How fast must a nonrelativistic electron move so that its de Broglie wavelength is the same as the wavelength of a 3.4-eV photon? (mass e = 9.11 x...

Thursday, May 19, 2016

How fast must a nonrelativistic electron move so that its de Broglie wavelength is the same as the wavelength of a 3.4-eV photon? (mass e = 9.11 x...

First, determine the wavelength of the photon. To do so, apply the formula of energy of photon.

$\displaystyle{E}={h}{f{}}$

Since the frequency of light is $\displaystyle{f{=}}\frac{{c}}{\lambda}$ , the formula of photon's energy can be re-written as:

$\displaystyle{E}=\frac{{{h}{c}}}{\lambda}$

Then, isolate the wavelength of the photon.

$\displaystyle\lambda=\frac{{{h}{c}}}{{E}}$

Plugging in the values, the formula becomes:

$\displaystyle\lambda=\frac{{{\left({6.63}\times{{10}}^{{-{34}}}{J}\cdot{s}\right)}{\left({3}\times{{10}}^{{8}}\frac{{m}}{{s}}\right)}}}{{{3.4}{e}{V}\cdot\frac{{{1.60}\times{{10}}^{{-{19}}}{J}}}{{{1}{e}{V}}}}}$

$\displaystyle\lambda={3.65625}\times{{10}}^{{-{7}}}$

So, the wavelength of the photon is $\displaystyle{3.65625}\times{{10}}^{{-{7}}}$ m.

Next, consider the formula for DeBroglie wavelength.

$\displaystyle\lambda=\frac{{h}}{{{m}{v}}}$

Then, isolate the speed of the particle.

$\displaystyle{v}=\frac{{h}}{{{m}\lambda}}$

Since it is given that the wavelength of the photon is the same as the DeBroglie wavelength of the electron, plug in $\displaystyle\lambda={3.65625}\times{{10}}^{{-{7}}}.'{m}.{A}{l}{s}{o},{p}{l}{u}{g{\in}}{t}{h}{e}{v}{a}{l}{u}{e}{o}{f{{t}}}{h}{e}{P}{l}{a}{n}{c}{k}'{s}{c}{o}{n}{s}{\tan{{t}}}{\quad\text{and}\quad}{t}{h}{e}{m}{a}{s}{s}{o}{f{{e}}}\le{c}{t}{r}{o}{n}.$

$\displaystyle{v}=\frac{{{6.63}\times{{10}}^{{-{34}}}{J}\cdot{s}}}{{{\left({9.11}\times{{10}}^{{-{31}}}{k}{g}\right)}{\left({3.65625}\times{{10}}^{{-{7}}}{m}\right)}}}$

$\displaystyle{v}={1990.47}\frac{{m}}{{s}}$

Therefore, the speed of the electron is 1990.47 m/s.

Ninenotes

Thursday, May 19, 2016

How fast must a nonrelativistic electron move so that its de Broglie wavelength is the same as the wavelength of a 3.4-eV photon? (mass e = 9.11 x...

No comments:

Post a Comment

find square roots of -1+2i