Speed of Free Electron Moving in Space

In summary, the total energy of a free electron moving through empty space is given by the formula E=1.5mc^2, where m is the mass of the electron and c is the speed of light. The proper relativistic kinetic energy equation is E_k = (\gamma - 1)mc^2. The potential energy of the electron is zero since it is free. The total energy of the electron is the sum of its kinetic energy and energy from the mass-energy equivalence.
  • #1
quickclick330
83
0
The total energy of a free electron moving through empty space is E=1.5mc^2, where m is the mass of the electron and c is the speed of light. What is this electron’s speed?

1. c (the speed of light)
2. 0.7453c
3. 0.8660c
4. 0.9428c
5. 0.9950c
6. 0.9999c


I thought since that Etotal = 1.5mc^2 then...

Etotal = KE = .5mv^2

1.5mc^2 = .5mv^2 and solve for v (masses would cancel which is what I was looking for)

What is wrong about this approach?
 
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  • #2
Using KE = .5mv^2 is only good when v << c.

Hint:
[tex]E = \gamma m c^2[/tex]
 
  • #3
I believe that the proper relativistic kinetic energy equation would be

[tex]E_k = (\gamma - 1)mc^2[/tex]

Or did you make that mistake on purpose, Doc Al?
 
  • #4
oh shoot...i forgot about that. Do I need to use the non-approximated version of KE? (I can never remember it.. but I'm looking it up right now)
 
  • #5
E is total energy, of course. :wink:

(There's no need to find kinetic energy.)
 
  • #6
okay
 
  • #7
so mc^2 would cancel out on both sides and I would be solving 1.5 = gamma right?
 
  • #8
quickclick330 said:
so mc^2 would cancel out on both sides and I would be solving 1.5 = gamma right?
Yep. That's all you need to do.
 
  • #9
ahh...got it, thank you :-)
 
  • #10
Oh, I get it... the total energy of the free electron is the sum of its kinetic energy and the energy from the mass-energy equivalence? The potential energy of the electron is zero, since it's free, right?

[tex]E_{total}= (\gamma - 1)mc^2 + mc^2[/tex]
[tex]\ \ \ = \gamma mc^2[/tex]

Is this so?
 
  • #11
kudoushinichi88 said:
Oh, I get it... the total energy of the free electron is the sum of its kinetic energy and the energy from the mass-energy equivalence? The potential energy of the electron is zero, since it's free, right?

[tex]E_{total}= (\gamma - 1)mc^2 + mc^2[/tex]
[tex]\ \ \ = \gamma mc^2[/tex]

Is this so?
Yep.
 
  • #12
Ah, thanks...
 

What is the speed of a free electron in space?

The speed of a free electron in space can vary, but on average, it is around 1,500 kilometers per second. However, this speed can change depending on factors such as the electron's energy and the presence of electromagnetic fields.

How does the speed of a free electron in space compare to the speed of light?

The speed of a free electron in space is much slower than the speed of light. In fact, the speed of light is approximately 299,792 kilometers per second, which is about 200 times faster than the average speed of a free electron in space.

What factors can affect the speed of a free electron in space?

The speed of a free electron in space can be influenced by several factors, including the electron's energy, the strength and direction of electromagnetic fields, and collisions with other particles. Additionally, the type of material the electron is traveling through can also impact its speed.

Can the speed of a free electron in space exceed the speed of light?

No, according to Einstein's theory of relativity, nothing can travel faster than the speed of light. As such, the speed of a free electron in space cannot exceed the speed of light.

Why is the speed of a free electron in space important for scientific research?

The speed of a free electron in space is important for understanding various physical phenomena, such as the behavior of particles in magnetic fields and the formation of celestial objects. It also plays a crucial role in technologies such as particle accelerators and space propulsion systems.

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