Question

An electron moving with velocity \(0.6 \mathrm{c}\), then de-brogly wavelength associated with is \(\ldots \ldots \ldots\) (rest mars of electron, \(\mathrm{m}_{0}=9.1 \times 10^{-31}(\mathrm{k} / \mathrm{s})\) \(\mathrm{h}=6.63 \times 10^{-34} \mathrm{Js}\) (A) \(3.24 \times 10^{-12} \mathrm{~m}\) (B) \(32.4 \times 10^{-12} \mathrm{~m}\) (C) \(320 \times 10^{-12} \mathrm{~m}\) (D) \(3.29 \times 10^{-14} \mathrm{~m}\)

Short answer

The de Broglie wavelength associated with the electron moving with velocity \(0.6c\) is approximately \(3.24 \times 10^{-12} \, m\).

Step-by-step solution

Identify the de Broglie wavelength formula

The de Broglie wavelength formula is given by: \[ \lambda = \frac{h}{p} \] where \(\lambda\) is the de Broglie wavelength, \(h\) is the Planck's constant, and \(p\) is the momentum of the particle.

Calculate the electron's momentum

The momentum formula for a relativistic particle is given by: \[ p = \frac{m_0 v}{\sqrt{1 - \frac{v^2}{c^2}}} \] Let's plug the values of mass of electron (\(m_0 = 9.1 \times 10^{-31} \, kg\)), velocity (\(v = 0.6c\)), and the speed of light (\(c = 3 \times 10^8 \, m/s\)) into the formula. \[ p = \frac{9.1 \times 10^{-31} \times 0.6 \times (3 \times 10^8)}{\sqrt{1 - 0.6^2}} \]

Solve for the momentum

After simplifying and calculating the denominator, we find: \[ p = \frac{9.1 \times 10^{-31} \times 0.6 \times (3 \times 10^8)}{\sqrt{1 - 0.36}} = \frac{9.1 \times 10^{-31} \times 0.6 \times (3 \times 10^8)}{\sqrt{0.64}} = \frac{9.1 \times 10^{-31} \times 0.6 \times (3 \times 10^8)}{0.8} \] Further calculation gives: \[ p \approx 2.051 \times 10^{-22} \, kg \cdot m/s \]

Calculate de Broglie wavelength

Now, we'll substitute the values of Planck's constant (\(h = 6.63 \times 10^{-34} \, Js\)) and the calculated momentum into the de Broglie wavelength formula: \[ \lambda = \frac{6.63 \times 10^{-34}}{2.051 \times 10^{-22}} \approx 3.23 \times 10^{-12} \, m \]

Match the calculated wavelength with the given options

Comparing our calculated wavelength with the given options, it's closest to option (A): \[ \lambda = 3.24 \times 10^{-12} \, m \]