Question

Protons can be accelerated to speeds near that of light in particle accelerators. Estimate the wavelength (in nm) of such a proton moving at 2.91 × 108 m/s (mass of a proton = 1.673 × 10−27 kg). Enter your answer in scientific notation.

Answer 1

Answer:

The wavelength is $3.31\times 10^{-7}\ nm$.

Explanation:

Given:

Rest mass of the proton (m₀) = 1.673 × 10⁻²⁷ kg

Velocity of the proton (v) = 2.91 × 10⁸ m/s

Wavelength of the proton (λ) = ?

Here, the proton is accelerated nearly to speed of light. So, the mass is no more a constant value. It's value is calculated using the relativistic mass equation which is given as:

$m=\frac{m_0}{\sqrt{1-\frac{v^2}{c^2}}}$

Plug in the values given and solve for 'm'. This gives,

$m=\frac{1.673\times 10^{-27}\ kg}{\sqrt{1-\frac{(2.91\times 10^{8})^2}{(3\times 10^8)^2}}}\\\\m=6.882\times 10^{-27}\ kg$

In order to determine the wavelength of proton, we have to make use of the formula of de Broglie wavelength.

The de Broglie wavelength is given as:

$\lambda=\frac{h}{mv}\\Where,h\to\ Planck's\ constant=6.626\times 10^{-34}\ Js$

Now, plug in the values given and solve for λ. This gives,

$\lambda=\frac{6.626\times 10^{-34}\ Js}{6.882\times 10^{-27}\ kg\times 2.91\times 10^{8}\ m/s}\\\\\lambda=3.31\times 10^{-16}\ m$

Now, 1 m = 10⁹ nm

Therefore,

$3.31\times 10^{-16}\ m = 3.31\times 10^{-16}\times 10^9\ nm\\\\3.31\times 10^{-16}\ m= 3.31\times 10^{-7}\ nm$

Hence, the wavelength is 3.31 × 10⁻⁷ nm.