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Chapter 34: Problem 30

How slowly must an electron be moving for its de Broglie wavelength to be \(1 \mathrm{mm} ?\)

Short Answer

Expert verified
The electron must be moving at a speed of approximately \(728.1 ms^{-1}\) for its de Broglie wavelength to be \(1 mm\).

Step by step solution

01

Identify the given variables and constants

The de Broglie wavelength \(\lambda\) is given as \(1mm = 1 \times 10^{-3} m\). The mass of an electron \(m_e\) is \(9.11 \times 10^{-31} kg\), and the Planck's constant \(h\) is \(6.63 \times 10^{-34} Js\).
02

Rearrange the de Broglie wavelength formula to solve for the velocity

Rearranging the de Broglie formula to solve for velocity (\(v\)), we get \(v = \frac{h}{\lambda m_e}\). This will give us the velocity in terms of the de Broglie wavelength, the Planck constant, and the mass of the electron.
03

Substitute the known values into the rearranged formula

Substituting the known values into the equation \(v = \frac{h}{\lambda m_e}\), we get \(v = \frac{6.63 \times 10^{-34} Js}{1 \times 10^{-3} m \times 9.11 \times 10^{-31} kg}\).
04

Solve the equation

Solving the equation, we find \(v \approx 728.1 ms^{-1}\). Therefore, the electron must be moving at a speed of approximately \(728.1 ms^{-1}\) for its de Broglie wavelength to be \(1mm\).

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