Question 15: Dual Nature of Matter and Radiation

Question

A particle of mass $$2 \mathrm{mg}$$ has the same wavelength as a neutron moving with a velocity of $$3 	imes 10^5 \mathrm{~ms}^{-1}$$. The velocity of the particle is (mass of neutron is $$1.67 	imes 10^{-27} \mathrm{Kg}$$)

Accepted Answer

To find the velocity of the particle of mass $$2 \mathrm{mg}$$ that has the same wavelength as a neutron moving with a velocity of $$3 	imes 10^5 \mathrm{~ms}^{-1}$$, we can use the de Broglie wavelength formula:
$$ \lambda = \frac{h}{mv} $$
Where:
$$ \lambda $$ is the wavelength
$$ h $$ is Planck's constant (approximately $$6.626 	imes 10^{-34} \mathrm{~Js}$$)
$$ m $$ is the mass of the particle
$$ v $$ is the velocity of the particle
First, let's find the wavelength of the neutron. The mass of the neutron is given as $$1.67 	imes 10^{-27} \mathrm{Kg}$$ and its velocity is $$3 	imes 10^5 \mathrm{~ms}^{-1}$$.
$$ \lambda = \frac{h}{m_{	ext{neutron}} \cdot v_{	ext{neutron}}} $$
Plugging in the values:
$$ \lambda = \frac{6.626 	imes 10^{-34}}{1.67 	imes 10^{-27} \cdot 3 	imes 10^5} $$
Let's calculate this:
$$ \lambda = \frac{6.626 	imes 10^{-34}}{5.01 	imes 10^{-22}} $$
$$ \lambda \approx 1.322 	imes 10^{-12} \mathrm{~m} $$
Since the particle of mass $$2 \mathrm{mg}$$ has the same wavelength, we can use this value to find its velocity. The mass of the particle is:
$$ m_{	ext{particle}} = 2 \mathrm{mg} = 2 	imes 10^{-3} \mathrm{g} = 2 	imes 10^{-6} \mathrm{Kg} $$
Now, using the de Broglie formula again:
$$ v_{	ext{particle}} = \frac{h}{\lambda \cdot m_{	ext{particle}}} $$
Plugging in the values:
$$ v_{	ext{particle}} = \frac{6.626 	imes 10^{-34}}{1.322 	imes 10^{-12} \cdot 2 	imes 10^{-6}} $$
Let's calculate this:
$$ v_{	ext{particle}} = \frac{6.626 	imes 10^{-34}}{2.644 	imes 10^{-18}} $$
$$ v_{	ext{particle}} \approx 2.507 	imes 10^{-16} \mathrm{~ms}^{-1} $$
Thus, the velocity of the particle is approximately $$2.5 	imes 10^{-16} \mathrm{~ms}^{-1}$$, which corresponds to option A.
Option A: $$2.5 	imes 10^{-16} \mathrm{~ms}^{-1}$$

Answer

$$2.5 	imes 10^{-16} \mathrm{~ms}^{-1}$$

Answer

$$1.5 	imes 10^{-13} \mathrm{~ms}^{-1}$$

Answer

$$2.5 	imes 10^{-13} \mathrm{~ms}^{-1}$$

Answer

$$1.5 	imes 10^{-16} \mathrm{~ms}^{-1}$$

Dual Nature of Matter and Radiation

A particle of mass $2 \mathrm{mg} has the same wavelength as a neutron moving with a velocity of 3 \times 10^5 \mathrm{~ms}^{-1}. The velocity of the particle is (mass of neutron is 1.67 \times 10^{-27} \mathrm{Kg}$)

Choose the correct answer: