DP IB Physics: SL
C. Wave Behaviour
C.5 Doppler effect
DP IB Physics: SLC. Wave BehaviourC.5 Doppler effectLinking questions: | |
|---|---|
| a) | What are the similarities and difference between light and sound waves? |
| b) | How can the Doppler effect be utilized to measure the rotational speed of extended bodies? |
| c) | What happens if the speed of light is not much larger that the relative speed between the source and the observer? |
| d) | What gives rise to emission spectra and how can they be used to determine astronomical distances? |
| e) | How can the use of Doppler effect for light be used to calculate speed? (NOS) |
a) What are the similarities and difference between light and sound waves?
- Solution:
- The properties of wave behaviour, such as interference, refraction, and reflection, are shared by sound and light waves. Their characteristics and nature, however, are very different.
- Sound waves are mechanical and longitudinal, needing a medium like air or water to propagate, whereas light waves are electromagnetic and transverse, moving through a vacuum or matter.
- ⇒ Similarities:
- Wave properties:
- Both display basic wave properties such interference, diffraction, refraction, and reflection.
- Wavelength and Frequency:
- Wave speed, wavelength, and frequency are used to characterize both.
- Transfer of Energy:
- Both move energy instead of substance from one location to another.
- The Doppler Effect
- The Doppler effect, in which the source or observer’s motion causes the frequency to seem to vary, affects both.
- ⇒ Differences between light and sound waves:
| Feature | Light Waves | Sound Waves |
|---|---|---|
| Type of Wave | Electromagnetic (Transverse) | Mechanical (Longitudinal) |
| Medium Required | Do not require a medium (can travel in vacuum) | Require a medium (air, water, solids) |
| Speed | ~3 × 10⁸ m/s in vacuum | ~343 m/s in air at room temperature |
| Direction of Vibration | Vibrations are perpendicular to wave direction | Vibrations are parallel to wave direction |
| Can Travel in Vacuum? | Yes | No |
| Nature of Propagation | Electric and magnetic fields oscillating | Compression and rarefaction of particles |
| Detectable by | Human eyes (visible range only) | Human ears (audible range only) |
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b) How can the Doppler effect be utilized to measure the rotational speed of extended bodies?
- Solution:
- By examining the frequency variations in the waves reflected from the spinning item, the Doppler effect may be utilized to calculate the rotational speed of extended bodies. with particular, a variety of Doppler shifts are produced when waves are reflected off a spinning object because various sections of the object move at different speeds with relation to the source.
- It is possible to calculate the rotational speed by examining the distribution of these frequency changes.
- The shift in a wave’s apparent frequency or wavelength caused by the relative motion of the source and the observer is known as the Doppler effect.
- Planets, stars, and spinning machines can all have their rotational speeds measured using this technique.
- ⇒ The Doppler effect:
- The shift in a wave’s frequency or wavelength with respect to an observer travelling in relation to the wave source is known as the Doppler effect.
- The observed frequency rises (blue shift) when an observer moves closer to a source of waves (such as light or sound) and falls (red shift) as they move farther apart.
- Figure 1 Doppler effect
- ⇒ Rotational doppler effect:
- As an object rotates, its various points have varying velocities towards the source of the waves.
- Doppler shift will be positive at places on the side of the item spinning in the direction of the source and negative at points on the other side.
- Figure 2 Rotational doppler effect
- ⇒ Measuring Rotational Speed:
- Spectral Line Broadening:
- Astronomers utilise the broadening of spectral lines to detect the rotation of objects such as stars in order to determine their rotational speed.
- The range of Doppler shifts from rapidly spinning stars’ varying rotational velocities results in larger spectral lines.
- Vortex Beams:
- Light beams with a helical phase structure, or vortex beams, are employed in different applications.
- When these beams come into contact with a spinning object, the rotational Doppler effect causes a frequency shift that is precisely proportional to the angular velocity of the object.
- These beams carry orbital angular momentum.
- Doppler Radar:
- To measure the speed of moving objects, Doppler radar systems send out electromagnetic waves and examine the frequency change of the reflected waves. By examining the frequency changes throughout the rotating object, this approach may be modified to estimate rotational speed.
