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Step-by-Step Solution
Step 1: Understand the given data
We are given the following information about light traveling in a material:
Frequency, $ \nu = 2 \times 10^{14}\text{ Hz}$
Wavelength, $\lambda = 5000\,\text{\AA} = 5000 \times 10^{-10}\text{ m} = 5 \times 10^{-7}\text{ m}$
We need to find the refractive index ($\mu$) of the material.
Step 2: Write down the relevant formula
The speed $v$ of the light wave in the material can be calculated using:
$$ v = \nu \cdot \lambda $$
Once we have $v$, we can find the refractive index using:
$$ \mu = \frac{c}{v}, $$
where $c$ is the speed of light in vacuum ($3 \times 10^8\,\text{m/s}$).
Step 3: Calculate the speed of light in the material
Substitute the values for frequency and wavelength:
$$ v = (2 \times 10^{14}\,\text{Hz}) \times \bigl(5 \times 10^{-7}\,\text{m}\bigr) $$
$$ v = (2 \times 5)\times (10^{14} \times 10^{-7})\,\text{m/s} $$
$$ v = 10 \times 10^{7}\,\text{m/s} = 10^{8}\,\text{m/s} $$
Step 4: Calculate the refractive index
$$ \mu = \frac{c}{v} = \frac{3 \times 10^8\,\text{m/s}}{10^8\,\text{m/s}} = 3. $$
Thus, the refractive index of the material is $3.00.$
Answer: 3.00
