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Chapter 39: Problem 14

If the energy of the virtual photon mediating an electronproton scattering, \(e^{-}+p \rightarrow e^{-}+p\), is given by \(E\), what is the range of this electromagnetic interaction in terms of \(E ?\)

Short Answer

Expert verified
Answer: The range of electron-proton scattering mediated by a virtual photon with energy E is given by the following expression: $$R = \frac{c\hbar}{E}$$, where \(c\) is the speed of light and \(\hbar\) is the reduced Planck constant.

Step by step solution

01

Apply Uncertainty Principle in Energy and Time

According to the uncertainty principle in energy and time, we have \(\Delta E \Delta t \approx \hbar\), where \(\Delta E\) is the energy of the virtual photon and \(\Delta t\) is the time duration of the interaction. In this case, we are given \(E\), so let's write the equation in terms of \(E\) and \(\Delta t\): $$E \Delta t \approx \hbar$$
02

Find Time Duration of Interaction

To find the time duration of the interaction, we can rearrange the equation from step 1 and solve for \(\Delta t\): $$\Delta t \approx \frac{\hbar}{E}$$
03

Calculate Spatial Range

Now, we can use the speed of light \(c\) to convert the time duration into a spatial range. We can write the spatial range \(R\) as: $$R = c \Delta t$$ Substitute the expression for \(\Delta t\) from step 2 into this equation: $$R = c \cdot \frac{\hbar}{E}$$
04

Final Answer

The range of the electromagnetic interaction in terms of the energy \(E\) is given by the following expression: $$R = \frac{c\hbar}{E}$$

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Most popular questions from this chapter

The de Broglie wavelength, \(\lambda\), of a 5-MeV alpha particle is \(6.4 \mathrm{fm}\), as shown in this chapter, and the closest distance, \(r_{\text {min }}\), to the gold nucleus this alpha particle can get is \(45.5 \mathrm{fm}\) (calculated in Example 39.1). Based on the fact that \(\lambda \ll r_{\text {min }}\), one can conclude that, for this Rutherford scattering experiment, it is adequate to treat the alpha particle as a a) particle. b) wave.

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