Question

An electron's rest mass is \(0.511 \mathrm{MeV} / \mathrm{c}^{2}\) a) How fast must an electron be moving if its energy is to be 10 times its rest energy? b) What is the momentum of the electron at this speed?

Short answer

Question: Determine the speed and momentum of an electron when its total energy is 10 times its rest energy. The rest mass of the electron is 0.511 MeV/c². Answer: The speed of the electron is approximately 0.968c, and its momentum is approximately 2.955 MeV/c.

Step-by-step solution

Find the total energy of the electron when its energy is 10 times its rest energy

Since the total energy is 10 times its rest energy, \(E = 10 * E_0\)

Write down the rest energy formula

The rest energy of an electron is given by: \(E_0 = m_0 c^2\) Where \(m_0\) is the rest mass of the electron and \(c\) is the speed of light in a vacuum.

Calculate the rest energy

Given the rest mass of the electron, \(m_0 = 0.511 \mathrm{MeV} / \mathrm{c}^{2}\), we can find the rest energy as, \(E_0 = 0.511 \mathrm{MeV}\)

Calculate the total energy

Now that we have the rest energy, find the total energy by multiplying it by 10: \(E = 10 * E_0 = 10 * 0.511 \mathrm{MeV} = 5.11 \mathrm{MeV}\)

Determine the speed of the electron

Using the total energy formula, rearrange it to solve for the speed of the electron \(v\): \(v = c\sqrt{1 - \frac{m_0^2c^4}{E^2}}\) Substitute the values of \(m_0 = 0.511 \mathrm{MeV} / \mathrm{c}^{2}\) and \(E = 5.11 \mathrm{MeV}\): \(v = c\sqrt{1 - \frac{(0.511)^2}{(5.11)^2}}\) Solve for \(v\): \(v \approx 0.968c\) The speed of the electron when its energy is 10 times its rest energy is approximately \(0.968c\), where \(c\) is the speed of light.

Find the momentum of the electron

Use the momentum formula to find the momentum \(p\) of the moving electron: \(p = m_0 v \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}}\) Substitute the values of \(m_0 = 0.511 \mathrm{MeV} / \mathrm{c}^{2}\) and \(v = 0.968c\) and solve for \(p\): \(p = (0.511)(0.968) \frac{1}{\sqrt{1 - (0.968)^2}} \approx 2.955\, \mathrm{MeV/c}\) The momentum of the electron at this speed is approximately \(2.955\, \mathrm{MeV/c}\).

Key concepts

Relativistic Energy

Relativistic energy refers to the total energy of an object when accounting for the effects of Einstein's theory of relativity. Specifically, as an object's velocity approaches the speed of light, its kinetic energy increases to a point where classical mechanics no longer applies, and relativistic effects must be considered.

In classical physics, the kinetic energy (\(E_k\)) of an object is given by the equation \(E_k = \frac{1}{2}mv^2\), where \(m\) is the mass and \(v\) is the velocity. However, at speeds close to the speed of light, this equation doesn't provide accurate results. Instead, we use the relativistic energy formula, \(E = \sqrt{(p^2c^2) + (m_0^2c^4)}\), where \(E\) is the total energy, \(p\) is the relativistic momentum, and \(m_0\) is the object's rest mass.

Momentum of an Electron

The momentum of an electron, just like for any other particle, indicates how much 'oomph' it carries as it moves. In classical physics, momentum (\(p\)) is a product of mass and velocity (\(p = mv\)). However, an electron moving at high velocities exhibits relativistic effects. The relativistic momentum is modified to account for these effects and is described by the formula \(p = \frac{m_0v}{\sqrt{1 - \frac{v^2}{c^2}}}\).

This formula shows how momentum increases as the electron's speed (\(v\)) gets closer to the speed of light (\(c\))—to such an extent that it diverges as \(v\) approaches \(c\). Because electrons have such a small rest mass, they can reach high speeds where these relativistic effects become very apparent, necessitating corrections for predictions to match actual observations.

Speed of Light

The speed of light (\(c\)) in a vacuum is one of the most fundamental constants in physics. It is exactly 299,792,458 meters per second and is of paramount importance in the theory of relativity. The steadfast nature of \(c\) implies that no matter how fast you're moving when you measure the speed of light, you'll still get the same number.

This is a tenet of Einstein's theory of relativity and leads to fascinating consequences, such as time dilation and length contraction. As objects move faster, getting closer to \(c\), the effects predicted by relativity become more significant—an important consideration when discussing high-speed particles like electrons.

Relativistic Momentum

Relativistic momentum is an extension of the conventional concept of momentum to accommodate the high-speed regime close to the speed of light. While regular momentum is simply the product of mass and velocity, relativistic momentum includes a factor that accounts for the relativistic effects--the Lorentz factor, \(\gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}}\).

The relativistic momentum of a particle is given by \(p = \gamma m_0 v\). As \(v\) approaches \(c\), \(\gamma\) increases dramatically and so does \(p\), reflecting the greater resistance to acceleration. Relativistic momentum is key for understanding how particles move in high-energy environments, such as in particle accelerators or cosmic phenomena.