A helium-neon laser beam has a wavelength in air of 633 nm. It takes 1.38 ns for the light to travel through 30 cm of an unknown liquid. What is the wavelength of the laser beam in the liquid?

Verified step by step guidance

1. Step 1: Start by identifying the relationship between the speed of light in the liquid and the speed of light in air. The speed of light in a medium is given by \( v = \frac{d}{t} \), where \( d \) is the distance traveled and \( t \) is the time taken. Use the given values: \( d = 30 \, \text{cm} = 0.30 \, \text{m} \) and \( t = 1.38 \, \text{ns} = 1.38 \times 10^{-9} \, \text{s} \).
2. Step 2: Calculate the speed of light in the liquid using \( v = \frac{d}{t} \). Substitute \( d \) and \( t \) into the formula to find \( v \), the speed of light in the liquid.
3. Step 3: Determine the refractive index of the liquid using the formula \( n = \frac{c}{v} \), where \( c \) is the speed of light in air (approximately \( 3.00 \times 10^{8} \, \text{m/s} \)) and \( v \) is the speed of light in the liquid calculated in Step 2.
4. Step 4: Use the relationship between wavelength, refractive index, and the wavelength in air: \( \lambda_{\text{liquid}} = \frac{\lambda_{\text{air}}}{n} \). Substitute \( \lambda_{\text{air}} = 633 \, \text{nm} = 633 \times 10^{-9} \, \text{m} \) and \( n \) (from Step 3) into the formula to find the wavelength of the laser beam in the liquid.
5. Step 5: Express the final wavelength in the liquid in nanometers (nm) for clarity, as the original wavelength in air was given in this unit.

Key Concepts

Wavelength and Frequency

Wavelength is the distance between successive peaks of a wave, while frequency is the number of peaks that pass a point in a given time. In a vacuum, the speed of light is constant, and the relationship between wavelength (λ), frequency (f), and speed (c) is given by the equation c = λf. When light travels through different media, its speed changes, affecting its wavelength while the frequency remains constant.

Refraction and Index of Refraction

Refraction is the bending of light as it passes from one medium to another due to a change in speed. The index of refraction (n) quantifies this change and is defined as the ratio of the speed of light in a vacuum to the speed of light in the medium. The relationship between the wavelength in a medium (λ') and the wavelength in a vacuum (λ) is given by λ' = λ/n, where n is the index of refraction of the medium.

Speed of Light in Different Media

The speed of light varies in different materials due to their optical properties. In a vacuum, light travels at approximately 3.00 x 10^8 m/s, but in a medium like water or glass, it travels slower. The time taken for light to travel a certain distance in a medium can be used to calculate the speed of light in that medium, which is essential for determining the new wavelength when light enters a different substance.