Lecture on Kirchhoff's Junction and Loop Rules

Overview

• Application of Kirchhoff's junction rule and loop rule to calculate current in complex circuits.
• Focus on a problem involving a circuit with resistors in series and parallel arrangements.

Circuit Description

• Resistors:
  • R1 = 3 ohms
  • R2 = 4 ohms
  • R3 = 12 ohms
• Objective: Calculate the current flowing through each resistor using Kirchhoff's rules.

Kirchhoff's Rules

Junction Rule (Current Law)

• At any junction, the sum of currents entering equals the sum of currents leaving.
• Equation: ( I_1 = I_2 + I_3 )

Loop Rule (Voltage Law)

• The sum of all voltages around a closed loop must equal zero.
• Consider voltage contributions from potential differences:
  • Voltage drop: High potential to low potential.
  • Voltage lift: Low potential to high potential.

Sign Conventions

• Direction of current affects voltage drop or lift.
• Traveling with current: Voltage drop.
• Traveling against current: Voltage lift.

Application - Circuit Problem

Loop 1 Analysis

• Voltage lift: 24 volts (battery)
• Voltage across resistors:
  • For R1: ( V_1 = I_1 \times 3 )
  • For R2: ( V_2 = I_2 \times 4 )
• Equation: ( 24 = 3I_1 + 4I_2 )

Loop 2 Analysis

• Voltage lift across resistors:
  • For R2: ( V_2 = I_2 \times 4 )
  • For R3: ( V_3 = I_3 \times 12 )
• Equation: ( 0 = 4I_2 - 12I_3 )

Solving Equations

• Combine the equations using substitution and elimination to find:
  • ( I_2 = 3 \text{ amps} )
  • ( I_3 = 1 \text{ amp} )
  • ( I_1 = 4 \text{ amps} )

Circuit Verification

• Redraw and verify current flow and potential:
  • Check that sums of currents at junction equal.
  • Verify potential differences and consistency with Ohm's law.

Further Examples

• Additional circuit problems with multiple batteries and resistors.
• Emphasis on current direction and solving for unknowns in larger, more complex circuits.

Concepts Illustrated

• Predicting current direction based on battery strength.
• Importance of careful sign convention in solving.
• Calculating potential at various points using reference voltages.

Conclusion

• Utilizing Kirchhoff's rules to calculate currents and potentials in circuits.
• Importance of systematic approach to avoid calculation errors.