[Physics Class Notes] on Mass – Energy Equivalence Pdf for Exam

In physics, the relationship between mass and energy in a rest frame of the system is the mass-energy equivalence, in which two values can only be different by the unit of measurement and a constant. 

Mass-energy equivalence implies that, even though the total mass of a system changes, the total energy and momentum remain constant. Consider the collision of an electron and a proton. It destroys the mass of both particles but generates a large amount of energy in the form of photons. The discovery of mass-energy equivalence proved crucial to the development of theories of atomic fusion and fission reactions.

Einstein’s Mass-Energy Relation

Mass-energy equivalence states that every object possesses certain energy even in a stationary position. A stationary body does not have kinetic energy. It only possesses potential energy and probable chemical and thermal energy. 

According to the field of applied mechanics, the sum of all these energies is smaller than the product of the mass of the object and the square of the speed of light.

When an object is at rest when it is not moving and shows no momentum, the mass, and energy results are equivalent and they can only be differentiated by one constant, that is, the square of the speed of the light (c2). 

Mass-energy equivalence means mass and energy are the same and can be converted into each other. Einstein put this idea forth but he was not the first to bring this into the light. He described the relationship between mass and energy accurately using his theory of relativity. The equation is known as Einstein’s mass-energy equation and is expressed as,

E=mc2

Where E= equivalent kinetic energy of the object,

m= mass of the object (Kg) and

c= speed of light (approximately = 3 x 108 m/s)

The formula states that a particle’s energy (e) in its rest state is the product of mass (m) with the square of the speed of light,c. 

It is because of the large numbers of the speed of light in everyday units. The formula says that the rest mass of a small amount resembles a large amount of energy even though it’s independent in the making of the matter. 

Let’s go deep into the topic and understand what is rest mass? 

Rest Mass

The mass that is calculated while the system is at rest is known as Rest mass, which is also known as invariant mass. 

It is a physical property that is not dependent on momentum, even when it’s approaching the speed of light at high speeds. 

The invariant mass of Photons which are massless particles is zero while free particles which are massless consist of both energy and momentum. 

The SI units of energy (E) are calculated in joules, mass (m) is calculated in kilograms, and speed of light ‘c’ is calculated in meters per second. 

Derivation of Einstein’s Equation

Derivation I

The simplest method to derive Einstein’s mass-energy equation is as follows,

Consider an object moving at a speed approximately of the speed of light.

A uniform force is acting on it. Due to the applied force, energy and momentum are induced in it.

As the force is constant, the increase in momentum of the object= mass x velocity of the body.

We know,

Energy gained= Force x Distance through which force acts

E= F x c ………………………………………… (1)

Also,

The momentum gained = force x Duration through which force acts

As, momentum = mass x velocity,

The momentum gained = m x c

Hence, Force= m x c ……………………………. (2)

Combining the equation (1) and (2) we get,

E= m c2

Derivation II

Whenever an object is in speed, it seems to get heavier. The following equation gives the increase in mass due to speed.

$m = frac{m_0}{sqrt{frac{(1-v^2 }{c^2}}}$

Where,

m- the mass of the object at the traveling speed

m₀– the mass of the object at a stationary position

v- speed of the object

c- speed of the light

We know, in motion, an object possesses kinetic energy and it is given by

E= ½ (mv2)

Total energy possessed by the object is approximately equal to kinetic energy and increases in mass due to speed. 

E≅ (mc2) + ½ (mv2)

E- (mc2) = ½ (mv2)   , for small v/c

E= Relativistic kinetic energy + mc2

The relativistic kinetic energy depends on the kinetic energy and speed of the object. We can simplify the equation by setting the speed of the object as zero. Hence the equations become as follows,

E= 0+mc2

E= mc2

Applications of Einstein’s Equation

The first person to put forth the word that the mass and energy’s equivalence as one of the general principles and the outcome of symmetry of time and space was Einstein. Einstein’s theory was used to understand nuclear fission and fusion reactions. Using the formula, it was revealed that a large amount of energy is liberated during nuclear fission and fusion processes. This phenomenon is used in creating nuclear power and nuclear weapons.

To find out binding energy in an atomic nucleus, the equation is used. By measuring the masses of various nuclei and subtracting them from the sum of masses of protons and neutrons, Binding energy is calculated. Measurement of binding energy is used to calculate the energy released during nuclear reactions. 

These energies seem much smaller as compared to the mass of the object that is multiplied by the square of the speed of the light. Because of this principle, atoms after a nuclear reaction have less mass than the atoms before the nuclear reaction. The difference in the before and after mass shapes up as heat and light with the same energy used as the difference. 

Einstein’s equation is used to find out the change in mass during the chemical reactions. Whenever there is a chemical reaction, breakage and the formation of new bonds take place. During the exchange of molecules, a change in mass takes place. For chemical energy, Einstein’s equation can be written as 

E= Δm x c2

Where Δm- change in mass

The formula provided by Einstein can be written with E as the energy which is released and removed and m can be written as the change in mass. 

It is explained in relativity, all the energy that an object moves with, provides a contribution to the total mass of that body, which is used in measuring how much it can resist accelerating. 

When the observer is at rest, the removal of energy is the same as the removal of mass which goes by the formula m = e/ c2. 

The radioactivity of various elements is based on the theory of mass-energy equivalence. Radioactivity produces X-rays, gamma rays. So in many radiotherapy equipment, the same principle is used. 

To understand the effect of gravity on all-stars, moon, and planet, and to measure the age of fossil fuels.

In many surgeries, where opening and stitching of body parts is not done, Cath lab is used. It works on Einstein’s equation.

To understand the universe, its constituents, and the age of planets, The equation is used.