The graph which shows the variation of the de Broglie wavelength (λ) of a particle and its associated momentum (p) is
The Correct Option is D
Solution and Explanation
To determine the graph that represents the variation of the de Broglie wavelength (λ) with respect to momentum (p), we need to analyze the relationship between these two quantities. According to the de Broglie hypothesis, the wavelength of a particle is given by:
\(λ = \frac hp\)
where:
- λ is the de Broglie wavelength.
- h is the Planck's constant.
- p is the momentum of the particle.
This equation indicates that the de Broglie wavelength is inversely proportional to the momentum. Hence, as momentum increases, the wavelength decreases, and vice versa. Graphically, this inverse relationship is represented by a hyperbolic curve when plotting λ versus p.
Given the options and the inverse relationship, the correct graph will demonstrate a hyperbolic decrease in λ as p increases, consistent with an inverse proportionality relation. Therefore, the correct answer corresponds to the graph with option (D).
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Concepts Used:
Wave Characteristics
The main properties of waves are as follows –
- Amplitude - The maximum displacement of the wave from the mean position is called the amplitude of the wave. It is the maximum height from the centre line to the crest or the trough. The crest is the highest point of the wave and the trough is the lowest point of the wave. Amplitude is measured in metres.
- Frequency - The number of vibrations passing a fixed point in a given amount of time is called frequency. The unit of frequency is Hertz.
- Wavelength - Wavelength is the distance between two identical points (adjacent crests or troughs). It is measured in metres. Frequency and wavelength are inversely proportional to each other.
- Time Period - The time taken by a complete wave to pass through a particular point is called the time period. The time period is measured in seconds. The time period is the reciprocal of the frequency.
- Speed - For a wave, speed is the distance travelled by a particular point on the wave in the given interval of time. Speed is measured in metres per second.