Einstein's Mass-Energy Equivalence Relation

According to Newton's second law of motion, force acting on a body is defined as the rate of change of its momentum i.e.,

Now, if this force 'F' displaces the body through a distance dx, its energy increases by

According to Einstein's relation of relativistic mass,

or m²c² - m²v² = m₀²c²

Differentiating and noting that m₀ and 'c' are constants,

c². 2m dm - m². 2v dv - v². 2m dm = 0

c² dm = mv . dv + v² dm = dK [from equation (i)]

dK = dm . c²

If particle is accelerated from rest to a velocity v, let its mass m₀ increase to m.

K = (m - m₀) c²

K + m₀c² = mc²

Here, the quantity m₀c² is the energy associated with the rest mass of the body.

K is the kinetic energy of the body. Thus, the total energy of the body is given by E = mc²

This is Einstein's mass-energy equivalence relation.

Sub Topics

The important applications are the following
Annihilation of matter
Pair production
Nuclear bomb

The important applications are the following

i) Kinetic energy of a body moving with a velocity v << c.

We know that

Here, mc² is the total energy and m₀c² is the rest mass energy. Therefore, (mc² - m₀c²) represents the kinetic energy.

This is the same result that we obtain by Newtonian mechanics.

Annihilation of matter

A positron has the same mass as an electron but an opposite charge of +e. When an electron and a positron come close to each other, they annihilate or destroy each other. Their masses are converted into energy according to Einstein's relation, and the energy thus obtained is released in the form of g- rays. Thus, matter is annihilated to give an equivalent amount of energy.

Pair production

This process is the reverse of annihilation of matter. A photon of energy 1.02 MeV nearing a massive nucleus, under its intense electric field, splits up into a pair of particles - an electron and a positron. Thus, energy has been materialised by converting it into matter.

Nuclear bomb

When protons and neutrons in the nucleus combine together to form the nucleus, the total mass of all the protons and neutrons (also called nucleons here, because these are the particles forming a nucleus) is just a little more than the mass of the nucleus itself. The deficiency in the mass, known as mass defect, is transformed into the binding energy of the nucleus. It is this energy which keeps the nucleus bound. If the nucleus is split up, as in an atomic bomb (strictly a nuclear bomb), this energy is released in the form of heat and g radiations.