Question

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

Answer 1

Answer:

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):

$1eV = 1.60x10^{-19} J$

$8.19x10^{-14} J . \frac{1 eV}{1.60x10^{-19} J}$ ⇒ $511.875 eV$

$E_{R} = 511.875 eV$

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:

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

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

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

$E = \frac{511.875 eV}{sqrt{1-\frac{0.1225c^{2}}{c^{2}}}}$

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

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

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

$E = 546.875 eV$

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).