Lecture Notes on Circuits and DC Instruments

21.1 Resistors in Series and Parallel

  • Series Circuits

    • Definition: Resistors are in series when the flow of charge (current) must flow through devices sequentially.

    • Characteristics: The total resistance in a series circuit is the sum of all resistances, and the current is the same through each component.

  • Example 21.1: Analysis of a Series Circuit

    • Given Data:

    • Voltage output of battery: $V = 12.0 ext{ V}$

    • Resistances:

      • $R_1 = 1.00 ext{ Ω}$

      • $R_2 = 6.0 ext{ Ω}$

      • $R_3 = 13.0 ext{ Ω}$

    • Questions to Analyze:

    • (a) Find the total resistance.

    • (b) Find the current.

    • (c) Calculate the voltage drop in each resistor and verify that these add up to the voltage output of the source.

    • (d) Calculate the power dissipated by each resistor.

    • (e) Find the power output of the source and confirm it equals the total power dissipated.

  • Parallel Circuits

    • Definition: A parallel connection splits the current; the voltage across each resistor is the same.

    • Total resistance in parallel:

    • Equation: rac{1}{R_{eq}} = rac{1}{R_1} + rac{1}{R_2} + rac{1}{R_3}

  • Example 21.2: Analysis of a Parallel Circuit

    • Given Data:

    • Voltage: $V = 12.0 ext{ V}$

    • Resistances:

      • $R_1 = 1.00 ext{ Ω}$

      • $R_2 = 6.00 ext{ Ω}$

      • $R_3 = 13.0 ext{ Ω}$

    • Questions for Analysis:

    • (a) Find the total resistance.

    • (b) Find the total current.

    • (c) Calculate the currents in each resistor to confirm they add to the total current.

    • (d) Calculate the power dissipated by each resistor.

    • (e) Find the power output of the source and check for equilibrium with total dissipated power.

  • Household Circuits

    • Generally, household circuits are wired in parallel to ensure all devices receive the same voltage and can operate independently.

  • Example: Will a Fuse Blow?

    • (a) Calculate the total current drawn by all devices in a household circuit.

    • (b) Determine if a 15A fuse will blow with all devices operating simultaneously.

  • Combination of Series and Parallel Resistors

    • If resistors are arranged in a combination of series and parallel, use appropriate strategies to find equivalent resistance.

    • Example 21.3:

    • Given a circuit with multiple resistors, analyze to find:

      • (a) Total resistance and current through the battery.

      • (b) Calculate the IR drop in each resistor.

      • (c) Find the current through each resistor.

      • (d) Determine power dissipated by each resistor.

21.2 Electromotive Force and Internal Resistance

  • Battery Characteristics:

    • Every voltage source has:

    • Electromotive Force (emf): Related to the source of potential difference.

    • Internal Resistance (r): Related to the construction of the source.

    • Terminal Voltage (V): Given by the equation:

    • V = ext{emf} - Ir

    • Here, $ ext{emf}$ represents the ideal voltage of the battery in an open circuit.

21.3 Kirchhoff’s Rules

  • Application of Kirchhoff’s Rules: Useful for analyzing complex circuits that cannot be reduced to series or parallel forms.

  • Kirchhoff's First Rule (Junction Rule):

    • The sum of the currents entering a junction equals the sum of the currents leaving the junction.

    • Equation: I_1 = I_2 + I_3

  • Kirchhoff's Second Rule (Loop Rule):

    • The sum of the voltages around any closed loop in a circuit must equal zero.

  • Resistor Potential Change:

    • When a resistor is encountered in the same direction as the current, the change in potential is negative:

    • Change in potential: -IR

    • When traversing against the current:

    • Change in potential: +IR

  • Emf Potential Change:

    • When traversing an emf from negative to positive terminal, the potential increases:

    • Change in potential: + ext{emf}

    • When moving from positive to negative terminal, the potential decreases:

    • Change in potential: - ext{emf}

  • Example 21.5: Calculating Current Using Kirchhoff's Rules:

    • Steps to follow:

    • Step 1: Label the circuit diagram with relevant values.

    • Step 2: Apply the junction rule to establish relationships between currents.

    • Step 3: Apply the loop rule for any closed loops present in the diagram.

    • Step 4: Solve the equations established from Step 2 and Step 3 to find unknown currents.

21.6 DC Circuits Containing Resistors and Capacitors

  • RC Circuits: Defined as circuits that contain a resistor ($R$) and a capacitor ($C$).

    • Function: Capacitors store electric charge.

  • Charging a Capacitor:

    • Voltage across a charging capacitor given by:

    • V = ext{emf} imes (1 - e^{-t/RC})

    • Time Constant ($ au$):

    • Given by: T = RC

    • Measured in seconds.

  • Discharging a Capacitor:

    • Voltage across a discharging capacitor is expressed as:

    • V = V_0 e^{-t/RC}

    • Where $V_0$ is the initial voltage across the capacitor.

  • Example Calculations of DC Circuits:

    • Problem inside the RC circuit involving resistance and capacitance.

    • Given values:

      • $R = 75.0 ext{ kΩ}$

      • $C = 25.0 ext{ μF}$

      • $E = 12.0 ext{ V}$

    • Determine the time constant and the charge on the capacitor after one time constant.

    • Time constant calculation:

      • T = RC = 75 imes 10^{3} imes 25 imes 10^{-6} = 1.875 ext{ s}

    • Charge at time $T$:

      • Q = CV = C V_{max}(1 - e^{-t/RC})

  • Another Example of Flash Circuit:

    • Parameters:

    • $V_0 = 275 ext{ V}$

    • $C = 125 ext{ μF}$

    • Tasks:

    • (a) Solve for resistance $R$ if capacitor charges to 90% in 15 seconds.

    • (b) Determine average current delivered during the discharge.

  • Complex Circuit Calculations:

    • Example: Given values to calculate voltages and currents at different points in the circuit, use voltage drops and current laws to find required variables.