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Chapter 26: Problem 50

The rest energy of an electron is \(0.511 \mathrm{MeV} .\) What momentum (in MeV/c) must an electron have in order that its total energy be 3.00 times its rest energy?

Short Answer

Expert verified
Answer: The momentum of the electron is approximately 1.44 MeV/c.

Step by step solution

01

Write down the relativistic energy-momentum relationship

The relativistic energy-momentum relationship is given by: E² = (mc²)² + (pc)² where E is the total energy, m is the mass, c is the speed of light, and p is the momentum.
02

Express the total energy in terms of rest energy

We are given that the total energy is 3 times the rest energy, so we can write: E = 3mc²
03

Substitute the total energy expression into the energy-momentum relationship

Now, substitute the total energy expression into the energy-momentum relationship: (3mc²)² = (mc²)² + (pc)²
04

Solve the equation for momentum (p)

To find the momentum, we can rearrange the equation and solve for p: (3mc²)² - (mc²)² = (pc)² (8m²c⁴) = (pc)² p = \sqrt{8m²c²}
05

Substitute the values and calculate the momentum

Now we can substitute the given rest energy value and the speed of light to find the momentum in MeV/c: Rest energy (E₀) = 0.511 MeV c = 1 (when working in units of MeV/c) E₀ = mc² m = \frac{E₀}{c²} = \frac{0.511}{(1)²} = 0.511 Now, substitute these values into the momentum equation: p = \sqrt{8(0.511)²} p ≈ 1.44 \, MeV/c The momentum of the electron when its total energy is 3 times its rest energy is approximately 1.44 MeV/c.

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Most popular questions from this chapter

A spaceship moves at a constant velocity of \(0.40 c\) relative to an Earth observer. The pilot of the spaceship is holding a rod, which he measures to be \(1.0 \mathrm{m}\) long. (a) The rod is held perpendicular to the direction of motion of the spaceship. How long is the rod according to the Earth observer? (b) After the pilot rotates the rod and holds it parallel to the direction of motion of the spaceship, how long is it according to the Earth observer?
Derivation of the Doppler formula for light. A source and receiver of EM waves move relative to one another at velocity \(v .\) Let \(v\) be positive if the receiver and source are moving apart from one another. The source emits an EM wave at frequency \(f_{\mathrm{s}}\) (in the source frame). The time between wavefronts as measured by the source is \(T_{\mathrm{s}}=1 / f_{\mathrm{s}}\) (a) In the receiver's frame, how much time elapses between the emission of wavefronts by the source? Call this \(T_{\mathrm{T}}^{\prime} .\) (b) \(T_{\mathrm{T}}^{\prime}\) is not the time that the receiver measures between the arrival of successive wavefronts because the wavefronts travel different distances. Say that, according to the receiver, one wavefront is emitted at \(t=0\) and the next at \(t=T_{\mathrm{o}}^{\prime} .\) When the first wavefront is emitted, the distance between source and receiver is \(d_{\mathrm{r}}\) When the second wavefront is emitted, the distance between source and receiver is \(d_{\mathrm{r}}+v T_{\mathrm{r}}^{\prime} .\) Each wavefront travels at speed \(c .\) Calculate the time \(T_{\mathrm{r}}\) between the arrival of these two wavefronts as measured by the receiver. (c) The frequency detected by the receiver is \(f_{\mathrm{r}}=1 / T_{\mathrm{r}} .\) Show that \(f_{\mathrm{r}}\) is given by $$f_{\mathrm{r}}=f_{\mathrm{s}} \sqrt{\frac{1-v / \mathrm{c}}{1+v / \mathrm{c}}}$$
An unstable particle called the pion has a mean lifetime of 25 ns in its own rest frame. A beam of pions travels through the laboratory at a speed of $0.60 c .$ (a) What is the mean lifetime of the pions as measured in the laboratory frame? (b) How far does a pion travel (as measured by laboratory observers) during this time?
A particle decays in flight into two pions, each having a rest energy of 140.0 MeV. The pions travel at right angles to each other with equal speeds of \(0.900 c .\) Find (a) the momentum magnitude of the original particle, (b) its kinetic energy, and (c) its mass in units of \(\mathrm{MeV} / c^{2}\).
When an electron travels at \(0.60 c,\) what is its total energy in mega- electron-volts?
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