Question

What is the de Broglie wavelength of an electron moving at speed $\frac{3}{5} c ?$

Short answer

Question: Determine the de Broglie wavelength of an electron moving at a speed of \(\frac{3}{5}c\), where \(c\) is the speed of light. Answer: The de Broglie wavelength of the electron moving at the given speed is approximately \(4.23 \times 10^{-11} \,\text{m}\).

Step-by-step solution

Write down the given values and constants

The given values and constants are: - Electron velocity, \(v = \frac{3}{5}c\) - Speed of light, \(c = 3.00 \times 10^8 \,\text{m/s}\) - Rest mass of electron, \(m_0 = 9.11 \times 10^{-31} \text{kg}\) - Planck's constant, \(h = 6.63 \times 10^{-34} \text{J s}\)

Calculate the momentum of the electron

To find the momentum of the electron, we'll use the relativistic momentum formula: \(p = \frac{m_0v}{\sqrt{1 - \frac{v^2}{c^2}}}\) Substitute \(v = \frac{3}{5}c\) and other given constants into the formula: \(p = \frac{(9.11 \times 10^{-31} \text{kg})(\frac{3}{5})(3.00 \times 10^8 \text{m/s})}{\sqrt{1 - \frac{(\frac{3}{5}c)^2}{c^2}}}\) Simplify the equation by canceling out \(c\) in the denominator of the fraction: \(p = \frac{(9.11 \times 10^{-31} \text{kg})(\frac{3}{5})(3.00 \times 10^8 \text{m/s})}{\sqrt{1 - (\frac{3}{5})^2}}\) Now, evaluate the expression to find the momentum: \(p \approx 1.566 \times 10^{-23} \,\text{kg m/s}\)

Calculate the de Broglie wavelength

Now, use the formula for the de Broglie wavelength and substitute the momentum: \(\lambda = \frac{h}{p}\) \(\lambda = \frac{6.63 \times 10^{-34} \text{J s}}{1.566 \times 10^{-23} \,\text{kg m/s}}\) Evaluate this expression to find the de Broglie wavelength: \(\lambda \approx 4.23 \times 10^{-11} \,\text{m}\) The de Broglie wavelength of the electron moving at the given speed is approximately \(4.23 \times 10^{-11} \,\text{m}\).