Question

The de-Broglie wavelength associated with a particle with rest mass \(\mathrm{m}_{0}\) and moving with speed of light in vacuum is..... (A) \(\left(\mathrm{h} / \mathrm{m}_{0} \mathrm{c}\right)\) (B) 0 (C) \(\infty\) (D) \(\left(\mathrm{m}_{0} \mathrm{c} / \mathrm{h}\right)\)

Short answer

The correct answer is option (C) \(\infty\) as the de-Broglie wavelength is undefined in this case.

Step-by-step solution

Recall the de-Broglie wavelength formula

The de-Broglie wavelength associated with any particle is given by \[ λ = \frac{h}{mv} \] Where \(h\) is Planck's constant, \(m\) is the mass of the particle, and \(v\) is the speed of the particle. Since the particle is moving with the speed of light, we'll use the relativistic mass formula, which takes into account the increase in mass as the particle's speed increases.

Recall the relativistic mass formula and apply it to the problem

The relativistic mass formula is given by \[ m = \frac{m_0}{\sqrt{1-\frac{v^2}{c^2}}} \] Where \(m_0\) is the rest mass of the particle, \(v\) is the speed of the particle, and \(c\) is the speed of light in a vacuum. However, since the problem states that the particle is moving with the speed of light in a vacuum, this means that \(v = c\). Substituting this value into the relativistic mass formula, we obtain \[ m = \frac{m_0}{\sqrt{1-\frac{c^2}{c^2}}} \] Simplifying the denominator, we find that \[ m = \frac{m_0}{\sqrt{1-1}} = \frac{m_0}{0} \]

Conclude that the de-Broglie wavelength is undefined in this case

Since the mass of the particle is given by \(\frac{m_0}{0}\), the mass becomes undefined as the speed of the particle approaches the speed of light. This means there is an asymptotic relationship between the speed and the mass of the particle, making it impossible to ever reach the speed of light. So, the de-Broglie wavelength formula is also undefined in this situation, as the mass has become undefined. Thus, the correct answer is option (C) \(\infty\) as the de-Broglie wavelength is undefined in this case.