Question

If kinetic energy of electron is $3.045\mathrm{MeV}$, then the total energy of electron is ______

Answer 1

Step 1. Given data:

It is given that the kinetic energy of electron is $3.045\mathrm{MeV}$.

We have to find the total energy.

Step 2. Find the rest mass energy.

The Einstein mass energy relation states that the energy produced by an object when it either combines or breaks itself is the product of the mass and the square of the speed of light.

So,

Loss in energy $\mathrm{E}=△{\mathrm{MC}}^2$

Here, $\mathrm{E}$ is the kinetic energy, $\mathrm{M}$ is the mass of body and $\mathrm{C}$ is the speed of light.

So,

$\mathrm{K}.\mathrm{E}.=(\mathrm{M}-{\mathrm{M}}_{\circ }){\mathrm{C}}^2$

$\mathrm{K}.\mathrm{E}.={\mathrm{MC}}^2-{\mathrm{M}}_0{\mathrm{C}}^2$

The Einstein equation is the most famous equation. It states that energy and mass are interchangeable and they are different forms of the same thing.
The rest mass energy of an electron is defined by ‘Einstein mass energy equivalence relation’.
According to relation of Einstein mass energy is
${\mathrm{E}}_0={\mathrm{M}}_0{\mathrm{C}}^2$
Where, $\mathrm{C}$ is velocity of light which is $3\times {10}^8\mathrm{m}/\mathrm{s}$, ${\mathrm{M}}_0$ is rest mass of an electron which is $9.11\times {10}^{-31}\mathrm{kg}$, and ${\mathrm{E}}_0$ is the rest mass energy
The Einstein equation is,
${\mathrm{E}}_0={\mathrm{M}}_0{\mathrm{C}}^2$
Substitute the value of rest mass of an electron and velocity of light in above equation.
$\Rightarrow$ ${\mathrm{E}}_0=9.11\times {10}^{-31}\times {\left(3\times {10}^8\right)}^2$

$\Rightarrow$ ${\mathrm{E}}_0=8.91\times {10}^{-14}\mathrm{J}$ ---------- $\left(1\right)$
Convert joule into $\mathrm{eV}$.
We know that,

$1\mathrm{eV}=1.6\times {10}^{-19}\mathrm{J}$

Therefore,

$1\mathrm{J}=\frac{1\mathrm{eV}}{1.6\times {10}^{-19}}$
Put above value in equation$\left(1\right)$.
$\Rightarrow$ ${\mathrm{E}}_0=\frac{8.91\times {10}^{-14}}{1.6\times {10}^{-19}}\mathrm{eV}$

$=0.511\times {10}^6\mathrm{eV}$

$=0.511\mathrm{MeV}$
So, the rest of mass energy is $0.511\mathrm{MeV}$.

Step 3. Calculate the total energy of electron.

Now, calculate the total energy of the electron.

Total energy $=$ Kinetic energy $+$ rest of mass energy

$=3.045+0.511$

$=3.556\mathrm{MeV}$

Hence, the total energy of the electron is $3.556\mathrm{MeV}$.