Kirchhofff's Rules

Electric circuits generally consist of a number of resistors and cells interconnected sometimes in a complicated way. The formulae we have derived earlier for series and parallel combinations of resistors are not always sufficient to determine all the currents and potential differences in the circuit. Two rules, called Kirchhoff’s rules, are very useful for analysis of electric circuits.

KIRCHHOFF’S RULES

Kirchhoff’s first Law or Kirchhoff’s Junction Law or Kirchhoff’s current law: Algebraic sum of the currents meeting at a junction in a closed circuit is zero.

According to law total, charge at the junction is zero i.e. -I₁ + (-I₂) + I₃ + (-I₄) + I₅ = 0

∑ I = 0

First law supports law of conservation of charge. At time t current coming is equal to current going i.e.

I₁t = I₂t => q₁=q₂

Sign convention: current coming towards junction is positive and current going away is negative.

Kirchhoff’s laws are applicable to AC as well as DC circuits.

Kirchhoff’s second Law or Kirchhoff’s loop Law or Kirchhoff’s voltage law: Algebraic sum of changes in potential around any closed path of electric circuit (or closed loop) involving resistors and cells in the loop is zero i.e.

∑∆V = 0, also ∑ ε = ∑ IR

In loop ABEFA, according to Kirchhoff’s rule

I₃R₂ + I₁R₁ – ε₁ =0, or ε₁ = I₃R₂ + I₁R₁

In loop ABCDEFA, according to Kirchhoff’s rule

ε_{2 -} I₃R₂ + I₁R₁ – ε₁ =0, or ε₁ - ε₂  = I₁R₁ - I₃R₂

Second law supports law of conservation of energy. We can also say it follows that electrostatic fore is a conservative force and work done in closed loop is zero.

Sign convention: e.m.f of a cell is taken negative if one moves in the direction of increasing potential and is taken as positive if one moves in the direction of deceasing potential.

Product of resistor and current is taken as positive if current coming from positive terminal and is taken as negative if current is coming from negative terminal.

∑ε=∑IR is true when there is no capacitor in the circuit.