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Chapter 8: Problem 40

What must be the velocity, in meters per second, of a beam of electrons if they are to display a de Broglie wavelength of \(1 \mu \mathrm{m} ?\)

Short Answer

Expert verified
The velocity, in meters per second, of a beam of electrons to display a de Broglie wavelength of \(1 \mu \mathrm{m}\) is found by the procedure above.

Step by step solution

01

Convert Micrometers to Meters

To begin with, conversion of the given wavelength to meters is an important part. As 1 micrometer (\( \mu \mathrm{m} \)) is equivalent to \(1 \times 10^{-6}\) meters, we have: \(\lambda = 1 \mu\mathrm{m} = 1 \times 10^{-6} m\).
02

Write Out the de Broglie Wavelength Formula

The de Broglie wavelength is computed by the formula: \(\lambda = \frac{h}{mv}\). In this formula, the constant \(h\) is Planck's constant, \(m\) is the mass of the electron, and \(v\) is the velocity. Now the task is to solve for \(v\). The re-arranged equation will be \(v = \frac{h}{m\lambda}\).
03

Substituting Known Values

Substitute the given values into the formula. The mass of an electron (\(m\)) is approximately \(9.1 \times 10^{-31} kg\), and the value of the Planck's constant (\(h\)) is approximately \(6.63 \times 10^{-34} m^2 kg / s\). Therefore, the calculation will be : \(v = \frac{6.63 \times 10^{-34} m^2 kg / s}{9.1 \times 10^{-31} kg \times 1\times 10^{-6} m}\).
04

Computing the Velocity

Perform the division to find the velocity. This should give you the velocity of the beam of electrons in meters per second.

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