Animated Solution for Physics - Current Electricity: Column I gives certain situations in which a straight metallic wire of resistance $R$ is used and Column II gives some resulting effects. Match the statements in Column I with the statements in Column II.

This matrix match problem is an absolute masterpiece. It takes a single, simple object—a straight metallic wire of resistance $R$—and subjects it to four entirely different physical realms: transient circuits, electromagnetic induction, electrostatics, and steady-state direct current. By analyzing how the wire responds to each environment, we can deeply solidify our understanding of core physics principles.

Let's embark on this journey and decode each situation step-by-step.

The Discharging Capacitor

Imagine connecting a fully charged capacitor across the ends of our metallic wire. What happens? The capacitor acts as a temporary reservoir of electrical energy. It immediately begins to discharge, driving a current through the wire.

However, this is an $RC$ circuit. The current is governed by the exponential decay equation:

Because the current decays over time, it is not a constant current. But, as long as any current is flowing through the resistance $R$, the wire will dissipate energy in the form of heat due to Joule heating ($H=\int I^{2}Rdt$). Therefore, thermal energy is generated.

Conclusion for (A): It matches perfectly with (q).

The Moving Conductor

Now, let's pull the wire through a uniform magnetic field $\vec{B}$ with a constant velocity $\vec{v}$, perpendicular to its length. This is the classic setup for motional EMF.

As the wire moves, every free electron inside it experiences a magnetic Lorentz force given by:

This force pushes the electrons to one end of the wire, leaving the other end deficient in electrons (positively charged). This separation of charges creates an internal electric field that eventually balances the magnetic force.

What are the observable effects? First, charges of the same magnitude appear at the two ends. Second, this charge separation establishes a constant potential difference (EMF) across the ends, calculated as $V=Bvl$. However, because the wire is just moving through space and is not part of a closed electrical loop, the charges have nowhere to go. Thus, no continuous current flows, and no thermal energy is generated.

Conclusion for (B): It matches with (r) and (s).

The Electrostatic Shield

In this scenario, we place the wire in a constant external electric field ${\vec{E}}_{\text{ext}}$ directed along its length.

Metals are conductors, meaning they have an abundance of free electrons. The moment the external field is applied, these electrons feel an electric force and drift in the opposite direction. They pile up at one end, leaving the other end positively charged.

This redistribution of charge creates an internal electric field ${\vec{E}}_{\text{int}}$ that opposes the external field. The electrons will continue to move until the internal field exactly cancels the external field. In electrostatic equilibrium, the net electric field inside the conductor is strictly zero (${\vec{E}}_{\text{net}}=0$).

Because the electric field inside is zero, it takes zero work to move a test charge from one end to the other. This means the entire wire is an equipotential volume, and the potential difference between the ends is exactly zero. However, just like in Situation B, charges of the same magnitude have accumulated at the ends to create that cancelling internal field.

Conclusion for (C): It matches with (s).

The Steady Battery

Finally, we connect a battery of constant EMF $\epsilon$ across the wire. This is the most familiar scenario from basic circuit theory.

The battery acts as an electron pump, maintaining a constant potential difference across the ends of the wire. According to Ohm's Law, this steady voltage drives a constant current ($I=\epsilon /R$) through the wire.

As this steady current flows through the resistance $R$, the electrons constantly collide with the atomic lattice, transferring kinetic energy. This results in continuous Joule heating, meaning thermal energy is generated at a constant rate ($P=I^{2}R$).

Conclusion for (D): It matches with (p), (q), and (r).

By carefully analyzing the physics of each environment, we have successfully decoded the matrix: - (A) $\to$ (q) - (B) $\to$ (r), (s) - (C) $\to$ (s) - (D) $\to$ (p), (q), (r)

This problem is a fantastic reminder that a single object can exhibit wildly different behaviors depending on the electromagnetic forces acting upon it!