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Chapter 33: Problem 58

Find the speed of an electron with kinetic energy (a) 100 eV, (b) \(100 \mathrm{keV},\) (c) \(1 \mathrm{MeV},\) and (d) \(1 \mathrm{GeV} .\) Use suitable approximations where possible.

Short Answer

Expert verified
The speed of the electron for the different energy levels can be computed using the approaches described in the steps. In cases where the kinetic energy is too high and relativistic effects need to be considered, the energy-momentum relation equation gives the correct speed estimation.

Step by step solution

01

Conversion of energy units

Convert the energy given in electronvolts to Joules by using the conversion factor\( 1 eV = 1.6 × 10^{-19} \) Joules. Similarly, \( 1 keV = 1 × 10^3 eV \), \( 1 MeV = 1 × 10^6 eV \), and \( 1 GeV = 1 × 10^9 eV \)
02

Calculation of speed for each energy level

Solve for \( v \) in the kinetic energy equation \( KE = 0.5 * m * v^2 \) for each part of the problem, (a) to (d). For part (d), check if the kinetic energy is too high for the non-relativistic approximation to hold (i.e. if it is greater than the electron's rest energy or 0.511 MeV). If so, use the energy-momentum relation \( KE = \sqrt{(p*c)^2 + (m_0*c^2)^2} - m_0*c^2 \) where \( p \) is the relativistic momentum, \( c \) is the speed of light, and \( m_0 \) is the rest mass of the electron.
03

Solving for the speeds

Now, with the information available, solve for the values of \( v \) in parts (a) to (d).

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Most popular questions from this chapter

The rest energy of an electron is 511 keV. What's the approximate speed of an electron whose total energy is 1 GeV? (Note: No calculations needed!)

You've been named captain of NASA's first interstellar mission since the Voyager robotic spacecraft. You board your spaceship, accelerate quickly to \(0.8 c,\) and cruise at constant speed toward Proxima Centauri, the closest star to our Sun. Proxima Centauri is 4 light-years distant as measured in the two stars' common rest frame. On the way, you conduct various medical experiments to determine the effects of a long space voyage on the human body. Back on Earth, Mission Control judges that your shipboard clocks run slow. What do you judge about clocks at Mission Control? a. They run fast. b. They keep time at the same rate as your clocks. c. They run slow. d. You can't tell anything about their clocks.

Why was it necessary to repeat the Michelson-Morley experiment throughout the year?

Earth and Sun are 8.33 light minutes apart. Event A occurs on Earth at time \(t=0\) and event \(\mathrm{B}\) on the Sun at \(t=2.45 \mathrm{min}\), as measured in the Earth-Sun frame. Find the time order and time difference between \(A\) and \(B\) for observers (a) moving on a line from Earth to Sun at \(0.750 c,\) (b) moving on a line from Sun to Earth at \(0.750 c,\) and (c) moving on a line from Earth to Sun at \(0.294 c.\)

You've been named captain of NASA's first interstellar mission since the Voyager robotic spacecraft. You board your spaceship, accelerate quickly to \(0.8 c,\) and cruise at constant speed toward Proxima Centauri, the closest star to our Sun. Proxima Centauri is 4 light-years distant as measured in the two stars' common rest frame. On the way, you conduct various medical experiments to determine the effects of a long space voyage on the human body. Taking your pulse, you find a. it's significantly slower than when you're on Earth. b. it's the same as when you're on Earth. c. it's significantly faster than when you're on Earth.
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