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# Mass Energy

A matter can be defined with the help of their extensive and intensive properties. Intensive properties can be defined as the properties of matter which do not change with change in the amount of substance. In other words these properties do not depend on quantity of the substance but on the quality of it. On the contrary, some of the properties vary with the amount of matter. These properties are called as extensive properties. Some common intensive properties are temperature, density and surface tension whereas mass and volume examples of extensive properties. These properties are inter-related to each other. Einstein relation provides a relation between energy and mass of matter. In 1905, first Einstein proposed the Special Theory of Relativity which was later proved and accepted by many scientists with the help of different experiments.

Nuclear weapons are based on the concept that mass can be turned into energy which further represented by mathematical relation; E = $mc^{2}$. This relation is well known as Einstein relation of mass and energy. It can be derived from Special Relativity theory. The special relativity theory is based on two rules. First postulate states that the laws of physics are the same in all inertial frames and second postulate is the principle of the constancy of the speed of light. In vacuum, the speed of light is a same constant value. The Special Relativity theory states that the mass appears to increase with speed; hence as the object moves faster, the mass of it must be increased.  If m is relativistic mass, and $m_0$ is rest mass of an object which moves with $v$ speed, change in mass of an object with its speed can be calculated with the help of given equation;

m = $\frac{m_{o}}{\sqrt{\frac{1 - v^{2}}{c^{2}}}}$.

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## Mass Energy Equivalence

The mass energy equivalence can be explained with the help of Einstein's famous equation. Let’s discuss the mass energy equivalence. We know that the particles of light are called as photons.  James Clerk Maxwell purposed that photons have momentum but no mass. But we know that momentum consists of two components; mass and velocity. Than how it is possible that photons would be mass less particles. Later Einstein purposed that energy of a photon must be equivalent to a quantity of mass. He conducted one experiment to prove that. Imagine a stationary box floating in deep space and emit a photon inside the box. To conserve the total momentum of the system the box will stop moving. In the absence of external force, the centre of mass will be at the same location.

Einstein proved that there must be a ‘mass equivalent’ to the energy of the photon and formulated his statement in terms of equation:
E = $mc^{2}$.

## Mass Energy Relation Derivation

Let us derive the mass energy relation. The momentum of photon can be calculated with the help of Maxwell’s expression which is also applicable for the momentum of an electromagnetic wave. The momentum of the photon can be given as;

$P_{photon}$ = $\frac{E}{c}$.

Here E is energy and c is the speed of light.

If the mass of box is M, and it will recoil slowly in the opposite direction to the photon whose speed is v; momentum of the box would be:
$p_{box}$ = $Mv$.

Let’s assume that photon reaches to the other side of the box in $|Delta t$ for a small distance, $\Delta x$. Hence speed of the box would be;
v = $\frac{\Delta x}{\Delta t}$

Let’s apply the rule of conservation of momentum;

M $\frac{\Delta x}{\Delta t}$ = $\frac{E}{c}$

If the length of box is L, then the time ∆t taken by a photon to reach the other side of the box would be;
$\Delta t$ = $\frac{L}{c}$

Now substitute it into the conservation of momentum equation and rearranging:

M $\Delta x$ = E  $\frac{L}{c^{2}}$

If the mass of photon is m and the box has position $x_1$ and the photon has position $x_2$;

$\vec{x}$ = $\frac{Mx_{1} + mx_{2}}{M + m}$

or,

$\frac{Mx_{1} + mx_{2}}{M + m}$ = $\frac{M(x_{1} - \Delta x) +mL}{M +m}$

or
If  $x_2$ = 0;
$mL$ = $M \Delta x$
or,
$mL$ = $\frac{EL}{c^{2}}$
or,
E = $mc^{2}$.

## Mass Energy Relation

The mass energy relation of Einstein suggests that mass of an object can be converted into energy.  If we consider a body which is moving with the speed of light, a uniform force acts on it.

This force pumps energy and momentum into the object. Since this force cannot change the speed of the object therefore it will increase the momentum of the object;

Change of momentum = mass $\times$ velocity of the object

There are two possible relations between energy, force and momentum. The energy gained by the object due to force would be equal to sum of force and distance through which force acts.

The mathematical expression can be written as;
E = Force $\times$ c.

Similarly momentum gained by the object would be sum of force and time.

Momentum = mass $\times$ velocity,

Force = m $\times$ c
.

Hence combination of both relations would be;

E = Force $\times$ c = (m $\times$ c) $\times$ c = $mc^{2}$.