Protons are being accelerated in a particle accelerator.When the energy of the protons is doubled,their de Broglie wavelength will
A)increase by a factor of 4.
B)increase by a factor of 2.
C)decrease by a factor of 2.
D)increase by a factor of $\sqrt { 2 }$
E)decrease by a factor of $\sqrt { 2 }$

Question

Quizplus · Accepted Answer

The de Broglie wavelength $\lambda$ is given by $\lambda = \frac{h}{p}$, where $h$ is Planck's constant and $p$ is the momentum of the particle. For a proton in a particle accelerator, its kinetic energy $E_k$ is related to its momentum by $E_k = \frac{p^2}{2m}$, where $m$ is the mass of the proton. If the energy of the protons is doubled, the new kinetic energy is $2E_k$, leading to a new momentum $p' = \sqrt{2m \cdot 2E_k} = \sqrt{2}p$. Thus, the new de Broglie wavelength $\lambda' = \frac{h}{p'} = \frac{h}{\sqrt{2}p} = \frac{\lambda}{\sqrt{2}}$, meaning the wavelength decreases by a factor of $\sqrt{2}$.

Protons Are Being Accelerated in a Particle Accelerator 2\sqrt { 2 }2​

Protons Are Being Accelerated in a Particle Accelerator $\sqrt { 2 }$