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Chapter 34: Problem 31

A proton and electron have the same de Broglie wavelength. How do their speeds compare, assuming \(v \ll c\) for both?

Short Answer

Expert verified
The speed of the proton is \( \frac{m_e}{m_p} \) times the speed of the electron where \(m_e\) and \(m_p\) are the electron and proton masses respectively.

Step by step solution

01

Write the de Broglie wavelength formula for both particles

Given the de Broglie wavelength formula \( \lambda = \frac{h}{mv} \), we write this formula for both the proton and the electron. For the proton \( \lambda_p = \frac{h}{m_pv_p} \) and for the electron \( \lambda_e = \frac{h}{m_ev_e} \). Here \(m_p\) and \(v_p\) are the mass and velocity of the proton respectively, and \(m_e\) and \(v_e\) are the mass and velocity of the electron respectively.
02

Set the two wavelengths equal

Since we are given that the de Broglie wavelengths of the proton and the electron are the same, we can set these two expressions equal to each other. This gives us \(\frac{h}{m_pv_p} = \frac{h}{m_ev_e}\).
03

Simplify the equation

From the equation in the last step, we can simplify to give \(v_p = \frac{m_e}{m_p}v_e\). This equation tells us the relationship between the velocities of the proton and electron.

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