Trying to work out the speed of a relativistic electron

In summary, the person is trying to calculate the speed of a relativistic electron with energy 10^10 eV. They are unsure if this is the total energy or just the kinetic energy. They attempt to use various equations to find the speed, but keep getting incorrect answers. Another person suggests using the equation β=pc/E and approximating p using a binomial expansion since the rest energy is much smaller than the total energy. Another suggestion is to use the equation v/c = √(1-(E_0/E)^2) as an alternative.
  • #1
Elfrae
10
0

Homework Statement



I have a relativistic electron with energy 10^10 eV and I want to work out its speed. I'm not told if that's total energy or just kinetic energy - which is the sensible assumption to make?


Homework Equations



(E_tot)^2 = p^2*c^2 + m0^2*c^4

p = gamma*mv

gamma = sqrt(1/1-beta^2)

beta = v/c

The Attempt at a Solution



I tried rearranging the above equations to get an expression for v, but I keep getting stupid answers (v = c, v > c).

I have converted the energy into J and assumed that it is the total energy. Can anyone tell me what I'm doing wrong?

Thanks!
 
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  • #2
That's the total energy, which is much greater than the rest energy, so your answer should come out very close to c. I suggest you calculate the velocity using β=pc/E. You also might want to use a binomial expansion to approximate p since m0<<E.
 
  • #3
I think that it is better to try this (not to use p, almost the same as vela wrote)
[tex] E = {\gamma}mc^2 = \frac{mc^2}{\sqrt{1-v^2/c^2}}[/tex]
[tex]\frac{v^2}{c^2}=1- \frac{{E_0}^2}{E^2}[/tex]
[tex]\frac{v}{c}=\sqrt{1- \frac{{E_0}^2}{E^2}} \approx 1-\frac{1}{2} \frac{{E_0}^2}{E^2}[/tex]
 
Last edited:

1. How do you even begin to calculate the speed of a relativistic electron?

To calculate the speed of a relativistic electron, you would need to use the formula for relativistic kinetic energy: E = mc^2 / sqrt(1 - v^2 / c^2), where m is the mass of the electron, c is the speed of light, and v is the velocity of the electron. From this equation, you can solve for v to find the speed of the electron.

2. Can you explain what it means for an electron to be "relativistic"?

An electron is considered "relativistic" when its velocity is a significant fraction of the speed of light, which is approximately 3 x 10^8 meters per second. At these high speeds, the classical equations for motion and energy no longer apply, and we must use the principles of special relativity to accurately describe the behavior of the electron.

3. What units are typically used to measure the speed of a relativistic electron?

The speed of a relativistic electron is typically measured in units of meters per second (m/s). However, it can also be expressed in units of the speed of light, with c = 1 representing the speed of light. This is often used in equations to simplify calculations.

4. How does the speed of a relativistic electron affect its properties?

As an electron's speed approaches the speed of light, its mass increases and its length contracts in the direction of motion. This phenomenon, known as time dilation, also affects the electron's internal properties, such as its energy and momentum. Additionally, the electron may emit radiation due to its high speed.

5. Are there any practical applications of calculating the speed of a relativistic electron?

Yes, there are several practical applications of calculating the speed of a relativistic electron. For example, understanding the behavior of relativistic electrons is crucial in particle accelerators and nuclear reactors. It also plays a role in medical imaging techniques such as positron emission tomography (PET). Additionally, the principles of special relativity are fundamental in modern technologies such as GPS systems and satellite communications.

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