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Chapter 15: Problem 2192

At room temperature, if the relative permittivity of water is 80 and the relative permeability be \(0.0222\) then the velocity of light in water is \(\mathrm{m} / \mathrm{s}\) (A) \(2.5 \times 10^{8}\) (B) \(2.26 \times 10^{8}\) (C) \(3.5 \times 10^{8}\) (D) \(3 \times 10^{8}\)

Short Answer

Expert verified
The velocity of light in water at room temperature, with a relative permittivity of 80 and relative permeability of 0.0222, can be calculated using the formula \(v = \frac{c}{\sqrt{\epsilon_r \mu_r}}\). Plugging in the given values, we find that the velocity of light in water is approximately \(2.26 \times 10^8\ \mathrm{m/s}\) (Option B).

Step by step solution

01

Identify the given values

We are given: Relative permittivity of water, \(\epsilon_r = 80\) Relative permeability of water, \(\mu_r = 0.0222\)
02

Write the formula for velocity of light in a medium

The formula for the velocity of light in a medium is given by: \(v = \frac{c}{\sqrt{\epsilon_r \mu_r}}\) where \(v\) is the velocity of light in the medium, \(c\) is the speed of light in a vacuum (approximately \(3 \times 10^8\ m/s\)), \(\epsilon_r\) is the relative permittivity, and \(\mu_r\) is the relative permeability.
03

Plug in the given values and calculate the velocity

Now we plug in the given values and the constant speed of light in vacuum: \(v = \frac{3 \times 10^8}{\sqrt{80 \times 0.0222}}\) Compute the value under the square root: \(v = \frac{3 \times 10^8}{\sqrt{1.776}}\) Now, calculate the square root and divide: \(v \approx \frac{3 \times 10^8}{1.3329} \approx 2.25 \times 10^8\ \mathrm{m/s}\)
04

Compare the calculated value with given options

Comparing our calculated value with the given options, we find that it is closest to: (B) \(2.26 \times 10^8\ \mathrm{m/s}\) So, the correct answer is (B) \(2.26 \times 10^8\ \mathrm{m/s}\).

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