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.

 

[sir-pinski]

The equation (1) is known as the classical kinetic energy formula and is used to calculate the kinetic energy of an object with mass m and velocity v. This equation is applicable for objects moving at speeds much slower than the speed of light.

On the other hand, equation (2) is known as the relativistic energy formula and takes into account the effects of special relativity at high speeds. This equation is applicable for objects moving at speeds close to the speed of light, such as electrons.

Therefore, if you are dealing with an electron moving at a speed close to the speed of light, it is more accurate to use equation (2) to calculate its energy. However, if the electron is moving at a much slower speed, equation (1) will provide a good approximation.

It is important to note that both equations are valid and can be used depending on the context of the problem. It is always best to use the appropriate equation based on the speed of the electron in order to obtain a more accurate result.