\)) and sapphire (\(v_s\)) as:
- Speed of light in quartz: \(v_q = \frac{c}{n_q} = \frac{c}{1.46}\)
- Speed of light in sapphire: \(v_s = \frac{c}{n_s} = \frac{c}{1.77}\)
Now, we need to compare \(v_q\) and \(v_s\). We can see that \(n_s = 1.77\) is greater than \(n_q = 1.46\). Since the speed is inversely proportional to the refractive index, a higher refractive index means a lower speed of light. Therefore, the speed of light in sapphire (\(v_s\)) must be less than the speed of light in quartz (\(v_q\)).
Mathematically, because \(n_s > n_q\), it follows that:
\[\frac{c}{n_s} < \frac{c}{n_q}\]
Which means:
\[v_s < v_q\]
This can also be written as \(v_q > v_s\).
Analyzing the Options
Let's look at the given options:
- \(v_q > v_s\): This matches our finding that the speed of light in quartz is greater than the speed of light in sapphire.
- \(v_s > v_q\): This contradicts our finding.
- \(v_s = v_q\): This would only be true if \(n_s = n_q\), which is not the case.
- \(v_s = \frac{v_q}{2}\): This suggests a specific numerical relationship that is not generally derived from the given refractive indices alone, and it also implies \(v_s < v_q\), but option 1 gives the direct inequality.
Based on our analysis, the correct relationship is \(v_q > v_s\).
Conclusion
The speed of light is inversely proportional to the refractive index of the medium. Since the refractive index of sapphire (1.77) is higher than that of fused quartz (1.46), the speed of light in sapphire is slower than the speed of light in fused quartz.
Therefore, the correct relationship is \(v_q > v_s\).
Revision Table: Refractive Index and Speed
| Concept | Definition/Relation | Impact on Speed |
|---|---|---|
| Refractive Index (n) | \(n = \frac{\text{Speed of light in vacuum (c)}}{\text{Speed of light in medium (v)}}\) | Higher n means lower speed (v) |
| Speed of Light in Medium (v) | \(v = \frac{\text{c}}{n}\) | Inversely proportional to n |
Additional Information: Factors Affecting Speed of Light
The speed of light is maximum in a vacuum (\(c \approx 3 \times 10^8\) m/s). When light enters any transparent medium (like water, glass, quartz, sapphire), its speed decreases. The refractive index quantifies this decrease. Different wavelengths of light can also have slightly different speeds in a medium, a phenomenon known as dispersion, which is why a prism can split white light into its component colors.
Key takeaways:
- Light travels fastest in a vacuum.
- Light slows down in any material medium.
- The denser the optical medium (higher refractive index), the slower the light travels through it.
Important Questions from Polarization by Reflection
What is the absolute refractive index of kerosene?
The refractive index of water \(_a\mu_w=\frac{4}{3}\) and refractive index of glass \(_a\mu_g=\frac{3}{2}\) . A lens placed in air has focal length 10 cm. What will be its focal length if placed inside water?
The refractive index of water and dense flint glass are 1.33 and 1.65, respectively. A ray of light travels from dense flint glass to water. The refractive index of water with respect to dense flint glass is _______ and the light ray bends _______ the normal in water.
The refractive indices of quartz crystal for right handed and left handed circularly polarized light of wavelength 762.9 nm are 1.5391 and 1.5392 respectively. The angle of rotation produced by the crystal plate of thickness 0.5 mm is:
A ray of light is incident on a transparent medium at an angle of 60°. The reflected ray of light is found to be completely polarised. Then, the refractive index of the transparent medium is nearly:
