Calculating the energy of an electron

• Kahsi
In summary, when dealing with energy conservation in a system, we can use the simplified equation E=mv^2/2 if the particles in the system are preserving their rest mass. Otherwise, we need to use the more complex equation E=mc^2+mv^2/2. This is why classical theory works for low velocity cases, where the particles have a negligible change in their rest mass.

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.

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 Ex 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.

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.

What is the formula for calculating the energy of an electron?

The formula for calculating the energy of an electron is E = -13.6/n^2, where n is the number of the energy level.

What is the unit of measurement for the energy of an electron?

The unit of measurement for the energy of an electron is electron volts (eV).

How is the energy of an electron related to its position in an atom?

The energy of an electron is related to its position in an atom by its energy level. Electrons with higher energy levels are farther from the nucleus and have more energy.

What is the significance of calculating the energy of an electron?

Calculating the energy of an electron is important because it helps us understand the behavior and properties of atoms and molecules. It also allows us to predict the chemical reactivity of elements and compounds.

Can the energy of an electron be negative?

Yes, the energy of an electron can be negative. This simply means that the electron has less energy than a free electron at rest, which is defined as having an energy of 0 eV.