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Chapter 38: Problem 26

Through what voltage must an electron be accelerated from rest in order to increase its energy to \(101 \%\) of its rest energy?

Short Answer

Expert verified
The electron must be accelerated through 5125 Volts.

Step by step solution

01

Understand the rest energy of an electron

The rest energy of an electron is given by the famous equation from Einstein's theory of relativity: \[ E_0 = mc^2 \] where - \( m \) is the mass of the electron (\( m = 9.11 \times 10^{-31} \text{ kg} \)) - \( c \) is the speed of light in a vacuum (\( c = 3 \times 10^8 \text{ m/s} \)).
02

Calculate the rest energy of the electron

Substitute the given values into the equation for the rest energy: \[ E_0 = (9.11 \times 10^{-31} \text{ kg}) \times (3 \times 10^8 \text{ m/s})^2 \]Calculate this to get: \[ E_0 = 8.2 \times 10^{-14} \text{ Joules} \].
03

Determine the total energy to which the electron must be accelerated

Since the electron must reach 101% of its rest energy, multiply the rest energy by 1.01: \[ E = 1.01 \times E_0 = 1.01 \times 8.2 \times 10^{-14} \text{ Joules} \]Calculate this to get: \[ E = 8.282 \times 10^{-14} \text{ Joules} \].
04

Calculate the kinetic energy gained by the electron

The kinetic energy gained by the electron is the difference between the total energy and the rest energy: \[ K.E. = E - E_0 = 8.282 \times 10^{-14} \text{ Joules} - 8.2 \times 10^{-14} \text{ Joules} \]This results in: \[ K.E. = 8.2 \times 10^{-16} \text{ Joules} \].
05

Relate the kinetic energy to the accelerating voltage

The kinetic energy (K.E.) of an electron can be related to the accelerating voltage (V) using the equation: \[ K.E. = eV \] where \( e \) is the elementary charge (\( e = 1.6 \times 10^{-19} \text{ C} \)).
06

Solve for the accelerating voltage

Rearrange the equation for voltage and solve: \[ V = \frac{K.E.}{e} = \frac{8.2 \times 10^{-16} \text{ Joules}}{1.6 \times 10^{-19} \text{ C}} \]Calculate this to get: \[ V = 5125 \text{ Volts} \].

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Einstein's theory of relativity
Einstein's theory of relativity is a cornerstone of modern physics. It includes the famous equation \( E_0 = mc^2 \), where energy (\(E_0\)) is equal to mass (\(m\)) times the speed of light (\(c\)) squared. This equation shows that mass and energy are interchangeable. For an electron, this means it has a rest energy just due to its mass. The theory of relativity helps us understand how particles behave at high speeds and provides the underlying principle for calculating the rest energy in the given problem.
Kinetic energy
Kinetic energy is the energy a particle has due to its motion. For electrons accelerated by a voltage, their kinetic energy increases as they move faster. It's calculated using the difference between the total energy and rest energy. In the problem, the total energy was 101% of the rest energy. The kinetic energy gained by the electron is this extra 1%. The formula given is: \ K.E. = E - E_0 \ where \(E\) is the total energy and \(E_0\) is the rest energy.
Elementary charge
The elementary charge is the basic unit of electric charge in physics, symbolized as \(e\). Its value is approximately \(1.6 \times 10^{-19} \text{C}\). An electron carries a negative elementary charge. When electrons are accelerated by a voltage, their energy change is related to this charge. In the formula \(K.E. = eV\), \(e\) is the elementary charge and \(V\) is the accelerating voltage. This relation helps us to find how much voltage is needed to give the electron a specific amount of kinetic energy.
Rest energy
Rest energy is the energy a particle has due to its mass alone, without any motion. For an electron, the rest energy is calculated using Einstein鈥檚 equation, \(E_0 = mc^2\). In the original problem, the rest energy was calculated by substituting the electron's mass and the speed of light. Rest energy is a crucial concept because it's the baseline energy that doesn't change regardless of the electron's movement. The electron's total energy consists of its rest energy plus any additional kinetic energy.
Accelerating voltage
Accelerating voltage is the electrical potential difference that accelerates charged particles like electrons. The energy gained by the electron is directly related to this voltage, as shown by \(K.E. = eV\), where \(K.E.\) is the kinetic energy, \(e\) is the elementary charge, and \(V\) is the voltage. In the given exercise, solving for \(V\) involved rearranging the formula to \(V = \frac{K.E.}{e}\). By plugging in the kinetic energy and elementary charge, we find the required accelerating voltage. This voltage tells us how much potential difference is needed to accelerate the electron to 101% of its rest energy.

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Most popular questions from this chapter

You wish to make \(a\) round trip from Earth in a spaceship, traveling at constant speed in a straight line for 6 months on your watch and then returning at the same constant speed. You wish, further, to find Earth to be 1000 years older on your return. (a) What is the value of your constant speed with respect to Earth? (b) How much do you age during the trip? (c) Does it matter whether or not you travel in a straight line? For example, could you travel in a huge circle that loops back to Earth?

A meter stick lies at rest in the rocket frame and makes an angle \(\phi^{\prime}\) with the \(x^{\prime}\) axis as measured by the rocket observer. The laboratory observer measures the \(x\) - and \(y\) -components of the meter stick as it streaks past. From these components the laboratory observer computes the angle \(\phi\) that the stick makes with his \(x\) axis. (a) Find an expression for the angle \(\phi\) in terms of the angle \(\phi^{\prime}\) and the relative speed \(v^{\text {rel }}\) between rocket and laboratory frames. (b) What is the length of the "meter" stick measured by the laboratory observer? (c) Optional: Why is your expression in part (a) different from equations derived in Problems 34 and \(35 ?\)

An unpowered rocket moves past you in the positive \(x\) direction at speed \(v^{\text {rel }}=0.9 c\). This rocket fires a bullet out the back that you measure to be moving at speed \(v_{\text {bullet }}=0.3 c\) in the positive \(x\) direction. With what speed relative to the rocket did the rocket observer fire the bullet out the back of her ship?

The Giant Shower Array detector, spread over 100 square kilometers in Japan, detects pulses of particles from cosmic rays. Each detected pulse is assumed to originate in a single high-energy cosmic proton that strikes the top of the Earth's atmosphere. The highest energy of a single cosmic ray proton inferred from the data is \(10^{20} \mathrm{eV}\). How long would it take that proton to cross our galaxy \(\left(10^{5}\right.\) light-years in diameter) as recorded on the wristwatch of the proton? (The answer is not zero!)

You are taking a trip from the solar system to our nearest visible neighbor, Alpha Centauri, approximately 4 light-years distant. At launch you experienced a period of acceleration that increased your speed with respect to Earth from zero to nearly half the speed of light. Now your spaceship is coasting in unpowered flight. Compare and contrast the observations you make now with those you made before the rocket took off from the Earth's surface. Be as specific and detailed as possible. Distinguish between observations made inside the cabin with the windows covered and those made looking out of uncovered windows at the front, side, and back of the cabin.
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