- Measurements of rotational speeds, such as air turbulence or the rotation of celestial planets, may be made remotely thanks to the rotational Doppler effect.
c) What happens if the speed of light is not much larger that the relative speed between the source and the observer?
- Solution:
- The principles of physics, especially those pertaining to special relativity, would be very different if the speed of light were not substantially faster than the relative speed between a source and an observer.
- Our knowledge of space, time, and causality would all collapse as a result of the violation of the speed of light’s constancy, which is a fundamental tenet of contemporary physics.
- The classical (non-relativistic) Doppler effect is no longer accurate if the relative speed between the source and the observer approaches the speed of light.
- The relativistic Doppler effect, which takes special relativity’s effects into account, must be used instead.
- Figure 3 Einstein’s Relativity
- ⇒ When relative Speed ≈ Speed:
- Time dilation becomes significant:
- The movable frame’s clocks run more slowly.
- Both motion and time dilation factors cause this to change the perceived frequency.
- Relativistic Doppler Effect:
- When the source and observer travel straight towards or away from one another, the observed frequency [math]f'[/math] may be calculated using the following formula:
- [math]f’ = f \sqrt{\frac{1 + \beta}{1 – \beta}} \quad \text{(for approach)} \\
f’ = f \sqrt{\frac{1 – \beta}{1 + \beta}} \quad \text{(for recession)}[/math] - Where:
- – [math]f[/math] = emitted frequency
- – [math]f'[/math] = observed frequency
- – [math]\beta = \frac{v}{c}[/math]
d) What gives rise to emission spectra and how can they be used to determine astronomical distances?
- Solution:
- When excited atoms’ electrons go to lower energy levels and release photons with certain wavelengths, they produce emission spectra, which are distinctive patterns of coloured lines.
- Redshift observations, which are wavelength shifts brought on by the Doppler effect as light moves across space, may be used to calculate astronomical distances from these spectra.
- ⇒ How emission spectra are formed:
- Electron Excitation:
- Electrons in an atom become excited and rise to higher energy levels as it absorbs energy, such as from heat, electricity, or radiation.
- Figure 4 Electron excitation:
- Electron De-excitation:
- Due to their instability, these excited electrons quickly return to their lower energy states.
- They emit photons of light with an energy equal to the difference between the energy levels as they descend:
- [math]E = hf = E_{\text{high}} – E_{\text{low}}[/math]
- Since every element has a different set of energy levels, the light that is released at particular wavelengths forms distinct spectral lines; this is known as the emission spectrum.
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e) How can the use of Doppler effect for light be used to calculate speed? (NOS)
- Solution:
- By examining the shift in wavelength of the light that an object emits, the Doppler effect for light—also referred to as redshift or blueshift—allows one to determine how fast an item is moving in relation to an observer.
- The light waves are compressed if the item is travelling in the direction of the viewer, producing a shorter wavelength (blueshift). The light waves are stretched if the object is moving away, giving rise to a longer wavelength (redshift).
- The shift in a wave’s apparent frequency (or wavelength) while the source and observer are moving relative to one other is known as the Doppler effect.
- This effect aids in determining how quickly an item is travelling towards or away from us in space when it comes to light waves.
- Figure 5 Doppler effect in light
- ⇒ Doppler shift:
- The measured wavelength of light looks shorter (blue-shifted) when a light source approaches an observer. On the other hand, the wavelength seems longer (redshifted) when the source becomes farther away.
- Redshift:
- The wavelengths are stretched when an object generating light travels away from the viewer, and this is seen as a shift towards red.
- Blueshift:
- Wavelengths are compressed and shift towards blue when an item approaches the viewer.
- Figure 6 Doppler effect in blue and red shift
- ⇒ Doppler shift formula for light (Non-relativistic Speeds)
- [math]\frac{\Delta \lambda}{\lambda} = \frac{v}{c}[/math]
- Where:
- – [math]\Delta \lambda = \lambda_{\text{observed}} – \lambda_{\text{rest}}[/math]
- – [math] \lambda[/math] = rest wavelength (know from lab measurements)
- – [math]v[/math] = speed of the source relative to the observer