What is the difference in energy between a beta particle at rest and one traveling 0.35c?

They have a difference in energy of 35 eV.

Explanation:

The energy at rest of a particle is given by:

  (1)

Where is the mass of the particle at rest and c is the speed of light.

Beta particles are high energy and high velocity electrons or positrons ejected from the nucleus of an atom as a consequence of a radioactive decay. Either if the beta particle is an electron¹ or a positron² it will have the same mass.

Hence, the mass of the beta particle at rest in equation (1) will be equal to the mass of an electron:

Replacing the values of and c in equation (1) it is gotten:

But , therefore:

It is better to express the rest energy in electronvolts (eV):

⇒

So the energy of the beta particle at rest is 511.875 eV.

Case for the one traveling at 0.35c:

Since it is traveling at 35% of the speed of light it is necessary to express equation (1) in a relativistic way, that can be done adding the Lorentz factor to it:

  (2)

Where v is the velocity of the particle (for this case 0.35c).

The difference in energy between the two particles can be determined using the relativistic form of the kinetic energy:

 (3)

Where E is the energy of the particle traveling at 0.35c and is the energy of the beta particle at rest.

They have a difference in energy of 35 eV.

Key terms:

¹Electron: Fundamental particle of negative electric charge.

²Positron: Is an electron with positive electric charge (similar to an electron in all its properties except in electric charge and magnetic moment).

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