Animated Solution for Physics - Thermodynamics: Column II gives certain systems undergoing a process. Column I suggests changes in some of the parameters related to the system. Match the statements in Column I to the appropriate process (es) from Column II.

The beauty of physics lies in its profound interconnectedness. This matrix match problem is a brilliant showcase of that unity, bringing together concepts from thermodynamics, electrodynamics, and modern nuclear physics into a single analytical challenge. To solve it, we must act as physical detectives, investigating five distinct processes to determine exactly how their energy and mass evolve.

The Charging Capacitor

Imagine an initially uncharged capacitor connected to a battery. The battery acts as an electrical pump, doing work to move charges from one plate to the other. This work does not vanish; it is stored within the electric field between the plates as electrical potential energy.

The energy of the system transitions from zero to a final state of $U=\frac{1}{2}C{V}^2$. Because the total energy of the system has clearly increased, this process perfectly matches Statement (A).

Adiabatic Compression

Next, we consider a gas enclosed in an adiabatic container, being compressed by a piston. The term 'adiabatic' is our critical clue—it dictates that no heat can enter or leave the system ($Q=0$).

When the piston is pushed down, mechanical work is done on the gas ($W<0$). According to the first law of thermodynamics, $\Delta U=Q-W$, this mechanical work directly increases the internal energy of the gas. The gas molecules accelerate, meaning their random kinetic motion intensifies. Thus, the total energy increases (matching Statement A), and mechanical energy is explicitly converted into the energy of random motion (matching Statement B).

Rigid Container Cooling

In process (r), a gas is kept in a rigid container and cools down due to a colder surrounding atmosphere. The word 'rigid' implies an isochoric process where the volume remains strictly constant. Consequently, the mechanical work done is zero ($W=0$).

Because the gas is cooling, heat is flowing out of the system ($Q<0$). This results in a decrease in the internal energy of the gas. Since the energy is decreasing and no mechanical work is involved, this process serves as a distractor and does not match any of the given statements.

Nuclear Fission

Process (s) takes us into the quantum realm. A heavy nucleus, initially at rest, undergoes nuclear fission, splitting into two nearly equal fragments and emitting neutrons.

During this violent split, a phenomenon known as 'mass defect' occurs. The total mass of the resulting products is slightly less than the mass of the original heavy nucleus. This means the mass of the system decreases, which perfectly aligns with Statement (D).

Where does this lost mass go? Albert Einstein's iconic equation, $E=\Delta m{c}^2$, provides the answer. The lost mass is converted into energy. Specifically, the internal nuclear binding energy is transformed into the mechanical kinetic energy of the flying fragments. This transformation matches Statement (C). Furthermore, because the system now possesses this new kinetic energy, its overall energy has increased, matching Statement (A).

The Wire Loop

Finally, we examine a resistive wire loop placed in a time-varying magnetic field. Here, Faraday's law of electromagnetic induction takes center stage. The changing magnetic flux induces an electromotive force (EMF), $\epsilon =-\frac{d\Phi }{dt}$, which drives a current through the loop.

Because the wire has electrical resistance, this induced current produces Joule heating ($H=I^{2}Rt$). Electrical energy is continuously converted into thermal energy, thereby increasing the total internal energy of the loop. This increase in energy matches Statement (A).

By systematically evaluating the thermodynamic and physical constraints of each system, we successfully map the statements to their corresponding processes. Statement A maps to p, q, s, and t. Statement B maps to q. Statement C maps to s. And Statement D maps to s. This problem beautifully illustrates how the fundamental laws of conservation govern everything from simple circuits to the splitting of atoms.