Electric Resistance, Ohm’s Law, and Resistor Networks

Resistance: Concept and Classification

• Resistance = opposition within any material to movement/flow of charge
  • Analogous to friction, air resistance, viscous drag → motion is opposed
• Classification of materials
  • Conductors: “almost no” resistance (e.g.
  • Copper core of wire)
  • Insulators: very high resistance (e.g.
  • Plastic wire coating)
  • Resistors: intermediate, purposely used to control current

Properties of Resistors

• Four governing variables: resistivity $\rho$, length $l$, cross-sectional area $A$, and temperature $T$
• Fundamental formula (geometric factors): $R = \rho \frac{l}{A}$
  • $R$ = resistance (Ω)
  • $\rho$ = resistivity (Ω·m)
  • $l$ = length (m)
  • $A$ = cross-sectional area (m²)

Resistivity (intrinsic)

• Measures a material’s inherent opposition to current
• Lower $\rho$ → better conductor (Cu < Ag < Al)
• Higher $\rho$ → better insulator (glass, rubber, ceramics)
• SI unit: Ω·m
• Practical example: copper wire with plastic insulation → copper chosen for its low $\rho$

Length

• Direct proportionality $R \propto l$
• Doubling $l$ doubles $R$
• Physical picture: electrons travel farther through resistive lattice → more collisions

Cross-Sectional Area

• Inverse proportionality $R \propto \frac{1}{A}$
• Doubling $A$ halves $R$
• Interpretation: more conduction pathways (like widening a river → less water resistance)
• Important caveat: electric current does NOT obey fluid continuity $A<em>1v</em>1=A<em>2v</em>2$; instead uses Kirchhoff’s laws

Temperature Dependence

• Most conductors: $R$ increases with $T$ (thermal oscillations hinder electron flow)
  • Expressed by taking $\rho=\rho(T)$
• Notable exceptions (reverse/complex behaviour):
  • Glass, pure Si, most semiconductors
  • Superconductors: $R \rightarrow 0$ below critical $T_c$

Ohm’s Law

• Relates V (potential drop), I (current), R (resistance)
  $V = IR$
• Implications
  • For fixed $R$: $V \propto I$; doubling current doubles voltage drop
  • Applies to a single resistor, any segment, or entire circuit (after replacing by equivalent resistance)
• Energy perspective: resistance creates energy loss → drop in electrical potential

Internal Resistance & Real Voltage Sources

• Real batteries/cells possess small internal resistance $r_{int}$
  • Effective delivered voltage:
    $V = \mathcal{E}<em>{cell} - I r</em>{int}$
  • $\mathcal{E}_{cell}$: electromotive force (EMF, open-circuit voltage)
• Special cases
  • Open switch ($I=0$) → $V = \mathcal{E}_{cell}$
  • Discharging battery: current exits (+) high-potential terminal, returns to (–) terminal
  • Rechargeable (secondary) cells: act as galvanic (discharge) vs. electrolytic (charge) systems

Electrical Power in Circuits

• General definition: $P = \frac{W}{t} = \frac{\Delta E}{t}$
• For resistors (energy dissipation): $P = IV = I^2R = \frac{V^2}{R}$
  • Inter-conversion via Ohm’s law
• Example application: toaster coils glow red → convert electrical energy → thermal due to high R

Resistor Configurations

Series Connection

• Single path; same current flows through every resistor $I<em>{series} = I</em>1 = I_2 = …$
• Voltage drops add:
  $V<em>s = V</em>1 + V<em>2 + … + V</em>n$
• Resistances add:
  $R<em>s = R</em>1 + R<em>2 + … + R</em>n$
• Equivalent resistance $R_s$ ALWAYS grows when adding more series resistors
• Kirchhoff’s loop rule enforces net potential drop = EMF
• Worked series example (5 V cell, 3 Ω, 5 Ω, 7 Ω):
  • $R_s = 3+5+7 = 15\,\Omega$
  • $I = \frac{5\,V}{15\,\Omega}=0.33\,A$ (through every element)
  • Individual drops: $V<em>{3\Omega}=1.0\,V$; $V</em>{5\Omega}=1.67\,V$; $V_{7\Omega}=2.33\,V$ (sum 5 V)

Parallel Connection

• Common high-potential node & low-potential node → identical voltage across each branch:
  $V<em>p = V</em>1 = V<em>2 = … = V</em>n$
• Equivalent resistance (reciprocal sum): $\frac{1}{R<em>p}=\frac{1}{R</em>1}+\frac{1}{R<em>2}+…+\frac{1}{R</em>n}$
  • $R_p$ is always LOWER than the smallest individual resistor
• Current division inversely proportional to branch resistance (Ohm + Kirchhoff’s junction rule)
  • If $R<em>2=2R</em>1$ → $I<em>2 = \frac{1}{2}I</em>1$
• Special cases
  • Two equal resistors $R$ in parallel → $R_p = \frac{R}{2}$
  • $n$ identical resistors $R$ in parallel → $R_p = \frac{R}{n}$
• River/waterfall analogy: multiple streams drop same height although paths differ
• Worked unequal-parallel example (10 V source, 5 Ω & 10 Ω):
  • $\frac{1}{R<em>p}=\frac{1}{5}+\frac{1}{10}=\frac{3}{10}$ ⇒ $R</em>p=\frac{10}{3}\,\Omega$
  • Total current $I_{tot}=\frac{10}{10/3}=3\,A$
  • Branch currents: $I<em>{5\Omega}=\frac{10}{5}=2\,A$; $I</em>{10\Omega}=\frac{10}{10}=1\,A$ (sum 3 A)

Additional Conceptual Links

• Kirchhoff’s Laws
  • Junction rule: $\sum I<em>{in} = \sum I</em>{out}$ (charge conservation)
  • Loop rule: $\sum V = 0$ around closed loop (energy conservation)
• Continuity vs. circuit current
  • Fluids: $A v = \text{const}$ (incompressibility) – NOT valid for electrical branching
• Engineering/medical analogies
  • Parallel resistors ≈ extra traffic lanes/cardiac bypass → lowers overall “congestion” (R)
• Ethical & practical implications
  • Proper conductor sizing critical for safety (overheating, fire)
  • Efficient power delivery requires minimizing unnecessary R (e.g., high-voltage transmission lines)