Question

Light travels slower in water than it does in air; however, its frequency remains the same. How does the wavelength of light change as it travels from air to water?

Short answer

As light travels from air to water, its wavelength decreases by a factor of 0.75 (or 75%). This means that the wavelength of light is shorter in water compared to air, while its frequency remains the same.

Step-by-step solution

Identify the given information

We are given the following information: 1. The speed of light is slower in water than in air. 2. The frequency of light remains the same when it travels from air to water.

Write the wave speed formula for air and water

In air, the wave speed is \(v_{air}\), the frequency is \(f\), and the wavelength is \(λ_{air}\). So, the formula for wave speed in air is: \(v_{air} = fλ_{air}\) Similarly, in water, the wave speed is \(v_{water}\), the frequency is \(f\), and the wavelength is \(λ_{water}\). So, the formula for wave speed in water is: \(v_{water} = fλ_{water}\)

Find the relationship between the wavelengths in air and water

As mentioned in the exercise, the frequency remains the same in both air and water. Therefore, we can rearrange the wave speed formula in both media to find the relationship between the wavelengths. Divide the wave speed formula in water by the wave speed formula in air: \(\frac{v_{water}}{v_{air}} = \frac{fλ_{water}}{fλ_{air}}\) Since the frequency f is the same in both media, it cancels out: \(\frac{v_{water}}{v_{air}} = \frac{λ_{water}}{λ_{air}}\) Now, we have the relationship between the wavelengths in air and water.

Determine the change in wavelength

To determine how the wavelength of light changes as it travels from air to water, we need to know the ratio of the speed of light in water to the speed of light in air. The speed of light in a vacuum (or air, for simplicity) is approximately \(3.0 × 10^8 m/s\). The speed of light in water is about \(2.25 × 10^8 m/s\). Therefore, the ratio of the speeds is: \(\frac{v_{water}}{v_{air}} = \frac{2.25 × 10^8 m/s}{3.0 × 10^8 m/s} = 0.75\) Now, plug this ratio into the relationship we found in step 3: \(0.75 = \frac{λ_{water}}{λ_{air}}\) To find how the wavelength changes as light travels from air to water, multiply the wavelength in air by the ratio of the speeds: \(λ_{water} = 0.75 × λ_{air}\)

Conclusion

As the light travels from air to water, its wavelength decreases by a factor of 0.75 (or 75%). This means that the wavelength of light is shorter in water compared to air, while its frequency remains the same.

Key concepts

Speed of Light

The speed of light is a fundamental concept in physics, often denoted by the symbol "c" when considering its speed in a vacuum. However, light doesn't always travel through a vacuum; it can move through various media such as air, water, or even glass. When light enters different media, its speed changes. This change is because different materials have different optical properties which affect how quickly light can travel through them.

In a vacuum, the speed of light is approximately \[3.0 \times 10^8 \text{ m/s} \].

• In air, the speed is almost the same as in a vacuum, since air has very low density and offers little resistance.
• In water, the speed is reduced to about \[2.25 \times 10^8 \text{ m/s} \], indicating more resistance encountered by the light wave.

Remember, the speed of light is crucial in understanding other aspects of light behavior, like refraction and color splitting. When the speed of light changes, as it does from air to water, it impacts other properties of the wave too, such as wavelength (but not frequency). This forms the basis of Snell's Law, which explains how light bends when it moves from one medium to another.

Wavelength Change

When light travels from one medium to another, such as from air into water, its speed changes, and so does its wavelength. Though the frequency of light remains constant no matter the medium, the wavelength does not. This change in wavelength is a direct result of the change in the speed of the light wave as it moves through different materials.

In air, we can express the speed of light using the formula:\[ v_{\text{air}} = f \lambda_{\text{air}} \]Where: \( v_{\text{air}} \) is the speed of light in air, \( f \) is the frequency (which doesn't change), and \( \lambda_{\text{air}} \) is the wavelength in air.

Similarly, in water, the formula will be:\[ v_{\text{water}} = f \lambda_{\text{water}} \] The relationship between the wavelengths in different media can be expressed as: \[ \frac{\lambda_{\text{water}}}{\lambda_{\text{air}}} = \frac{v_{\text{water}}}{v_{\text{air}}} \]

• When light moves from a faster medium (like air) to a slower one (like water), \( \lambda_{\text{water}} \) becomes smaller. This means the wavelength gets shorter in water.

If the speed of light in water is 0.75 times what it is in air, then the wavelength in water is also 0.75 times the wavelength in air. This shrinking of the wavelength affects how we perceive the color of objects when viewed underwater.

Frequency of Light

The frequency of light is an important characteristic that remains constant irrespective of the medium it travels through. Unlike speed and wavelength, frequency does not change when light transitions from one medium to another. This constancy is crucial because it ensures that the energy of the light wave remains the same, no matter where it travels.

The frequency of a wave is defined as the number of oscillations or cycles per second, measured in hertz (Hz). It can be calculated by the formula: \[ f = \frac{v}{\lambda} \]Where: \( v \) is the wave speed in the medium, and \( \lambda \) is the wavelength in that medium.

• For a given source of light, the frequency remains constant as it travels through different media like air or water.
• This constancy also ensures that the color of the light doesn't change as it moves through various media, as color is determined by frequency.

Understanding that frequency remains unchanged helps in solving problems related to light refraction, as it allows focusing on how the speed and wavelength vary across media. It gives us a firm anchor point when analyzing how light behaves in different environments.