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Found in: Page 1181

### Fundamentals Of Physics

Book edition 10th Edition
Author(s) David Halliday
Pages 1328 pages
ISBN 9781118230718

# Figure 38-24 shows an electron moving through several regions where uniform electric potentials V have been set up. Rank the three regions according to the de Broglie wavelength of the electron there, greatest first.

The rank is ${\lambda }_{2}>{\lambda }_{1}>{\lambda }_{3}$.

See the step by step solution

## Step 1: Describe the de Broglie wavelength:

The de Broglie wavelength is given by,

${\mathbit{\lambda }}{\mathbf{=}}\frac{\mathbf{h}}{\mathbf{p}}$

Here, h is the Planck's constant, ${\mathbit{\lambda }}$ is wavelength, and p is the momentum.

The kinetic energy in terms of momentum is given by,

role="math" localid="1663142299185" ${\mathbit{K}}{\mathbf{=}}\frac{{\mathbf{p}}^{\mathbf{2}}}{\mathbf{2}\mathbf{m}}\phantom{\rule{0ex}{0ex}}{\mathbit{p}}{\mathbf{=}}\sqrt{\mathbf{2}\mathbf{m}\mathbf{K}}$

Here, m is the mass.

Consider an electron, accelerated with a potential difference of V, the kinetic energy is given by,

${\mathbit{K}}{\mathbf{}}{\mathbf{=}}{\mathbf{}}{\mathbit{e}}{\mathbit{V}}$

The momentum is given by,

${\mathbit{p}}{\mathbf{=}}\sqrt{\mathbf{2}\mathbf{m}\mathbf{e}\mathbf{V}}$

The expression for wavelength will be as follows:

${\mathbit{\lambda }}{\mathbf{=}}\frac{\mathbf{h}}{\sqrt{\mathbf{2}\mathbf{m}\mathbf{e}\mathbf{V}}}$

## Step 2: Rank the three regions according to the de Broglie wavelength of the electron:

From the above de Broglie wavelength expression, the wavelength is inversely proportional to the $\sqrt{V}$, which means the wavelength decreases as the potential difference increases and vice-versa.

As given the potential difference,

${V}_{1}=-100\text{}V\phantom{\rule{0ex}{0ex}}{V}_{2}=-200\text{}V\phantom{\rule{0ex}{0ex}}{V}_{3}=+100\text{}V\phantom{\rule{0ex}{0ex}}$

So, the rank of de Broglie wavelength will be ${\lambda }_{2}>{\lambda }_{1}>{\lambda }_{3}$ .

Therefore, the rank is ${\lambda }_{2}>{\lambda }_{1}>{\lambda }_{3}$ .