Animated Solution for Physics - Current Electricity: The Kirchhoff's first law ($\sum i=0$) and second law ($\sum iR=\sum E$) where, the symbols have their usual meanings, are respectively based on

When we dive into the world of electrical circuits, two fundamental rules govern the behavior of currents and voltages: Kirchhoff's Current Law (KCL) and Kirchhoff's Voltage Law (KVL). While they might seem like abstract mathematical equations at first glance, they are actually deeply rooted in the most fundamental conservation laws of physics.

The Junction Rule (KCL)

Imagine a busy intersection where several roads meet. Cars enter the intersection from some roads and leave through others. If there is no traffic jam (no cars piling up in the middle), the number of cars entering the intersection must exactly equal the number of cars leaving it.

This is precisely what Kirchhoff's First Law states for electrical circuits. At any junction (or node) in a circuit, the algebraic sum of the currents is zero:

Since electric current is simply the rate of flow of electric charge ($i=\frac{dq}{dt}$), saying that the input current equals the output current is mathematically equivalent to saying that the total charge entering the junction equals the total charge leaving it. Wires and junctions are conductors; they do not have the capacity to store or accumulate charge like a capacitor does. Therefore, KCL is a direct manifestation of the Conservation of Charge.

The Loop Rule (KVL)

Now, let's shift our perspective from a single point to a complete journey. Imagine hiking up a mountain and then returning to your starting point. The total change in your gravitational potential energy over the entire round trip is exactly zero.

Kirchhoff's Second Law applies this same logic to electric charges moving in a closed loop. It states that the algebraic sum of all potential differences (voltages) around any closed loop in a circuit is zero:

Or, in terms of electromotive forces and voltage drops:

Electric potential is defined as the electric potential energy per unit charge ($V=\frac{U}{q}$). When a charge moves through a battery, it gains potential energy. As it moves through resistors, it loses that energy (usually as heat). KVL tells us that the total energy supplied by the sources is completely dissipated by the components in the loop by the time the charge returns to its starting point. No energy is magically created out of nowhere, and no energy vanishes without a trace. Thus, KVL is a direct consequence of the Conservation of Energy.

By understanding the physical realities behind the equations, we can confidently conclude that Kirchhoff's first law is based on the conservation of charge, and the second law is based on the conservation of energy. This makes option (d) the correct choice.