Question

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

Answer 1

They have a difference in energy of 35 eV.

Explanation:

The energy at rest of a particle is given by:

`E_(R) = m_(0)c^2` (1)

Where
`m_(0)` 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:

`m_(e) = 9.1095x10^(-31) Kg`

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

`E_(R) = (9.1095x10^(-31) Kg)(3.00x10^(8) m/s)^(2)`

`E_(R) = 8.19x10^(-14) Kg.m^(2)/s^(2)`

But
`1 J = Kg.m^(2)/s^(2)`, therefore:

`E_(R) = 8.19x10^(-14) J`

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

`1eV = 1.60x10^(-19) J`

`8.19x10^(-14) J . (1 eV)/(1.60x10^(-19) J)` ⇒
`511.875 eV`

`E_(R) = 511.875 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:

`E = \frac{m_(0)c^(2)}{sqrt{1-(v^(2))/(c^(2))}}` (2)

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

`E = \frac{511.875 eV}{sqrt{1-((0.35c)^(2))/(c^(2))}}`

`E = \frac{511.875 eV}{sqrt{1-(0.1225c^(2))/(c^(2))}}`

`E = \frac{511.875 eV}{sqrt{1-0.1225}}`

`E = \over{511.875 eV}{sqrt{0.8775}}`

`E = \over{511.875 eV}{0.936}`

`E = 546.875 eV`

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

`K = E – E_(R)` (3)

Where E is the energy of the particle traveling at 0.35c and
`E_(R)` is the energy of the beta particle at rest.

`K = 546.875 eV – 511.875 eV`

`K = 35 eV`

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).