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Chapter 34: Problem 73

A photon's wavelength is equal to the Compton wavelength of a particle with mass \(m .\) Show that the photon's energy is equal to the particle's rest energy.

Short Answer

Expert verified
The photon's energy is equal to the particle's rest energy.

Step by step solution

01

Identify the relevant equations

The energy of a photon, \(E\), is related to its wavelength, \(\lambda\), by the Planck-Einstein relation: \(E = \frac{h c}{\lambda}\), where \(h\) is Planck's constant, \(c\) is the speed of light and \(\lambda\) is the wavelength of the photon. On the other hand, the Compton wavelength of a particle is given by: \(\lambda = \frac{h}{m c}\), where \(m\) is the mass of the particle. The rest energy of a particle is given by Einstein's equation: \(E_0 =m c^2 \).
02

Insert Compton wavelength into Planck-Einstein relation

By plugging the \(\lambda\) from the Compton wavelength formula into the Planck-Einstein relation, you get: \(E = \frac{h c}{h / m c}= m c^2 \).
03

Compare photon's energy with particle's rest energy

Comparing this last obtained expression, we see that \(E = E_0 = m c^2 \). Which means photon's energy is equal to the particle's rest energy.

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Most popular questions from this chapter

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

A 150 -pm X-ray photon Compton-scatters off an electron and emerges at \(135^{\circ}\) to its original direction. Find (a) the wavelength of the scattered photon and (b) the electron's kinetic energy.

What would the constant in Equation 34.2 be if black body radiance were defined for fixed intervals of frequency rather than wavelength? (Hint: Use \(\lambda=c / f\) to express the radiance as \(R(f, T),\) then differentiate to find the maximum, and solve the resulting relation numerically. Express your answer in a form like Equations \(34.2 \mathrm{a}\) and \(\mathrm{b} .\) )

Electrons in a photoelectric experiment emerge from an aluminum surface with maximum kinetic energy \(1.3 \mathrm{eV} .\) Find the wavelength of the illuminating radiation.

A microwave oven uses electromagnetic radiation at \(2.4 \mathrm{GHz}\). (a) What's the energy of each microwave photon? (b) At what rate does a \(900-\) W oven produce photons?
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