Question

State whether the following is true or false and explain your answer: The wavelength of He-Ne laser light in water is less than its wavelength in air. (The index of refraction of water is \(1.33 .)\)

Short answer

Answer: Yes, the wavelength of He-Ne laser light in water is shorter than its wavelength in air.

Step-by-step solution

Write down the known values

Index of refraction of water (n₁) = 1.33 Index of refraction of air (n₂) ≈ 1 (since air has a very low index of refraction) Wavelength of He-Ne laser light in air (λ₂) = ?

Understand the relationship between index of refraction, speed of light, and wavelength

The formula for the index of refraction (n) is: n = \(\frac{c}{v}\) where 'c' is the speed of light in vacuum and 'v' is the speed of light in the medium. Furthermore, the speed of light in a medium is related to its wavelength (λ) by the equation: v = \(\frac{c}{n} = \frac{cλ}{λ₀}\) where 'λ₀' is the wavelength in vacuum, and 'λ' is the wavelength in the medium. Rearranging the above equation for the wavelength in medium, we get: λ = \(\frac{λ₀}{n}\)

Compare the wavelengths in water and air

Using the formula λ = \(\frac{λ₀}{n}\), compare the wavelengths of He-Ne laser light in water (λ₁) and air (λ₂): λ₁ = \(\frac{λ₀}{n₁}\), where n₁ is the index of refraction of water λ₂ = \(\frac{λ₀}{n₂}\), where n₂ is the index of refraction of air Now, divide λ₁ by λ₂: \(\frac{λ₁}{λ₂} = \frac{\frac{λ₀}{n₁}}{\frac{λ₀}{n₂}}\) Simplify the equation: \(\frac{λ₁}{λ₂} = \frac{n₂}{n₁}\) Since we know that \(n₁ > n₂\), we can conclude that: \(\frac{λ₁}{λ₂} < 1\) Therefore, λ₁ < λ₂.

Conclude the result

The wavelength of He-Ne laser light in water (λ₁) is less than its wavelength in air (λ₂). Thus, the statement is true.