Resistor Networks, RC Time Constants, and Torque Explained
Three foundational engineering formulas — parallel resistance, RC charging time, and torque — worked through with real numbers.
These three formulas show up across nearly every basic circuits and mechanics course, each reducing a physical system to a single clean number.
Parallel resistance: less than either resistor
For two resistors in parallel: R_total = (R1 × R2) / (R1 + R2). A 220Ω and 330Ω resistor combined in parallel: (220 × 330) / (220 + 330) = 72,600 / 550 = 132Ω — notice this is lower than even the smaller 220Ω resistor alone, because a parallel path always gives current more ways to flow, reducing total resistance.
RC time constant: how fast a capacitor charges
τ = R × C. A 1,000Ω resistor paired with a 1 microfarad (0.000001F) capacitor: τ = 1,000 × 0.000001 = 0.001 seconds (1 millisecond). This time constant represents the time for the capacitor to charge to about 63.2% of its final voltage — full practical charging is generally considered complete after about 5 time constants (5ms in this case).
Torque: force times leverage
τ = F × r, where r is the perpendicular distance from the pivot point. A 50N force applied at a 0.3-meter lever arm: τ = 50 × 0.3 = 15 N·m. This is exactly why a longer wrench makes a bolt easier to turn — the same applied force produces more torque at a greater distance from the pivot.
Why these formulas all describe a trade-off
Each formula reflects a trade-off familiar from everyday tools: parallel resistors trade individual resistance for combined current capacity, RC circuits trade resistor/capacitor size for charging speed, and torque trades force for leverage distance. Understanding the underlying formula makes the "why" behind each trade-off concrete rather than intuitive guesswork.
Common mistakes to avoid
- Applying the two-resistor parallel formula to three or more resistors without adjusting it (the general form uses reciprocals: 1/R_total = 1/R1 + 1/R2 + 1/R3...)
- Forgetting that "5 time constants" for full RC charging is a practical approximation, not an exact endpoint — technically, full charge is only approached asymptotically, never fully reached
- Assuming torque only depends on force, ignoring that the same force applied off-perpendicular to the lever arm produces less effective torque (scaled by the sine of the angle)
Calculate your own circuits and mechanics with the resistor series & parallel calculator, RC time constant calculator, and torque calculator.