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 A1v1=A2v2; 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}{cell} - I r{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{series} = I1 = I_2 = …
• Voltage drops add:
  Vs = V1 + V2 + … + Vn
• Resistances add:
  Rs = R1 + R2 + … + Rn
• 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{3\Omega}=1.0\,V; V{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:
  Vp = V1 = V2 = … = Vn
• Equivalent resistance (reciprocal sum): \frac{1}{Rp}=\frac{1}{R1}+\frac{1}{R2}+…+\frac{1}{Rn}
  • R_p is always LOWER than the smallest individual resistor
• Current division inversely proportional to branch resistance (Ohm + Kirchhoff’s junction rule)
  • If R2=2R1 → I2 = \frac{1}{2}I1
• 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}{Rp}=\frac{1}{5}+\frac{1}{10}=\frac{3}{10} ⇒ Rp=\frac{10}{3}\,\Omega
  • Total current I_{tot}=\frac{10}{10/3}=3\,A
  • Branch currents: I{5\Omega}=\frac{10}{5}=2\,A; I{10\Omega}=\frac{10}{10}=1\,A (sum 3 A)

Additional Conceptual Links

• Kirchhoff’s Laws
  • Junction rule: \sum I{in} = \sum I{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)