Study Notes on DC Circuits

Chapter 19: DC Circuits

Section 19-1: EMF and Terminal Voltage

  • Electric circuits require batteries or generators to produce current, referred to as sources of electromotive force (emf) denoted as ε.
  • A battery is described as a nearly constant voltage source but has a small internal resistance, which diminishes the actual output voltage.
  • The relationship between the ideal emf and terminal voltage is expressed as:
    • V_{ab} = E - I r
    • Where:
    • V_{ab} = terminal voltage
    • E = ideal emf
    • I = current
    • r = internal resistance.
  • The emf is equivalent to the battery voltage under conditions when no current flows through the circuit.

Section 19-2: Resistors in Series and in Parallel

Resistors in Series

  • All resistors in series have the same current flowing through them.
  • Voltage across each resistor can be expressed as:
    • V = V₁ + V₂ + V₃
  • Each voltage across a resistor can be represented as:
    • V₁ = IR₁
    • V₂ = IR₂
    • V₃ = IR₃
  • Therefore, the overall voltage in a series circuit can be summarized as:
    • V = I(R₁ + R₂ + R₃) = IR_S
  • The equivalent resistance for resistors in series (R_S) is calculated as:
    • R_S = R₁ + R₂ + R₃
  • Adding more resistors in a series configuration increases the total resistance.
  • Implication: If one light bulb in a series circuit fails, the entire circuit is broken, leading to all bulbs going out.
    • This configuration is not practical for household wiring due to the dependency on the continuity of all components.

Resistors in Parallel

  • Resistors in parallel experience the same voltage across each:
    • V = V₁ = V₂ = V₃
  • The total current (I) across the parallel circuit is the sum of the currents through each parallel path:
    • I = I₁ + I₂ + I₃
  • The relationship with resistances in parallel can be given by:
    • rac{1}{R_{P}} = rac{1}{R₁} + rac{1}{R₂} + rac{1}{R₃}
  • This indicates that adding more resistors in a parallel setting leads to a decrease in total (equivalent) resistance:
    • Note: The special case for just two resistors in parallel gives:
    • R_P = rac{R₁ R₂}{R₁ + R₂}
  • Household circuits typically use a parallel configuration to ensure that each appliance receives full voltage and one appliance's failure does not affect others.

Section 19-3: Kirchhoff's Rules

  • Useful for circuits that can't be simply reduced to series or parallel configurations.

Junction Rule

  • The sum of currents entering any junction is equal to the sum of currents leaving that junction:
    • Example: I1 = I2 + I_3

Loop Rule

  • The sum of the voltages around a closed loop is zero:
    • Sign conventions for battery and resistors are critical here:
    • For a battery moving against the emf, utilize positive; with current through resistor, use negative.

Problem Solving with Kirchhoff's Rules

  1. Label currents with directions across different branches.
  2. Apply junction and loop rules, ensuring the equations amount to the number of unknowns.
  3. Resolve for unknown currents, adjusting if a negative current value arises, indicating an opposite flow direction.

Example Problem Using Kirchhoff's Rules

  • Current through given circuit branches can be evaluated via established rules and solving techniques, leading to results that can then be analyzed.

Section 19-5: Circuits Containing Capacitors in Series and Parallel

Characteristics of Capacitors

  • Capacitors in series maintain the same charge, while those in parallel share a uniform voltage across all.
  • The equivalent capacitance for capacitors in series is represented by:
    • rac{1}{C_S} = rac{1}{C₁} + rac{1}{C₂} + …
  • Conversely, in parallel:
    • C_P = C₁ + C₂ + C₃ + …
  • Capacitors exhibit a decline in total capacitance when arranged in series, contrasting with resistors.

Section 19-6: RC Circuits - Resistor and Capacitor in Series

  • An RC circuit consists of a resistor and a capacitor in series, affecting the current and voltage behavior.
  • On closing the switch at time t=0:
    • I = rac{E}{R} (initial current)
    • V_C = 0 (initial voltage across capacitor)
  • As time progresses, the capacitor begins to charge exponentially:
    • Relationship to charge, current, voltage:
    • I = rac{E}{R} e^{-t/RC}
    • V_C = E (1 - e^{-t/RC})
    • Q = C imes V_C
  • Time constant for charging a capacitor defined as:
    • au = RC
  • Discharging follows an exponential decay pattern:
    • Relationships during discharge:
    • VC = V0 e^{-t/ au}
    • Q = Q_0 e^{-t/ au}
    • I = rac{V_0}{R} e^{-t/ au}

Section 19-7: Electric Hazards

  • Currents inflicting damage arise from sufficiently high voltages allowing dangerous current flows, particularly through water.
  • Currents of 10 to 100 mA can lead to disruption or greater bodily harm.
  • Household wiring occurs in parallel and is safeguarded via fuses or circuit breakers to avoid excess heat from overloads.
  • Circuit safety is enhanced through the use of dedicated ground lines, with identifiable three-prong plugs indicating their use.
  • Solid precautions are necessary to ensure electrical safety in environments where exposure to moisture is plausible, highlighting the need for Ground Fault Interrupter (GFI) outlets in wet areas.