Question 108: Dual Nature of Matter and Radiation

Question

A proton and an $$\alpha$$-particle are accelerated from rest by $$2 \mathrm{~V}$$ and $$4 \mathrm{~V}$$ potentials, respectively. The ratio of their de-Broglie wavelength is :

Accepted Answer

The de-Broglie wavelength of a particle is given by:
$$\lambda = \frac{h}{p}$$
where $$h$$ is the Planck's constant and $$p$$ is the momentum of the particle.
The momentum of a particle of mass $$m$$ and charge $$q$$ accelerated through a potential difference $$V$$ is given by:
$$p = \sqrt{2 m q V}$$
For a proton, $$m = 1.67 	imes 10^{-27} \mathrm{~kg}$$ and $$q = 1.6 	imes 10^{-19} \mathrm{~C}$$, and for an $$\alpha$$-particle, $$m = 6.64 	imes 10^{-27} \mathrm{~kg}$$ and $$q = 2 	imes 1.6 	imes 10^{-19} \mathrm{~C}$$.
Therefore, the ratio of their de-Broglie wavelengths is:
$$\frac{\lambda_p}{\lambda_\alpha} = \frac{p_\alpha}{p_p} = \sqrt{\frac{m_\alpha}{m_p} \cdot \frac{q_\alpha V_\alpha}{q_p V_p}} = \sqrt{\frac{6.64 	imes 10^{-27}}{1.67 	imes 10^{-27}} \cdot \frac{2 	imes 1.6 	imes 10^{-19} 	imes 4}{1.6 	imes 10^{-19} 	imes 2}} = \sqrt{16} = 4$$
Therefore, the ratio of their de-Broglie wavelengths is 4 : 1.

Answer

4 : 1

Answer

2 : 1

Answer

8 : 1

Answer

16 : 1

Dual Nature of Matter and Radiation

A proton and an $\alpha-particle are accelerated from rest by 2 \mathrm{~V} and 4 \mathrm{~V}$ potentials, respectively. The ratio of their de-Broglie wavelength is :

Choose the correct answer: