E equal to mc squared derivation

Here is how we derive mass energy equivalence

Force is defined as rate of change of momentum i.e.,

F = \frac{d}{dt}(m\nu) ….. (1)

According to the theory of relativity, both mass and velocity are variable. Therefore

F = \frac{d}{dt}(m\nu) = m \frac{d\nu}{dt} + \nu \frac{dm}{dt} ….. (2)

Let the force F displace the body through a distance dx. Then, the increase in the kinetic energy (dE_k) of the body is equal to the work done (Fdx) . Hence,

dE_k = F dx = m \frac{d\nu}{dt}dx + \nu \frac{dm}{dt}dx

or, dE_k = m\nu d\nu + \nu^2 dm ….. (3)

According to the law of variation of mass of velocity

m = \frac{m_0}{\sqrt{1-\frac{\nu^2}{c^2}}} ….. (4)

Squaring both sides, m^2 = \frac{m_0^2}{1-\frac{\nu^2}{c^2}}

or, m^2c^2 = m_0^2c^2 + m^2\nu^2

Differentiating,

c^2 2m dm = m^2 2\nu d\nu + \nu^2 2m dm

or, c^2 dm = m\nu d\nu + \nu^2 dm ….. (5)

From equation (3) and (5)

dE_k = c^2 dm ….. (6)

Thus, a change in K.E. dE_k is directly proportional to a change in mass  dm. When a body is at rest, its velocity is zero, (K.E. = 0) and m = m_0 . When its velocity is \nu , its mass becomes m. Therefore, integrating equation (6)

E_k = \int_0^{E_k}dE_k = c^2 \int_{m_0}^{m} dm = c^2(m-m_0)

Therefore, E_k = mc^2 - m_0c^2 ….. (7)

This is the relativistic formula for K.E. When the body is at rest, the internal energy stored in the body is m_0c^2 . m_0c^2 is called the rest mass energy. The total energy (E) of the body is the sum of K.E. (E_k) and rest mass energy m_0c^2

So, E = E_k + m_0c^2 = (mc^2 - m_0c^2) + m_0c^2 = mc^2

This is Einstein’s mass energy relation.

This relation states a universal equivalence between mass and energy. It means that mass may appear as energy and energy as mass.

The relationship E = mc^2 between energy and mass forms the basis of understanding nuclear reactions such as fission and fusion. These reactions take place in nuclear bombs and reactors. When a uranium nucleus is split up, the decrease in its total rest mass appears in the form of an equivalent amount of K.E. of its fragments.

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