Chapter 6: Problem 116

Find the wavelength of a proton that is moving at \(1.00 \%\) of the speed of light (when \(\beta=0.01\) ).

Short Answer

Expert verified

The wavelength of a proton moving at 1.00% of the speed of light is approximately \(1.33 \times 10^{-14} m\).

Step by step solution

Find the speed of the proton

Given the proton is moving at 1.00% of the speed of light, we can find its speed (v) by multiplying the speed of light (c) by 0.01: \(v = 0.01c\)

Determine the rest mass of a proton

The rest mass of a proton (m₀) is approximately \(1.67\times10^{-27}\: kg \).

Calculate the relativistic mass of the proton

To calculate the relativistic mass (m) of the proton, we use the relativistic mass formula: \(m = \frac{m_0}{\sqrt{1 - \beta^2}} \) Here, β is the ratio of the proton's speed to the speed of light (β = v/c). Given β = 0.01, we can calculate the relativistic mass: \(m = \frac{1.67\times10^{-27} kg}{\sqrt{1 - (0.01)^2}} \)

Solve for the relativistic mass

Evaluating the expression above, we find the relativistic mass of the proton: \(m = 1.67\times10^{-27} kg\cdot\frac{\sqrt{1 - (0.01)^2}}{1 - (0.01)^2}=1.67\times10^{-27} kg\cdot\frac{1}{\sqrt{1 - (0.01)^2}}=1.67\times10^{-27} kg\cdot \frac{1}{\sqrt{1 - 0.0001}}\) \(m \approx 1.67\times10^{-27} kg\cdot\frac{1}{0.99995} \approx 1.67\times10^{-27}\: kg \)

Use the de Broglie wavelength formula

Now that we have the relativistic mass of the proton, we can use the de Broglie wavelength formula to find the wavelength (λ) of the proton: \(\lambda = \frac{h}{mv}\) Here, h is Planck's constant, approximately \(6.63\times10^{-34} Js\).

Calculate the wavelength

Substitute the values of h, m, and v in the formula to find the wavelength of the proton: \(\lambda = \frac{6.63\times10^{-34} Js}{(1.67\times10^{-27} kg)(0.01c)} \) Since c is the speed of light, approximately \(3\times10^8 m/s\), we have: \(\lambda = \frac{6.63\times10^{-34} Js}{(1.67\times10^{-27} kg)(0.01\times3\times10^8 m/s)}\)

Solve for the wavelength

Evaluating the expression above, we find the wavelength of the proton: \(\lambda = \frac{6.63\times10^{-34} Js}{(1.67\times10^{-27} kg)(3\times10^6 m/s)} = 0.0133 \times 10^{-12} m\) The wavelength of the proton moving at 1.00% of the speed of light is approximately \(1.33 \times 10^{-14} m\).

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Find the wavelength of a proton that is moving at \(1.00 \%\) of the speed of light (when \(\beta=0.01\) ).

Short Answer

Step by step solution

Find the speed of the proton

Determine the rest mass of a proton

Calculate the relativistic mass of the proton

Solve for the relativistic mass

Use the de Broglie wavelength formula

Calculate the wavelength

Solve for the wavelength

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