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)

*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 + v^{2}dm ………. (3)

m=m_{0}/root(1-(v^{2}/c^{2}))

m^{2} = m_{0}^{2}c^{2}
/ c^{2}-v^{2}

Differentiating we get

c^{2}dm = mv dv + v^{2}dm ...(4)

Comparing equations (3) and (4) we get

dE_{k} = c^{2}dm ……………(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} = c^{2}
/ ∫_{m0}^{m }dm

E_{k} = c^{2} (m-m_{o})
= mc^{2} - m_{o}c^{2}

This is the relativistic formula for kinetic
energy. m_{o} is the rest mass and m_{o}c^{2} 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_{o}c^{2 } = = mc 2 - m_{o}c ^{2} + m_{o}c
^{2}

E=mc^{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 m_{o}c^{2} 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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11th 12th std standard Class Physics sciense Higher secondary school College Notes : Einstein's mass-energy equivalence |

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