Calculating the energy of an electron

[Kahsi]

Let's say that an electron is having the speed of X m/s.

So the energy is sometimes calculated like this:

(1)E= \frac{mv^2}{2}

and sometimes like this

(2)E = mc^2 + \frac{mv^2}{2}

When should I use 1 and when should I use 2?

Thank you for any help.

 

[SpaceTiger]

When we talk about energy, we usually talk about conserving it in a system. This means that the total energy at one time is equal to the total energy at another time. A particle's total energy is given by:

E=\gamma mc^2

where \gamma is the Lorentz factor:

\gamma=\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}

In the limit of small v, the first equation goes to your second one:

E=mc^2+\frac{1}{2}mv^2

Now, if we have some hypothetical set of particles and we want to conserve total energy, we can write:

\sum_{i=1}^{N}E_{i,1}=E_x+\sum_{i=1}^{N}E_{i,2}

where E_x includes energy in radiation, fields, etc. This can be expanded to

\sum_{i=1}^{N}(m_{i,1}c^2+\frac{1}{2}m_{i,1}v_{i,1}^2)=E_x+\sum_{i=1}^{N}(m_{i,2}c^2+\frac{1}{2}m_{i,2}v_{i,2}^2)

If, for all particles:

m_{i,1}=m_{i,2}

then the equation simplifies to

\sum_{i=1}^{N}\frac{1}{2}m_{i,1}v_{i,1}^2=E_x+\sum_{i=1}^{N}\frac{1}{2}m_{i,2}v_{i,2}^2

because the mc² terms are the same on both sides.

In words, as long as the components of your system are preserving their rest mass, then the mc² term is not needed. In practice, this corresponds to cases in which the velocities are well below that of light. This is why classical theory works in the low-velocity limit.