Problem 34 - Master Resource Book in JEE Main Physics

Question

In \(\mathrm{H}\) spectrum, the wavelength of \(\mathrm{H}_{\alpha}\) line is \(656 \mathrm{~nm}\) whereas in a distance galaxy, the wavelength of \(\mathrm{H}_{\alpha}\) line is \(706 \mathrm{~nm}\). Estimate the speed of galaxy with respect to earth. (a) \(2 \times 10^{8} \mathrm{~ms}^{-1}\) (b) \(2 \times 10^{7} \mathrm{~ms}^{-1}\) (c) \(2 \times 10^{6} \mathrm{~ms}^{-1}\) (d) \(2 \times 10^{5} \mathrm{~ms}^{-1}\)

Step-by-Step Solution

Answer

The speed of the galaxy with respect to Earth is approximately \( 2 \times 10^7 \text{ m/s} \). The correct option is (b).

Step 1: Understand the Concept of Redshift

The shift in the wavelength of spectral lines due to the Doppler effect is known as redshift or blueshift. If the wavelength is observed to be longer than the rest wavelength, then the source is moving away, causing a redshift.

Step 2: Calculate the Redshift (z)

The redshift \( z \) is calculated using the formula: \[ z = \frac{\lambda_{observed} - \lambda_{rest}}{\lambda_{rest}} \]In this problem, \( \lambda_{rest} = 656 \text{ nm} \) and \( \lambda_{observed} = 706 \text{ nm} \).So, \[ z = \frac{706 - 656}{656} = \frac{50}{656} \approx 0.0762 \]

Step 3: Calculate the Velocity Using Redshift

The velocity \( v \) of a galaxy moving with respect to Earth can be estimated using: \[ v = zc \]where \( c = 3 \times 10^8 \text{ m/s} \) is the speed of light.So, \[ v = 0.0762 \times 3 \times 10^8 \approx 2.29 \times 10^7 \text{ m/s} \]

Step 4: Compare and Select the Closest Answer

The calculated velocity \( 2.29 \times 10^7 \text{ m/s} \) is closest to option (b) \( 2 \times 10^7 \text{ m/s} \). Therefore, the correct answer is (b).

Key Concepts

Doppler Effect

The Doppler effect is a phenomenon that occurs when the source of a wave (like sound or light) moves relative to an observer. This effect is named after Christian Doppler, who first proposed it in 1842. The Doppler effect can cause a change in frequency or wavelength of the wave depending on the relative movement. When a source moves towards an observer, the waves are compressed, resulting in a higher frequency or shorter wavelength. This is known as a blueshift.

• Conversely, when the source moves away, the waves are stretched, giving a lower frequency or longer wavelength, known as a redshift.
• This effect is not only observable in sound waves but also in light waves from astronomical objects.
• In astronomy, the Doppler effect is crucial for understanding the movement of distant stars and galaxies.

By examining the shifts in the spectra, astronomers can deduce whether an object is moving towards or away from us and calculate its speed.

Galaxy Velocity

The velocity of a galaxy indicates how fast it is moving relative to the Earth. Calculating this velocity often involves understanding and using the concept of redshift. When analyzing distant galaxies, scientists look at the shift in spectral lines, such as the hydrogen spectral lines, to determine the galaxy's velocity.

• Using the redshift value, one can calculate the velocity using the formula: \[ v = zc \]where \(v\) is velocity, \(z\) is redshift, and \(c\) is the speed of light.
• The speed of light, \( c \), is approximately \( 3 \times 10^8 \) meters per second.
• The calculated velocity can indicate whether the galaxy is moving towards or away from us.

This information is valuable for understanding the expansion of the universe and the dynamics of galaxies.

Hydrogen Spectral Lines

Hydrogen spectral lines are specific wavelengths of light emitted or absorbed by hydrogen atoms. These lines appear in the electromagnetic spectrum and are significant in astronomical studies. Due to their simple structure, hydrogen spectral lines serve as excellent markers for studying the universe.

• In astronomy, one of the common lines observed is the \(H_\alpha\) line, which occurs at around \(656 \text{ nm}\) when at rest.
• This line can shift towards longer wavelengths (red-shift) or shorter wavelengths (blue-shift) depending on the velocity of the source relative to the observer.
• By observing shifts in hydrogen spectral lines, scientists can gather important information about the motion and distance of celestial bodies.

The shifts in these spectral lines are a direct consequence of the Doppler effect, providing insight into the kinematics of galaxies and expansion of the universe.