Question

How slow would an electron have to be traveling for its wavelength to be at least$\mathbf{\text{1\hspace{0.17em}μm}}$?

Short answer

The electron speed is $\text{v}=\text{728\hspace{0.17em}m}/s$.

Step-by-step solution

Given data.

The wavelength of the electron.$\lambda =\text{1}\times {\text{10}}^{\text{-6}}\text{\hspace{0.17em}m}$

Mass of electron is$\text{m}=\text{9.1}\times {\text{10}}^{\text{-31}}\text{\hspace{0.17em}kg}$ .

Step 2: de Broglie wavelength.

The following equation can be used to describe the de Broglie wavelength.

$\mathbf{\text{p=}}\frac{\mathbf{\text{h}}}{\mathbf{\lambda }}\mathbf{\text{………………(1)}}$

where the kinetic energy is defined as

$\mathbf{\text{p=mv………………(2)}}$

Speed of the electron

Using Equation$\text{(1)}$ and$\text{(2)}$ ,

$\begin{array}{l}\text{mv}=\frac{\text{h}}{\lambda }\\ \text{v}=\frac{\text{h}}{\text{m}\lambda }\end{array}$

Applying the resulting formula for the electron's speed

$\begin{array}{c}\text{v}=\frac{\text{h}}{\text{m}\lambda }\\ =\frac{\text{6.626}\times {\text{10}}^{\text{-34}}J\cdot s}{(\text{9.1}\times {\text{10}}^{\text{-31}}kg)\text{×}(\text{1}\times {\text{10}}^{\text{-6}}m)}\\ =\text{728\hspace{0.17em}m}/s\end{array}$

Therefore the speed of the electron is $\text{728\hspace{0.17em}m}/s$.