Mass - Energy Equivalence

Introduction to Mass-Energy Equivalence

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 square of the speed of light.

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)

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 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= m0/$\sqrt{[(1-v^{2})/c^{2}]}$

Where,

m- mass of the object at the travelling speed

m0- mass of the object at stationary position

v- speed of the object

c- speed of the light

We know, in motion, 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 increase 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 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

1. 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.

2. To find out binding energy in an atomic nucleus, the equation is used. By measuring the masses of various nuclei and subtracting it 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.

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

E= Δm x c2

Where Δm- change in mass

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

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

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

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

Q1. Explain Einstein's Mass-energy Equivalence Equation?

Ans- Einstein’s mass-energy equivalence equation is the most basic formula that gives the relation between mass and energy. It states energy and mass are the same and interchangeable under the appropriate situations. The equation is given as

E=mc2

• In Einstein's equation mass is multiplied by the speed of light because whenever energy in any kind is converted into another form of pure energy, the resulting energy is moving at the speed of light. Pure energy is nothing but electromagnetic radiation. And the electromagnetic radiations always travel at a constant speed of light.

• Whenever an object is moving at the speed four times relative to another object, it does not necessarily have four times energy. It contains 16 times the energy. It means the magnitude is squared. Hence, the speed of light is squared as a conversion factor to decide the energy possessed by an object.

Q2. How Mass-energy Equivalence is Related to Gravity?

Ans-

• There are two different types of mass, the gravitational mass and the inertial mass. The gravitational mass is nothing but the gravitational force acting on the object and the strength of the gravitational field created by an object.

• The inertial mass is the magnitude of acceleration when force is applied to the object.

• The mass-energy equivalence is related to the inertial mass.

• Newton's gravity principle is based on the weak equivalence theory. This theory states that the gravitational and inertial masses of an object are the same.

• But practically, they are not the same. Due to gravity, energies, forces act on an object in motion thus causes the change in mass.

• Application of the Mass-energy equivalence principle gives all the energies associated because gravity is considered.