^Einstein’s massenergy equivalence

^Einstein’s mass-energy equivalence

The relation

is called relativistic energy or Einstein’s mass-energy equivalence as it reveals that mass & energy are interconvertible.  Here the quantity m₀ c² is the energy associated with the rest mass of the body, K is the kinetic energy of the body & the sum of KE & rest mass energy is called the total energy (E) of the body.

Differentiating this relation we get

c².2 m dm – m².2v dv – v².2m dm = 0

or c² dm = mv.dv + v² dm            _ _ _ _ (1)

Also 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 by a distance dx, its energy increases by

Equating (1) & (2) we get

dK = dm.c²                                            _ _ _ _ (3)

If particle is accelerated from rest to a velocity v, let its mass m₀ increase to m, then its total increase in KE can be obtained by integrating equation (3) , i.e.

Suppose a body of rest mass m₀ moves with a velocity v << c.  Its mass at speed v is

As mc² is the total energy and m₀ c² is the rest mass energy. Therefore the term (mc² – m₀ c²) represents the kinetic energy, thus KE of the body is

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

Facts

1. Zero rest mass i.e. m₀ = 0 means massless particle & for such particles, E = pc i.e. even a massless particle can possess energy.

2. An electron and photon have same de-broglie wavelength. Which has more total energy?

Ans. Same λ means same p (as, p = h/λ) & quantity having more m₀ will have more E, i.e. Electron.

3. An electron and proton have same total energy. Which has more de-broglie wavelength?

Ans. In order to have same E the quantity having less m₀ must have more p or less λ (as, p = h/λ), i.e. Electron.

4. Also KE of a particle is

5. If a photon and electron has same de Broglie wavelength, then the KE of the photon is  times the kinetic energy of the electron.

Commonly used values

1. Charge of electrons = e = – 1.6 x 10^-19 C

2. Rest mass of electrons = m₀ = 9.1 x 10^-31

3. Sp. charge of electron = e/m₀ = 1.76 x 10¹¹ C/ kg

4. hc = 1.989 x 10^-25  » 2 x 10^-25

5.