If an optical medium possesses a relative permeability of $ \frac{10}{\pi} $ and relative permittivity of $ \frac{1}{0.0885} $, then the velocity of light is greater in vacuum than in that medium by ________ times. $ (\mu_0 = 4\pi \times 10^{-7} \, \text{H/m}, \quad \epsilon_0 = 8.85 \times 10^{-12} \, \text{F/m}, \quad c = 3 \times 10^8 \, \text{m/s}) $
If an optical medium possesses a relative permeability of $ \frac{10}{\pi} $ and relative permittivity of $ \frac{1}{0.0885} $, then the velocity of light is greater in vacuum than in that medium by ________ times. $ (\mu_0 = 4\pi \times 10^{-7} \, \text{H/m}, \quad \epsilon_0 = 8.85 \times 10^{-12} \, \text{F/m}, \quad c = 3 \times 10^8 \, \text{m/s}) $
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Correct Answer: 6
Approach Solution - 1
To find how many times the velocity of light in vacuum is greater than in the given medium, we first need to determine the velocity of light in the medium using its relative permeability (μr) and permittivity (εr). The velocity of light in a medium is given by:
$$ v = \frac{1}{\sqrt{\mu \epsilon}} $$
where:
- μ = μ0μr = \( (4π \times 10^{-7}) \times \frac{10}{\pi} = 4 \times 10^{-7} \, \text{H/m} \)
- ε = ε0εr = \( (8.85 \times 10^{-12}) \times 0.0885 = 7.82775 \times 10^{-13} \, \text{F/m} \)
Plug these values into the velocity formula:
$$ v = \frac{1}{\sqrt{(4 \times 10^{-7}) \times (7.82775 \times 10^{-13})}} $$
Calculate the product:
$$ (4 \times 10^{-7}) \times (7.82775 \times 10^{-13}) = 3.1311 \times 10^{-19} $$
Thus:
$$ v = \frac{1}{\sqrt{3.1311 \times 10^{-19}}} $$
Find the square root:
$$ \sqrt{3.1311 \times 10^{-19}} \approx 5.593 \times 10^{-10} $$
Calculate v:
$$ v \approx \frac{1}{5.593 \times 10^{-10}} \approx 1.788 \times 10^{9} \, \text{m/s} $$
Now, the velocity of light in vacuum, c = \(3 \times 10^8\) m/s. Hence, the ratio of velocities is:
$$ \frac{c}{v} = \frac{3 \times 10^8}{1.788 \times 10^9} $$
Calculate this ratio:
$$ \frac{3 \times 10^8}{1.788 \times 10^9} \approx 0.1678 $$
Therefore, the velocity of light in vacuum is greater than that in the medium by:
$$ \frac{1}{0.1678} \approx 5.96 $$
This rounds to approximately 6, confirming our solution is within the expected range (6,6).
Approach Solution -2
The velocity of light in any medium is given by the formula: \[ V = \frac{C}{\sqrt{\mu \epsilon}} \] where:
- \( C \) is the speed of light in vacuum,
- \( \mu \) is the permeability of the medium,
- \( \epsilon \) is the permittivity of the medium.
In this case:
- The relative permeability \( \mu_r = \frac{10}{\pi} \),
- The relative permittivity \( \epsilon_r = \frac{1}{0.0885} \),
- The permeability \( \mu = \mu_0 \mu_r \),
- The permittivity \( \epsilon = \epsilon_0 \epsilon_r \).
Substituting the values: \[ \mu = (4\pi \times 10^{-7}) \times \frac{10}{\pi} = 4 \times 10^{-6} \, \text{H/m} \] \[ \epsilon = 8.85 \times 10^{-12} \times \frac{1}{0.0885} = 1 \times 10^{-10} \, \text{F/m} \] Now, substitute these values into the formula for velocity: \[ V = \frac{3 \times 10^8}{\sqrt{(4 \times 10^{-6})(1 \times 10^{-10})}} = \frac{3 \times 10^8}{\sqrt{4 \times 10^{-16}}} = \frac{3 \times 10^8}{2 \times 10^{-8}} = 1.5 \times 10^{16} \, \text{m/s} \]
Thus, the velocity of light in the medium is \( \frac{1}{6} \) times the velocity in a vacuum, so the correct answer is 6.
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