Einstein's mass-energy equivalence

Einstein's mass-energy equivalence

Consider a body of rest mass m_o. A force F is acting on it in X-direction. According to Newton's second law of motion, force is defined as the rate of change of momentum.

i.e. F = d/dt(mv)   ………..(1)

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

F = m(dv/dt )  + v(dm/dt)  ………….(2)

If a body is displaced through a distance dx due to the force F then, the increase in kinetic energy dEk of the body is

dE_k = Fdx

= ( m (dv/dt) + v(dm/dt ) ) dx

dE_k = mv dv + v²dm   ………. (3)

m=m₀/root(1-(v²/c²))

m² = m₀²c² / c²-v²

Differentiating we get

c²dm = mv dv + v²dm           ...(4)

Comparing equations (3) and (4) we get

dE_k = c²dm      ……………(5)

Thus the change in kinetic energy dE_k is directly proportional to the change in mass dm

When a body is at rest, its velocity is zero and m = m_o. When its velocity is v its mass becomes m. Therefore integrating equation (5)

∫₀^E dE_k = c² /  ∫_m0^m dm

E_k = c² (m-m_o) = mc² - m_oc²

This is the relativistic formula for kinetic energy. m_o is the rest mass and m_oc² is the internal energy (rest mass energy or rest energy).

Total energy = kinetic energy of the moving body + rest mass energy

E = E_k + m_oc²  = = mc 2 - m_oc ² + m_oc ²

E=mc²

This is Einstein's mass-energy equivalence.

Implications of the equivalence between mass and energy

Particles like electron, proton, neutron have mass. If a particle has mass, it has rest energy m_oc² and may or may not have other forms of energy such as kinetic energy and potential energy. The particle of light, the photon has zero mass but has energy.

It is possible to convert an isolated system of particles with mass into a system of particles with less mass, even zero mass. Similarly, it is possible to convert a particle of zero mass into a particle with mass.

(i.e) Rest energy is converted into other types of energy (mass is converted into energy) or other types of energy are converted into rest energy (energy is converted into mass). Hence, the statement 'mass energy equivalence' comes true and the total energy is conserved in the isolated system.

Example : When an electron meets its antiparticle the positron, both of them annihilate and form two photons. Since, the total energy is conserved, the total energy associated with the electron - positron pair (kinetic energy + rest energy) is transferred to the photons that have no rest energy (no mass). Also, an energetic photon (zero mass) can create an electron - positron pair (particles with mass).

Other examples such as nuclear fission and fusion process are discussed in the next chapter nuclear physics.