Question

According to Kirchhoff's Loop Rule (Second Law) applied across a closed network mesh, the algebraic sum of all potential variations around any closed loop must equal zero. This fundamental circuit law is a direct consequence of which physical conservation principle?

• A. \( \text{Conservation of Linear Momentum} \)
• B. \( \text{Conservation of Electric Charge} \)
• C. \( \text{Conservation of Mass} \)
• D. \( \text{Conservation of Energy} \)

Hint:

Kirchhoff had two separate rules built on two separate conservation laws: the Junction Rule comes from charge conservation, while the Loop Rule comes from the fact that electric potential at any single point can only ever have one value. Think about which quantity is being tracked as the charge completes a full trip around the loop.

Answer 1

The Correct Option is D

Concept: Kirchhoff's circuit laws keep track of currents and voltages within electrical networks. The loop rule states that the algebraic sum of the electromotive forces (\( \text{EMFs} \)) plus the algebraic sum of the potential drops across all resistors in any closed loop must equal zero: \[ \sum \Delta V = 0 \]

Step 1: Link electrostatic potential to work and energy definitions. Electric potential difference is defined as the work done per unit charge by electrostatic forces to move a test charge between two points. The electrostatic field is conservative, meaning the total work done moving a charge around any closed loop path, starting and ending at the exact same point, is always zero. \[ W_{\text{loop}} = \oint q \cdot \vec{E} \cdot d\vec{r} = 0 \]

Step 2: Match the loop rule to its conservation law. Since the net work done on a unit charge around a closed loop is zero, the charge gains no net energy, and loses no net energy once it completes a full loop. This makes Kirchhoff's Loop Rule a direct statement of the Conservation of Energy.

Answer 2

Concept:

• Electric potential at a given point has one single, fixed value, so returning to the same point after a full trip around a loop must bring the potential back to exactly where it started.
• Potential difference is energy per unit charge, so this return-to-the-same-value behaviour of potential describes a bookkeeping of energy, not of charge or momentum.

Step 1: Track potential as a charge moves around the loop.
Say a unit charge starts at point A with potential $V_A$, travels through several cells and resistors, gaining or losing potential at each component, and finally returns to point A.

Step 2: Apply the single valued nature of potential.
Because potential at point A can only have one value, the potential the charge ends with on returning to A must equal $V_A$ again, so the total, algebraic change over the full trip is zero: $\sum \Delta V = 0$.

Step 3: Convert the potential statement into an energy statement.
Since potential difference equals work done per unit charge, a net potential change of zero over the loop means the net work done on the charge, and so its net energy gain, over one complete loop is also zero. Energy supplied by sources exactly balances energy absorbed by resistors.

Step 4: Compare with the other conservation laws.
Charge conservation governs the Junction Rule, current in equals current out at a node, not the Loop Rule. Mass and momentum play no role in this static circuit calculation, leaving energy conservation as the only principle that matches the Loop Rule statement.

Final Answer: Conservation of Energy.