Electric Current & DC Circuit Summary

Attention-Grabbers & Context

  • “ELECTRIFYING!!!, SHOCKING BUT TRUE!!!!, You will be ex-static!” – motivational phrases used to introduce the topic and signal excitement in Physics 11 (Mr. Stephenson, 2019).
  • Focus: Direct-current (DC) electricity, its physical basis, and quantitative laws used for circuit analysis.

Fundamental Law of Electric Charges

  • Key postulates:
    • Opposite charges attract; like charges repel.
    • Charged objects may attract neutral objects (polarization effects).
  • Elementary charges:
    • Electron charge: q_e=-1.6\times10^{-19}\,\text{C}
    • Proton charge: q_p=+1.6\times10^{-19}\,\text{C}
    • Neutron: q_n = 0
  • All matter comprises protons, neutrons, electrons.
  • Coulomb (C) = unit of electric charge.

Voltage (Electric Potential Difference)

  • A battery’s internal chemical reaction separates charge, creating positive & negative terminals.
  • Definition: Voltage (V) = electric potential energy per unit charge.
    • V = \dfrac{E_{\text{potential}}}{Q}
    • Unit: volt (V); 1\,\text{V}=1\,\text{J}/\text{C}
  • Provides electrons in the external circuit with electric potential energy that can be converted to other forms by loads.

Electric Current

  • Two conventions:
    • Conventional current: flow of positive charge from + to –.
    • Electron flow (the physical reality): electrons move from – to +.
  • Definition: I = \dfrac{Q}{t}
    • Unit: ampere (A); 1\,\text{A}=1\,\text{C}/\text{s}
  • Charge–electron conversion example:
    • Number of electrons in -1\,\text{C}:
      \dfrac{-1\,\text{C}}{-1.6\times10^{-19}\,\text{C/e¯}} \approx 6.2\times10^{18}\,\text{electrons}

Resistance (R)

  • Opposes the flow of electric charge; arises from collisions between charge carriers and lattice atoms.
  • Consequences: charges lose electric potential energy → thermal energy, light, etc.
  • Unit: ohm (Ω).
  • Physical examples: light-bulb filament, stove heating element; commercial resistors with color-code bands.

Ohm’s Law

  • Relationship between voltage, current, resistance:
    • V = IR
    • Linear for ohmic materials; slope in an I–V graph equals resistance.

Worked Examples (Ohm’s Law)

  • Light bulb, R=20\,\Omega,\;V=5.0\,\text{V}:
    • I = \dfrac{V}{R} = \dfrac{5.0}{20}=0.25\,\text{A}
  • Motor, R=75\,\Omega,\;V=12\,\text{V}:
    • I = \dfrac{12}{75}=0.16\,\text{A}
  • Unknown battery voltage, I=0.80\,\text{A},\;R=25\,\Omega:
    • V = IR =0.80\times25 = 20\,\text{V}

Standard Circuit Symbols (Table 3.1 excerpts)

  • Cell: long line = +, short line = – ; represents a source of electric potential.
  • Battery: several cells in series.
  • Conducting wire: straight line.
  • Load/Resistor: zig-zag line (R, Ω).
  • Switch (open/closed): break or continuous line with pivot.
  • Voltmeter: circle with V (measures V).
  • Ammeter: circle with A (measures I).

Types of Circuits

Series Circuits

  • Single path for current.
  • Laws:
    • Current: IS = I1 = I2 = I3 (same everywhere).
    • Voltage: VS = V1 + V2 + V3.
    • Equivalent resistance: RT = R1+R2+R3.
    • Overall: IS = \dfrac{VS}{R_T}.

Parallel Circuits

  • Two or more independent paths; current splits at junctions.
  • Laws:
    • Voltage: VS = V1 = V2 = V3 (same across each branch).
    • Current: IS = I1 + I2 + I3.
    • Equivalent resistance: \dfrac{1}{RP}=\dfrac{1}{R1}+\dfrac{1}{R2}+\dfrac{1}{R3}.
    • Overall: IS = \dfrac{VS}{R_P}.

