CAN YOU derive E=MC^2 ?

Question

PLEASE HELP ME OUT FOR DERIVATION OF E=MC^2

Answer 1

The derivation of E=mc^2 is although not standard(By standard i mean it can be derived from many logics) but basic approach begins from

Newtons second law with other formulae i.e,, work and energy.

You are aware that

F=m.a (This is valid only in classical mechanics but in relativity) it goes

F= dp/dt (rate of change of angular momentum)

and

F = d/dt(m v)

in relativity both mass and velocity are variables so

F = m dv/dt + v dm/dt -Eq.1

where m is moX Gamma

gamma is 1/sqrt ( 1- (v^2/c^2) ) -Eq.2

and mo is rest mass

Put Eq2 in Eq.1 and differentiate to get( by doing simple differential mathematics and little logics

F = mo.gamma dv/dt - mox (v^2/c^2) x dv/dtx gamma^3
------------------------------------------ ----------------Eq.3
Just check the '-' sign

Looks Good.

now you are aware that when the force acts there is a work done.

so

work done is Force x elemental displacement

dW = F x dS ( where S is displacement) -Eq4

Now you must integrate this to get the total work done. so from velocitys of lower limit of '0' to upper limit of some v

so total work done is change in kinetic energy as

W = Integration(FxdS) = change in Kinetic Energy --Eq.5


You please integrate by putting Eq.3 in Eq.4 and Ea5. Also use the boundary or integration limits.

You will end up with

Change in Kinetic Energy as

= mxc^2- moxc^2 -Eq.6


But you know Total energy of any system is

E = Kinetic Energy + Potential Energy

some how einstein assumed that Potential energy is

= moxc^2

so putting this in Eq.6 you will end up with

E=mxc^2

If you face any problem write to me.