Kirchhoff’s Rules 🚀

1. Why do we need them?

  • Series-and-parallel shortcuts break down when circuits look like spaghetti. 🌀
  • Kirchhoff’s two rules let you track every current and voltage, no matter how tangled the connections. :contentReference[oaicite:0]{index=0}

2. First, label the circuit ✍️

  • Draw an arrow on each resistor and mark that current as I. If your final answer for I is positive, the real flow matches the arrow; if it is negative, current actually runs opposite to the arrow. :contentReference[oaicite:1]{index=1}
  • On every source, mark the positive (P) and negative (N) terminals and choose a direction for the current through the cell. :contentReference[oaicite:2]{index=2}
  • Moving from N → P through a cell gives the voltage jump \(V = \mathcal{E} – I r\). Moving from P → N gives \(V = \mathcal{E} + I r\). :contentReference[oaicite:3]{index=3}

3. The two golden rules 🌟

Junction Rule (Current Rule) 🚦

The total current flowing into a junction equals the total current flowing out:

\[\sum I_{\text{in}} \;=\; \sum I_{\text{out}}\]

No charge piles up, so whatever flows in must flow out immediately. :contentReference[oaicite:4]{index=4}

Loop Rule (Energy Rule) 🔄

As you walk once around any closed loop, the algebraic sum of all potential changes is zero:

\[\sum \Delta V \;=\; 0\]

You finish where you started, so your net energy change is zero. :contentReference[oaicite:5]{index=5}

4. Parallel cells shortcut 🔋

For n cells (emfs \(\mathcal{E}_1,\;\mathcal{E}_2,\dots\) and internal resistances \(r_1,r_2,\dots\)) wired in parallel:

\[\frac{1}{r_{\text{eq}}} \;=\; \frac{1}{r_1} + \frac{1}{r_2} + \dots\] \[\frac{\mathcal{E}_{\text{eq}}}{r_{\text{eq}}} \;=\; \frac{\mathcal{E}_1}{r_1} + \frac{\mathcal{E}_2}{r_2} + \dots\]

This pair lets you collapse many cells into one “super-cell.” :contentReference[oaicite:6]{index=6}

5. Worked examples 🧮

Cube of resistors

  • 12 identical \(1\;\Omega\) resistors form a cube.
  • Connect a \(10\;\text{V}\) battery across opposite corners.
  • Symmetry shows the three edges leaving the positive corner all carry the same current \(I\).
  • Loop rule around face ABCC′EA gives \(\mathcal{E} = \tfrac{5}{2} I R\) → \(I = 4\;\text{A}\).
  • Equivalent resistance \(R_{\text{eq}} = \tfrac{5}{6}\;\Omega\); total current \(12\;\text{A}\) splits as 4 A along each symmetric edge. :contentReference[oaicite:7]{index=7}

Three-loop network

  • Apply the junction rule to assign three unknown currents \(I_1, I_2, I_3\).
  • Write loop equations:
    \(7 I_1 – 6 I_2 – 2 I_3 = 10\)
    \(I_1 + 6 I_2 + 2 I_3 = 10\)
    \(2 I_1 – 4 I_2 – 4 I_3 = -5\)
  • Solve to find \(I_1 = 2.5\;\text{A},\; I_2 = \tfrac{5}{8}\;\text{A},\; I_3 = \tfrac{7}{8}\;\text{A}.\) :contentReference[oaicite:8]{index=8}

6. Quick tips 💡

  • Count unknown currents. Write the same number of independent equations (junction + loop) to solve them.
  • Choose loop directions consistently (clockwise or anticlockwise) to keep sign errors away.
  • If symmetry exists, use it to cut down on algebra. 🎯 :contentReference[oaicite:9]{index=9}

Important NEET Nuggets 🏆

  1. Junction rule embodies charge conservation—master it for combination circuits. :contentReference[oaicite:10]{index=10}
  2. Loop rule flows from energy conservation—essential for mixed sources and resistors. :contentReference[oaicite:11]{index=11}
  3. Cell voltage changes: \(V = \mathcal{E} \pm I r\) questions appear often. :contentReference[oaicite:12]{index=12}
  4. Know the parallel-cell formulas for emergency back-up battery questions. :contentReference[oaicite:13]{index=13}
  5. Balanced Wheatstone bridge ⇒ zero galvanometer current—spot this to simplify problems fast. :contentReference[oaicite:14]{index=14}