Q.Suppose India had a target of producing by 2020 AD, 200,000 MW of electric power, ten percent of which was to be obtained from nuclear power plants. Suppose we are given that, on an average, the efficiency of utilization (i.e. conversion to electric energy) of thermal energy produced in a reactor was 25%. How much amount of fissionable uranium would our country need per year by 2020? Take the heat energy per fission of 235U to be about 200MeV.
Mass Energy Equivalence: From Intuition to the Formula
Imagine you have a lump of coal. You know you can burn it to get heat, and that heat can run a steam engine. The energy you get out seems to come from the chemical bonds in the coal. But what if I told you that the coal itself — just sitting there, not burning — already contains a staggering amount of energy locked inside its very mass? That is the core idea of mass-energy equivalence.
The Intuition: Mass is Frozen Energy
Think of mass as a kind of "frozen" or "stored" energy. When you burn coal, you are only releasing a tiny fraction of this stored energy — the energy in the chemical bonds. The rest of the mass remains as matter. But if you could somehow completely convert that lump of coal into pure energy, you would get an unimaginable amount — enough to power a city for years.
This is not a metaphor. Mass and energy are not two separate things that can be converted into each other like dollars and rupees. They are the same fundamental thing, just in different forms. Mass is a highly concentrated form of energy. Energy, when concentrated enough, behaves like mass.
The Precise Statement
The relationship is given by the most famous equation in physics:
E=mc2
Where:
E is the energy equivalent of the mass (in joules, J)
m is the mass (in kilograms, kg)
c is the speed of light in vacuum (3×108 m/s)
The speed of light is a huge number. Squaring it makes it enormous. This is why a tiny amount of mass corresponds to a colossal amount of energy.
What This Equation Actually Means
The equation tells you exactly how much energy is "stored" inside any object with mass m. If you could annihilate that mass completely, you would get E joules of energy.
Example: A 1 kg mass (like a litre of water) contains:
E=1×(3×108)2=9×1016 J
That is 90 quadrillion joules — roughly the energy released by a 20-megaton nuclear bomb. This is not energy you can normally access; it is locked inside the nucleus of atoms.
Where Does This Show Up in Real Life?
Nuclear Reactions: In nuclear fission (splitting atoms) or fusion (joining atoms), a tiny fraction of the mass of the nucleus is converted into energy. The mass of the products is slightly less than the mass of the reactants. The "missing" mass has become energy — exactly as E=mc2 predicts. This is how the Sun works and how nuclear power plants generate electricity.
Particle Physics: When a particle and its antiparticle meet, they annihilate completely into pure energy (usually gamma rays). The energy produced equals mc2 for the two particles. …
Why this formula?
Why E=mc2 — The Reasoning Behind Mass-Energy Equivalence
The formula E=mc2 is not a random guess. Einstein arrived at it by thinking deeply about what happens to energy when you move an object. The core insight: if an object gains energy, it must behave as if it has gained mass.
The Starting Point: Relativistic Momentum
In special relativity, the momentum of a particle is not simply p=mv. Instead, it is:
p=1−v2/c2m0v
where m0 is the rest mass (mass measured when the object is at rest). This formula already tells us something strange: as speed approaches c, momentum shoots toward infinity — no object with mass can reach the speed of light.
The Energy-Momentum Relation
Einstein then asked: what is the correct expression for kinetic energy that matches this new momentum? In classical physics, kinetic energy is K=21mv2. But that formula fails at high speeds.
The relativistic kinetic energy turns out to be:
K=1−v2/c2m0c2−m0c2
This looks odd — why subtract m0c2? Because when v=0, the first term becomes m0c2, and we want K=0 at rest. So the subtraction gives zero kinetic energy when the object is stationary.
Note
The term m0c2 appears naturally as a rest energy — energy that an object has simply because it has mass, even when completely at rest.
The Crucial Step: What Happens When You Add Energy?
Now consider a box that emits light (photons) in opposite directions. The light carries away energy. Classical physics says the box loses energy but its mass stays the same. Einstein showed this cannot be true.
The argument (simplified): if the box emits a pulse of light with energy E, the light carries momentum p=E/c. By conservation of momentum, the box recoils. But after the light is absorbed by the opposite wall, the box stops. The net effect: the box has moved slightly. Its center of mass has shifted — unless the energy carried by the light also carried mass.
Important
For the center of mass of the entire system (box + light) to remain stationary, the light must behave as if it has an effective mass m=E/c2. Therefore, energy itself has inertia.
The Full Formula
The total energy of any object — moving or at rest — is:
E=1−v2/c2m0c2
For an object at rest (v=0), this reduces to:
E=m0c2
For a moving object, the total energy is the sum of rest energy and kinetic energy:
E=m0c2+K
E=mc2
where m is the relativistic massm=1−v2/c2m0, or equivalently:
Working backward from the required electrical output, through the stated conversion efficiency to get thermal power, then dividing by the energy released per fission, gives the number of fissions — and hence the mass of U-235 — needed each year. …
Find the required nuclear electric output (10% of 200,000 MW), convert to thermal power using the 25% efficiency, find the annual thermal energy, divide by 200 MeV/fission to get the number of fissions, and convert to mass. Result: about 3.08 × 10⁴ kg (roughly 30.8 tonnes) of U-235 per year.
Step 1 — Required nuclear electric power
Pelec=10%×200,000MW=20,000MW=2×1010W
Step 2 — Required thermal power
Since only 25% of thermal energy converts to electricity:
Pthermal=0.25Pelec=0.252×1010=8×1010W
Step 3 — Thermal energy needed per year
Eyear=Pthermal×(1yr in seconds)=8×1010×3.154×107=2.5232×1018J
Same / Similar Concept — real previous-year questions on the same or a closely similar concept, not this exact question.
AHSEC Higher Secondary (HS) Final Examination 2023Set ANNUAL2 marks
Q.Calculate the energy equivalent of 1 kg of a substance.
OR
Discuss conductor and insulator in terms of energy gap.
›Reveal solutionSolution
By Einstein's mass-energy relation, 1 kg of matter is equivalent to an enormous 9×10¹⁶ J of energy.
Einstein's mass-energy equivalence relation states that mass and energy are interconvertible: E = mc², where c is the speed of light in vacuum (c ≈ 3×10⁸ m/s).
For m = 1 kg:
E = 1 × (3×10⁸)² = 1 × 9×10¹⁶ = 9×10¹⁶ J.
This illustrates how much energy is 'locked' in even a small amount of mass — comparable to the energy released by tens of megatons of TNT — which is why nuclear reactions, which convert only a tiny fraction of mass into energy, release such enormous amounts of energy compared with ordinary chemical reactions.
AHSEC Higher Secondary (HS) Final Examination 2018Set ANNUAL2 marks
Q.Some scientists have predicted that global nuclear war on the earth would be followed a severe "Nuclear Winter". What might be the basis of this prediction? (2)
›Reveal solutionSolution
Nuclear winter: dust/smoke from nuclear war blocks sunlight, cooling the earth drastically. OR: B₀ = E₀/c = 1.6 × 10⁻⁷ T.
Main — basis of nuclear winter. A global nuclear war would release vast amounts of energy, igniting widespread fires and vaporising material. The resulting dust, smoke and soot would be thrown high into the atmosphere and spread around the globe. This thick layer would absorb and scatter incoming solar radiation, so far less sunlight reaches the earth's surface. With sunlight cut off, surface temperatures would fall sharply for a long period — a 'nuclear winter' — devastating plant life and the food chain. The basis of the prediction is therefore the blocking of sunlight by atmospheric dust and smoke produced in the war.