Current & Ohm's Law
Cylindrical Conductor — From Physics to Circuits
Electric Current
I = dQ/dt [A = C/s]
Current density (per unit area):
J = I/S [A/m²]
J = σ·E (conduction current)
J = σ·E (conduction current)
Ohm's Law
V = I·R [V]
R = L/(σ·S) = ρ·L/S [Ω]
R = L/(σ·S) = ρ·L/S [Ω]
σ = conductivity [S/m], ρ = resistivity = 1/σ [Ω·m]
💡 Valid only for uniform cross-section conductors.
Power
P = V·I = I²R = V²/R [W]
If P > 0 → power absorbed (load).
If P < 0 → power supplied (source).
Conductivity Table
| Material | σ [S/m] |
|---|---|
| Gold | 4.1×10⁷ |
| Aluminum | 4.0×10⁷ |
| Silicon | 4.4×10⁻⁴ |
| Glass | 10⁻¹² |
Series & Parallel Resistance
Series Resistors — same current
Parallel Resistors — same voltage
Series Combination
R_eq = R₁ + R₂ + R₃ + …
Same current through all. Total V = sum of individual V drops. R_eq > any single resistor.
Parallel Combination
1/R_eq = 1/R₁ + 1/R₂ + 1/R₃
Two resistors: R_eq = R₁R₂/(R₁+R₂)
Two resistors: R_eq = R₁R₂/(R₁+R₂)
Same voltage across all. Total I = sum of branch currents. R_eq < smallest resistor.
Open & Short Circuits
Open circuit: R = ∞, I = 0
Short circuit: R = 0, V = 0
Short circuit: R = 0, V = 0
💡 If a resistor is shorted (R=0) in parallel with others, the whole parallel combination = 0. All current flows through the short.
📝 Real-world Example — Transmission Line
AWG aluminum cable: σ=4×10⁷ S/m, A=107 mm², L=100 km. System delivers 10 MW at V=200 kV, I=50 A.
R_TL = L/(A·σ) = 100×10³ / (107×10⁻⁶ × 4×10⁷) = 23.36 Ω
P_loss = I²R = 50² × 23.36 = 58,400 W ≈ 58.4 kW
Loss% = 58.4 / (10000 + 58.4) = 0.58% ← High voltage = low loss!
P_loss = I²R = 50² × 23.36 = 58,400 W ≈ 58.4 kW
Loss% = 58.4 / (10000 + 58.4) = 0.58% ← High voltage = low loss!
EMF, Batteries & Internal Resistance
Real Battery with Internal Resistance r
Ideal vs Real Source
Ideal: V_terminal = V_emf
Real: V_terminal = V_emf − I·r
Real: V_terminal = V_emf − I·r
r is the internal resistance. Higher current drawn → larger voltage drop inside → lower terminal voltage.
Power Distribution
P_total = I·V_emf
P_load = I²·R_L
P_internal = I²·r
P_load = I²·R_L
P_internal = I²·r
Total power = load power + internal losses.
Special Cases
Open circuit: I = 0, V_terminal = V_emf (no drop).
Short circuit: V = 0, I = V_emf/r (maximum current).
💡 These are the two extremes of the V-I characteristic line for a source.