Introduction to Mass-Energy Equivalence
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.
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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.
FAQs on Mass - Energy Equivalence
1. What is the principle of mass-energy equivalence in simple terms?
The principle of mass-energy equivalence, famously expressed by Einstein's equation E=mc², states that mass and energy are two different forms of the same fundamental entity. It means that mass can be converted into energy, and energy can be converted into mass. A small amount of mass can be transformed into a very large amount of energy because the conversion factor is the square of the speed of light (c²), which is an enormous number.
2. What does each variable in the equation E=mc² stand for?
In the mass-energy equivalence equation, each variable represents a specific physical quantity:
- E stands for the equivalent energy, measured in Joules (J).
- m stands for the mass of the object, measured in kilograms (kg).
- c stands for the speed of light in a vacuum, which is a constant approximately equal to 3 x 10⁸ meters per second (m/s).
3. What are some real-world examples of mass-energy equivalence?
Mass-energy equivalence is not just a theoretical concept; it has several practical applications and observable examples:
- Nuclear Power Plants: In nuclear fission, the nucleus of a heavy atom like uranium splits into smaller parts. The total mass of the products is slightly less than the original mass. This 'lost' mass is converted into a huge amount of thermal energy, which is used to generate electricity.
- The Sun and Stars: Stars generate energy through nuclear fusion, where light nuclei (like hydrogen) combine to form a heavier nucleus (like helium). The resulting nucleus has less mass than the sum of the original nuclei, and the mass difference is released as light and heat.
- Particle Accelerators: Scientists can create new particles by colliding existing ones at very high speeds. The kinetic energy of the colliding particles is converted into the mass of new, often heavier, particles.
- Radioactive Decay: An unstable atomic nucleus loses energy by emitting radiation. This process results in a small decrease in the nucleus's mass, which is converted into the energy of the emitted particles.
4. Why is the speed of light (c) squared in the equation E=mc²?
The speed of light (c) is squared in the equation because it serves as a massive conversion factor that relates the units of mass to the units of energy. The squaring arises from the relativistic derivation of the formula, where an object's energy increases with its velocity. Because the value of 'c' is extremely large (approximately 300,000,000 m/s), squaring it makes the conversion factor immense. This explains why even a minuscule amount of mass can be converted into a tremendous quantity of energy, a key principle behind nuclear power and weapons.
5. How does the concept of binding energy relate to mass-energy equivalence?
Binding energy is the energy that holds the protons and neutrons (nucleons) together in an atomic nucleus. This energy is a direct manifestation of mass-energy equivalence. The mass of a stable nucleus is always slightly less than the sum of the individual masses of its constituent nucleons. This difference in mass is called the mass defect (Δm). According to E=mc², this mass defect is converted into the binding energy that holds the nucleus together. The higher the binding energy per nucleon, the more stable the nucleus is.
6. What is the difference between rest mass and relativistic mass?
The distinction between rest mass and relativistic mass is a key concept in special relativity:
- Rest Mass (m₀): This is the intrinsic, unchangeable mass of an object as measured in its own reference frame (i.e., when it is at rest). It is a fundamental property of the object.
- Relativistic Mass: This is the effective mass of an object when it is in motion relative to an observer. As an object's speed increases, its kinetic energy increases, and this added energy contributes to its inertia, making it behave as if its mass has increased. Relativistic mass is not a fundamental property but depends on the object's velocity.
For everyday speeds, the difference is negligible, but it becomes significant as an object approaches the speed of light.
7. Does mass-energy equivalence mean matter is destroyed to create energy?
This is a common misconception. Mass-energy equivalence does not mean matter is 'destroyed'. Instead, it demonstrates that matter is a highly concentrated form of energy. In processes like nuclear fission or fusion, mass is converted or transformed into other forms of energy (such as thermal or electromagnetic energy). The total amount of mass-energy in a closed system is always conserved. So, nothing is truly destroyed; it just changes form.
8. How is mass-energy equivalence applied in nuclear reactions like fission and fusion?
In both nuclear fission and fusion, the principle of mass-energy equivalence governs the energy release. The core concept is the mass defect. In a nuclear reaction, the total rest mass of the product particles is less than the total rest mass of the initial reactants. This 'missing' mass (Δm) has been converted directly into energy (E) according to the formula E = (Δm)c². Since c² is a very large number, even a tiny mass defect results in the release of an enormous amount of energy, which is characteristic of nuclear reactions.
