Question
Is $\text{p}=\frac{\text{E}}{\text{c}}$ valid for electrons?
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Answer
From Einstein's mass$-$energy equation, $\text{E}=\text{mc}^2$
$\Rightarrow\text{E}=\frac{\text{m}_0\text{c}^2}{\sqrt{1\frac{\text{v}^2}{\text{c}^2}}}$
Relativistic momentum,
$\text{p}=\text{mv}$
$\Rightarrow\text{p}=\frac{\text{m}_0\text{c}^2}{\sqrt{1\frac{\text{v}^2}{\text{c}^2}}}$
Combining the above equations, we get:
$\text{E}^2=\text{m}_0{}^2\text{c}^ 4+\text{p}^2\text{c}^2$
From the above equation, it is clear that for $\text{p}=\frac{\text{E}}{\text{c}}$ to be valid, the rest mass of the body should be zero.
As electrons do not have zero rest mass, this equation is not valid for electrons.
$\Rightarrow\text{E}=\frac{\text{m}_0\text{c}^2}{\sqrt{1\frac{\text{v}^2}{\text{c}^2}}}$
Relativistic momentum,
$\text{p}=\text{mv}$
$\Rightarrow\text{p}=\frac{\text{m}_0\text{c}^2}{\sqrt{1\frac{\text{v}^2}{\text{c}^2}}}$
Combining the above equations, we get:
$\text{E}^2=\text{m}_0{}^2\text{c}^ 4+\text{p}^2\text{c}^2$
From the above equation, it is clear that for $\text{p}=\frac{\text{E}}{\text{c}}$ to be valid, the rest mass of the body should be zero.
As electrons do not have zero rest mass, this equation is not valid for electrons.
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