Kirchhoff's Laws

• Overview: Continued discussion on Kirchhoff's laws in the context of electrical circuits.

Kirchhoff's Laws

• Kirchhoff’s Junction Law (KCL):

  • Definition: The total current entering a junction equals the total current leaving the junction. This is a statement of the conservation of electric charge.

  • Formula: $\sum I<em>{in} = \sum I</em>{out}$

• Kirchhoff's Loop Law (KVL):

  • Definition: The sum of the electrical potential differences (voltage) around any closed circuit loop is zero. This represents the conservation of energy.

  • Formula: $\sum V = 0$

Sign Conventions

• Voltage Drops:

  • Upstream: Voltage rises ($+V$), moving against the current.

  • Downstream: Voltage drops ($-V$), moving with the current.

• Usage of voltage changes in equations:

  • KVL and sign conventions must be followed to ensure correctness of calculations.

Ohm’s Law

• Definition: The relationship between voltage ($V$), current ($I$), and resistance ($R$) represented by the equation:
  $V = IR$

• Application: Identify how voltages and currents relate to resistances in a series or parallel circuit.

Circuit Analysis with Kirchhoff's Laws

• Multi-loop circuits must incorporate both KCL and KVL, leading to multiple equations:

  • KCL: $\sum I<em>{in} = \sum I</em>{out}$

  • KVL for voltage: $\sum V_{ab} = 0$

• Example of applying both rules in practice:

  • Given a circuit with resistors and current paths, draw loop equations and calculate unknown currents or resistances.

Example Problems

• Practice Problems:

  • Write down Kirchhoff’s loop equations to solve for currents in multi-loop circuits.

  • Discover equivalent resistances across different configurations.

Battery and Electromotive Force (EMF)

• Definition of EMF:

  • The voltage generated by a battery or power source not accounting for its internal resistance.

• Terminal Voltage:

  • Definition: The voltage available at the terminals of the battery for external use, calculated as:
    $V<em>{terminal} = \mathcal{E} - Ir</em>{int}$

  • Where:

  • $\mathcal{E}$ = EMF (electromotive force) of the battery

  • $I$ = current drawn from the battery

  • $r_{int}$ = internal resistance of the battery.

• Real vs. Ideal Batteries:

  • Ideal Battery: $V_{terminal} = \mathcal{E}$

  • Real Battery: \mathcal{E} > V_{terminal}

  • Heat losses due to internal resistance affect terminal voltage when current is drawn.

Electrical Measuring Instruments

Ammeter

• Purpose: Measures current flowing through a circuit element.

• Connection: Must be connected in series with the circuit.

• Requirements: Should have very low resistance to prevent significant alteration of current.

Voltmeter

• Purpose: Measures the voltage across a circuit element.

• Connection: Should be connected in parallel to the circuit.

• Requirements: Should have very high resistance to avoid drawing current from the circuit and thus altering the voltage reading.

Current Distribution and Power

• Principle: Current splits at junctions based on resistance (Ohm’s Law).

• Power Dissipation: Energy lost as heat due to internal resistance is calculated as:
  $P = I^2 r_{int}$

Summary of Formulas and Concepts

1. Ohm's Law ($V = IR$)

   • Concept: Defines the linear relationship between voltage and current through a conductor. It shows that current is proportional to voltage and inversely proportional to resistance.

2. Kirchhoff's Junction Law ($\sum I<em>{in} = \sum I</em>{out}$)

   • Concept: Based on the conservation of charge. It ensures that no charge is lost or accumulated at a single point (node) in a circuit.

3. Kirchhoff's Loop Law ($\sum V = 0$)

   • Concept: Based on the conservation of energy. It states that the total energy supplied by the source in a loop is exactly equal to the energy consumed by the components.

4. Terminal Voltage ($V<em>{terminal} = \mathcal{E} - Ir</em>{int}$)

   • Concept: Explains why the usable voltage from a real battery is lower than its theoretical EMF. The term $Ir_{int}$ represents the internal "lost" voltage due to the battery's own resistance.

5. Power Dissipation ($P = I^2 r$)

   • Concept: Describes the rate at which electrical energy is converted into thermal energy (heat) within a resistor or internal battery resistance.