Kirchhoff's LawsJEE Main

Two-loop circuit — junction & loop rules — interactive Physics simulation for IIT-JEE.

Concept

Two rules solve any circuit. Junction rule (KCL): charge conservation — currents into a node sum to zero. Loop rule (KVL): energy conservation — EMFs and IR drops around any closed loop sum to zero. Assign current directions freely; a negative answer just means the real current flows the other way.

Key formula

\sum_{node} I = 0, \qquad \sum_{loop} \varepsilon = \sum_{loop} IR, \qquad V_{node} = \frac{\sum \varepsilon_i/R_i}{\sum 1/R_i} \;(\text{Millman})

Derivation

For the two-loop circuit here: let the junction potential be V (bottom rail = 0).

Each source branch delivers $i = (\varepsilon - V)/R$ ; the middle branch drains $V/R_3$ . KCL: $\frac{\varepsilon_1 - V}{R_1} + \frac{\varepsilon_2 - V}{R_2} = \frac{V}{R_3}$ — solve once for V (Millman's theorem), then all three currents follow.

Scenarios to explore

Kirchhoff's Laws — Junction & loop rules on a live two-loop circuit.

Real-world applications

Any multi-source network: automotive wiring, battery banks.
Charging circuits — detect when a weaker battery is being charged (its power < 0).
Circuit simulators (SPICE) are industrial-scale KCL solvers.

JEE exam tips

Node-potential (Millman) method beats loop equations when many branches share two nodes.
If ε₂ < V_node, battery 2 is being CHARGED — its current runs backward through it.
Power check: Σ(power from sources) = Σ(I²R) — free verification of your algebra.

Common mistakes

Sign chaos — fix a traversal direction and stick to it: EMF + when going − → + inside the source.
Writing more loop equations than independent loops (branches − nodes + 1).
Forgetting a negative current answer is valid — direction was guessed wrong, magnitude is right.

Exam traps to avoid

KVL fails in changing magnetic fields (induced EMF) — that's Faraday territory.
Internal resistance belongs INSIDE the loop equation: ε − Ir − IR = 0.