Combination (Series–Parallel)

  • Real-world circuits often mix both configurations; analyze by reducing step-wise to a single equivalent resistance.

Kirchhoff’s Laws (Advanced Circuit Analysis)

  • Kirchhoff’s Current Law (KCL):
    • At any junction, \sum I{\text{in}} = \sum I{\text{out}}.
  • Kirchhoff’s Voltage Law (KVL):
    • For any closed loop, \sum (\Delta V) = 0 (sum of rises and drops is zero).
  • Together with Ohm’s law and series/parallel R rules, allow complete solution of complex circuits.

Effects of Adding Loads & Safety Considerations

  • Adding resistors in parallel ↓ R_T → ↑ total current.
  • Excessive current risks:
    • Device damage, wire overheating, fire.
    • Mitigation: fuses & circuit breakers (open the circuit if current exceeds a preset threshold).
  • Short circuit = unintended low-resistance path → very high I, causing shock & burns.

Electric Power

  • Definition: P = \dfrac{E}{t}.
  • Unit: watt (W); 1\,\text{W}=1\,\text{J/s}.
  • Electrical expressions:
    • Basic: P = VI.
    • Substitutions via Ohm’s Law:
    • P = I^2 R (replace V).
    • P = \dfrac{V^2}{R} (replace I).
  • Dimensional check: (J/C)\times(C/s)=J/s=W.

Power Examples

  • 9 V battery, I=0.20\,\text{A}:
    • P = VI = 9\times0.20 = 1.8\,\text{W}.
  • Wire loss, I=50\,\text{A},\;R=0.10\,\Omega:
    • P = I^2R=(50)^2\times0.10 = 250\,\text{W}.

Electric Energy & Billing

  • Kilowatt-hour (kWh): energy used when P=1\,\text{kW} for 1\,\text{h}.
    • 1\,\text{kWh} = 1000\,\text{W}\times3600\,\text{s}=3.6\times10^{6}\,\text{J}.
  • Energy calculation: E = P t (convert units as needed).
  • Example: Hair-dryer P=1200\,\text{W}=1.2\,\text{kW}, run 20 min = \tfrac{1}{3}\,\text{h}.
    • E = 1.2\,\text{kW}\times\tfrac{1}{3}\,\text{h}=0.40\,\text{kWh}.
    • Cost at 0.25\,\$/\text{kWh}: 0.40\times0.25=\$0.10.

Electromotive Force (emf), Internal Resistance & Terminal Voltage

  • emf (symbol \varepsilon or \xi): open-circuit voltage of a source (no current drawn).
  • Real sources possess internal resistance r.
  • When current I flows, internal drop I r lowers terminal voltage V_{ab}:
    • V_{ab}=\varepsilon - I r.
  • Diagram conventions: positive terminal at higher potential; internal r in series with ideal emf source.

emf Example (Page 22)

  • Given battery \varepsilon = 12.0\,\text{V}, internal r=0.50\,\Omega, external load R_L = 10.0\,\Omega (labeled 10.5 Ω total when including r):
    1. Total resistance: R{\text{tot}}=RL + r = 10.0 + 0.50 = 10.5\,\Omega.
    2. Current: I = \dfrac{\varepsilon}{R_{\text{tot}}} = \dfrac{12.0}{10.5}=1.14\,\text{A}.
    3. Terminal voltage: V_{ab}=\varepsilon - I r = 12.0 - (1.14)(0.50)=11.4\,\text{V}.

Practical & Ethical Considerations

  • Understanding circuit behavior is critical to safe household wiring, appliance design, and prevention of electrical hazards.
  • Engineers must weigh efficiency (minimizing I^2 R losses) against safety and cost.
  • Awareness of internal resistance helps in battery selection for sensitive electronics.

Quick Reference: Key Equations

  • I = \dfrac{Q}{t}  V = IR  RT^{\text{series}} = \sum Ri  \dfrac{1}{RP}=\sum \dfrac{1}{Ri}
  • P = VI = I^2 R = \dfrac{V^2}{R}  E = P t  V_{ab}=\varepsilon - I r