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Einstein's mass-energy equivalence
Consider a body of rest mass mo. 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
dEk = Fdx
= ( m (dv/dt) + v(dm/dt ) ) dx
dEk = mv dv + v2dm ………. (3)
m2 = m02c2 / c2-v2
Differentiating we get
c2dm = mv dv + v2dm ...(4)
Comparing equations (3) and (4) we get
dEk = c2dm ……………(5)
Thus the change in kinetic energy dEk is directly proportional to the change in mass dm
When a body is at rest, its velocity is zero and m = mo. When its velocity is v its mass becomes m. Therefore integrating equation (5)
∫0E dEk = c2 / ∫m0m dm
Ek = c2 (m-mo) = mc2 - moc2
This is the relativistic formula for kinetic energy. mo is the rest mass and moc2 is the internal energy (rest mass energy or rest energy).
Total energy = kinetic energy of the moving body + rest mass energy
E = Ek + moc2 = = mc 2 - moc 2 + moc 2
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 moc2 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.
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