The refractive index of crown glass is close to 3/2. If the speed of light in air is c, then the speed of light in the crown glass will be close to
(2 / 3)c
Understanding Refractive Index and Speed of Light
The question asks about the speed of light in crown glass, given its refractive index and the speed of light in air. This involves the fundamental relationship between the refractive index of a medium and the speed of light in that medium.
Defining Refractive Index
The refractive index ($\mu$) of a medium is a measure of how much the speed of light is reduced when passing through that medium. It is defined as the ratio of the speed of light in a vacuum (or approximately in air) to the speed of light in the medium. The formula is:
\(\mu = \frac{\text{Speed of light in vacuum (or air)}}{\text{Speed of light in the medium}}\)
In this problem, we are given the speed of light in air as \(c\), and the refractive index of crown glass as \(3/2\).
Calculating Speed of Light in Crown Glass
We can rearrange the formula to find the speed of light in the medium (crown glass):
Speed of light in the medium = \(\frac{\text{Speed of light in air}}{\text{Refractive index of the medium}}\)
Let \(v_{glass}\) be the speed of light in crown glass. Using the given values:
- Refractive index of crown glass, \(\mu_{glass} \approx 3/2\)
- Speed of light in air, \(c_{air} = c\)
Substituting these values into the formula:
\(v_{glass} = \frac{c_{air}}{\mu_{glass}}\)
\(v_{glass} = \frac{c}{3/2}\)
To divide by a fraction, we multiply by its reciprocal:
\(v_{glass} = c \times \frac{2}{3}\)
\(v_{glass} = \frac{2}{3}c\)
Analyzing the Options
Let's look at the given options based on our calculation:
- Option 1: \((3 / 2)c\). This would mean the speed of light in glass is faster than in air, which is incorrect. Refractive index is always \(\ge 1\) for any medium other than vacuum (or air), implying the speed of light in the medium is less than or equal to the speed in vacuum/air.
- Option 2: \((4 / 9)c\). This value is obtained by squaring the reciprocal of the refractive index, which is not the correct formula.
- Option 3: \((2 / 3)c\). Our calculation resulted in \(\frac{2}{3}c\). This matches this option.
- Option 4: \((9 / 4)c\). This value is obtained by squaring the refractive index, which is incorrect.
Based on the calculation, the speed of light in crown glass is approximately \((2/3)c\).
| Concept | Formula | Explanation |
|---|---|---|
| Refractive Index (\(\mu\)) | \(\mu = \frac{c_{air}}{v_{medium}}\) | Ratio of speed of light in air to speed in medium. Higher \(\mu\) means slower speed in the medium. |
| Speed in Medium (\(v_{medium}\)) | \(v_{medium} = \frac{c_{air}}{\mu}\) | Rearrangement to find the speed of light within a specific material. |
Conclusion on Speed of Light in Crown Glass
Given the refractive index of crown glass is close to \(3/2\) and the speed of light in air is \(c\), the speed of light in the crown glass is calculated using the formula \(v_{glass} = \frac{c_{air}}{\mu_{glass}}\). Substituting the values, we get \(v_{glass} = \frac{c}{3/2} = \frac{2}{3}c\).
Revision Table: Optics Basics
| Term | Definition/Formula |
|---|---|
| Speed of Light in Vacuum | Denoted by \(c\), approximately \(3 \times 10^8\) m/s. The maximum speed. |
| Refractive Index | \(\mu = c/v\), where \(v\) is speed of light in the medium. Dimensionless quantity. |
| Optical Density | Related to refractive index; higher optical density means higher refractive index and slower light speed. |
Additional Information: Properties of Crown Glass
Crown glass is a type of optical glass used in lenses and prisms. It is known for having a relatively low refractive index and low dispersion compared to other types of glass like flint glass. The refractive index value of \(3/2\) or 1.5 is a common approximate value used in many introductory physics problems for crown glass, although actual values can vary slightly depending on the exact composition and the wavelength of light.
The speed of light changes when it passes from one medium to another because the photons interact with the electrons of the atoms in the material. This interaction causes a delay, effectively reducing the overall speed at which the light energy propagates through the material.
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