Question

A helium-neon laser produces light of wavelength \(\lambda_{\mathrm{vac}}=632.8 \mathrm{nm}\) in vacuum. If this light passes into water, with index of refraction \(n=1.333\), what will each of the following characteristics be? a) speed b) frequency c) wavelength d) color

Short answer

Answer: When the helium-neon laser light passes into water, its characteristics are as follows: a) Speed: \(2.25 × 10^8 m/s\) b) Frequency: \(4.74 × 10^{14} Hz\) c) Wavelength: \(475.1 nm\) d) Color: Blue

Step-by-step solution

1) Find the speed of light in water

Use the formula for the speed of light in a medium: $$ v = \frac{c}{n} $$ where \(v\) is the speed of light in water, \(c\) is the speed of light in vacuum and \(n\) is the index of refraction. We are given \(c = 3.00 × 10^8 m/s\) in vacuum and \(n=1.333\). Plug in these values and find the speed: $$ v = \frac{3.00 × 10^8 m/s}{1.333} $$

2) Calculate the speed of light in water

After calculating the speed using the formula from step 1, we get: $$ v = 2.25 × 10^8 m/s $$ So, the speed of light in water is \(2.25 × 10^8 m/s\).

3) Find the frequency of light in water

The frequency of light does not change as it enters a different medium. So, we use the formula: $$ v_{vac} = c = f_{vac} \times \lambda_{vac} $$ Here, \(f_{vac}\) is the frequency in vacuum and \(\lambda_{vac}\) is the wavelength in vacuum. We are given \(\lambda_{vac} = 632.8 nm\) and \(c = 3.00 × 10^8 m/s\). Solve for the frequency: $$ f_{vac} = \frac{c}{\lambda_{vac}} $$

4) Calculate the frequency of light

Substituting the given values in the formula, we have: $$ f_{vac} = \frac{3.00 × 10^8 m/s}{632.8 × 10^{-9} m} $$ After calculating the frequency, we get: $$ f_{vac} = 4.74 × 10^{14} Hz $$ So, the frequency of light in vacuum is \(4.74 × 10^{14} Hz\), and it remains the same in water.

5) Find the wavelength of light in water

Now, we can use the frequency and the speed of light in water to find the new wavelength: $$ v_{water} = f_{vac} \times \lambda_{water} $$ Here, \(v_{water}\) is the speed of light in water and \(\lambda_{water}\) is the wavelength in water. Plug in the given values and solve for \(\lambda_{water}\): $$ \lambda_{water} = \frac{v_{water}}{f_{vac}} $$

6) Calculate the wavelength of light in water

Using the values from previous steps, we get: $$ \lambda_{water} = \frac{2.25 × 10^8 m/s}{4.74 × 10^{14} Hz} $$ After calculating the wavelength, we get: $$ \lambda_{water} = 475.1 nm $$ So, the wavelength of light in water is \(475.1 nm\).

7) Determine the color of light in water

The color of light is determined by its wavelength. In this case, the wavelength in water is \(475.1 nm\). Based on the visible light spectrum, this wavelength corresponds to a blue color. In conclusion, when the helium-neon laser light passes into water, its characteristics are as follows: a) Speed: \(2.25 × 10^8 m/s\) b) Frequency: \(4.74 × 10^{14} Hz\) c) Wavelength: \(475.1 nm\) d) Color: Blue