Sure, let's break these questions down step-by-step for a clear understanding.
### Question 1: Calculate the frequency of the wave in air
Given:
- Wavelength ([tex]\(\lambda\)[/tex]) = 2 cm
- Speed of light in air, [tex]\(c = 3 \times 10^{10}\)[/tex] cm/s
We can use the wave equation to find the frequency [tex]\(f\)[/tex]:
[tex]\[ f = \frac{c}{\lambda} \][/tex]
Substituting the given values:
[tex]\[ f = \frac{3 \times 10^{10} \, \text{cm/s}}{2 \, \text{cm}} \][/tex]
[tex]\[ f = 1.5 \times 10^{10} \, \text{Hz} \][/tex]
So, the frequency of the wave in air is [tex]\(1.5 \times 10^{10}\)[/tex] Hz.
### Question 2: Calculate the speed of light in the medium with a refractive index of 5
Given:
- Refractive index of the medium ([tex]\(n\)[/tex]) = 5
- Speed of light in air, [tex]\(c = 3 \times 10^{10}\)[/tex] cm/s
The speed of light in the medium ([tex]\(v\)[/tex]) can be calculated using the formula:
[tex]\[ v = \frac{c}{n} \][/tex]
Substituting the given values:
[tex]\[ v = \frac{3 \times 10^{10} \, \text{cm/s}}{5} \][/tex]
[tex]\[ v = 6 \times 10^9 \, \text{cm/s} \][/tex]
So, the speed of light in the medium is [tex]\(6 \times 10^9\)[/tex] cm/s.
### Question 3: Calculate the ratio of the speeds in two materials with given refractive indices
Given:
- Refractive index of material [tex]\(A\)[/tex] ([tex]\(n_A\)[/tex]) = [tex]\(\frac{5}{4}\)[/tex]
- Refractive index of material [tex]\(B\)[/tex] ([tex]\(n_B\)[/tex]) = [tex]\(\frac{2}{3}\)[/tex]
The speed of light in a material is inversely proportional to its refractive index. Thus, the ratio of the speeds [tex]\(v_B / v_A\)[/tex] can be calculated as:
[tex]\[ \frac{v_B}{v_A} = \frac{n_A}{n_B} \][/tex]
Substituting the given values:
[tex]\[ \frac{v_B}{v_A} = \frac{\frac{5}{4}}{\frac{2}{3}} \][/tex]
[tex]\[ \frac{v_B}{v_A} = \frac{5}{4} \times \frac{3}{2} \][/tex]
[tex]\[ \frac{v_B}{v_A} = \frac{15}{8} \][/tex]
[tex]\[ \frac{v_B}{v_A} = 1.875 \][/tex]
So, the ratio of the speeds in the two materials is 0.5333 (or [tex]\(\frac{16}{30}\)[/tex]) approximately.
In summary:
1. The frequency of the wave in air is [tex]\(1.5 \times 10^{10}\)[/tex] Hz.
2. The speed of light in the medium is [tex]\(6 \times 10^9\)[/tex] cm/s.
3. The ratio of the speeds in the two materials is approximately 0.5333.
