If de-Broglie wavelength is \(\lambda\) when energy is E. Find the wavelength at \(\frac{E}{4}\) (Kinetic Energy).
- \(2\lambda \)
- \(\sqrt{2}\lambda \)
- \(\lambda \)
- \(\frac{\lambda }{\sqrt{2}}\)
The Correct Option is A
Solution and Explanation
\(\lambda=\frac{h}{p}\)
where h is Planck's constant and p is the momentum of the particle. The momentum of a particle with kinetic energy E is given by:
\(p = β(2mE)\)
where m is the mass of the particle.
If the de Broglie wavelength of a particle with energy E is π, then we have:
\(\lambda = \frac{h}{β(2mE)}\)
To find the de Broglie wavelength of a particle with kinetic energy E/4, we first need to find the momentum of the particle. The momentum is given by:
\(p = \sqrt{(2m(\frac{E}{4}))} = \sqrt{(m\frac{E}{2})}\)
The de Broglie wavelength of the particle with momentum p is then given by:
\(\lambda^{'} = \frac{h}{p} = \frac{h}{\sqrt(m\frac{E}{2})}\)
Dividing this expression by the original expression for π, we get:
\(\frac{{\lambda}'}{\lambda} = \sqrt{2}\)
So, the wavelength of the particle with kinetic energy \(\frac{E}{4}\) is β2 times the wavelength of the particle with kinetic energy E. Therefore, the answer is option 2: β2π.
Answer. A
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Concepts Used:
De Broglie Hypothesis
One of the equations that are commonly used to define the wave properties of matter is the de Broglie equation. Basically, it describes the wave nature of the electron.
De Broglie Equation Derivation and de Broglie Wavelength
Very low mass particles moving at a speed less than that of light behave like a particle and waves. De Broglie derived an expression relating to the mass of such smaller particles and their wavelength.
Plankβs quantum theory relates the energy of an electromagnetic wave to its wavelength or frequency.
E = hΞ½ β¦β¦.(1)
E = mc2β¦β¦..(2)
As the smaller particle exhibits dual nature, and energy being the same, de Broglie equated both these relations for the particle moving with velocity βvβ as,
This equation relating the momentum of a particle with its wavelength is de Broglie equation and the wavelength calculated using this relation is the de Broglie wavelength.