Protons can be accelerated to speeds near that of light in particle accelerators. Estimate the wavelength (in nm) of such a proton moving at

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.

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Linh Đan 5 months 2021-08-17T06:54:50+00:00 1 Answers 12 views 0

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    2021-08-17T06:56:26+00:00

    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.

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