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: $I<em>S = I</em>1 = I<em>2 = I</em>3$ (same everywhere).
  • Voltage: $V<em>S = V</em>1 + V<em>2 + V</em>3$.
  • Equivalent resistance: $R<em>T = R</em>1+R<em>2+R</em>3$.
  • Overall: $I<em>S = \dfrac{V</em>S}{R_T}$.

Parallel Circuits

• Two or more independent paths; current splits at junctions.
• Laws:
  • Voltage: $V<em>S = V</em>1 = V<em>2 = V</em>3$ (same across each branch).
  • Current: $I<em>S = I</em>1 + I<em>2 + I</em>3$.
  • Equivalent resistance: $\dfrac{1}{R<em>P}=\dfrac{1}{R</em>1}+\dfrac{1}{R<em>2}+\dfrac{1}{R</em>3}$.
  • Overall: $I<em>S = \dfrac{V</em>S}{R_P}$.

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, $\sum I<em>{\text{in}} = \sum I</em>{\text{out}}$.
• 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: $R<em>{\text{tot}}=R</em>L + r = 10.0 + 0.50 = 10.5\,\Omega$.
  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

• $I = \dfrac{Q}{t}$  $V = IR$  $R<em>T^{\text{series}} = \sum R</em>i$  $\dfrac{1}{R<em>P}=\sum \dfrac{1}{R</em>i}$
• $P = VI = I^2 R = \dfrac{V^2}{R}$  $E = P t$  $V_{ab}=\varepsilon - I r$