Current, Resistance, and DC Circuits
Current is charge in motion, I=dQ/dt, and resistance is whatever opposes that motion. Kirchhoff's rules (charge conservation at junctions, energy conservation around loops) let you solve circuits too complicated for simple series/parallel reduction. The unit's calculus-specific payoff is the RC circuit: once a capacitor is in the loop, the current itself depends on how much charge has already accumulated, which turns the circuit equation into a differential equation with exponential solutions — charging and discharging that happen fast at first and slow down, never quite finishing in finite time.
What you'll learn
- Calculate current from the rate of charge flow, I=dQ/dt.
- Apply Ohm's law and calculate resistance from resistivity and geometry.
- Calculate power dissipated in a resistor using any of the equivalent power formulas.
- Apply Kirchhoff's junction and loop rules to solve multi-loop circuits.
- Analyze the exponential charging and discharging of a capacitor in an RC circuit.
- Calculate the RC time constant and interpret what it does and doesn't tell you about how "charged" a capacitor is.
1. Current, Resistance, and Ohm's Law
Current is the rate charge flows past a point: I=dQ/dt, measured in amperes (coulombs per second). A material's resistance to that flow depends on its resistivity ρ (an intrinsic material property) and its geometry: R=ρL/A — longer wires resist more, thicker wires resist less, exactly like a longer or narrower pipe restricting water flow more.
Ohm's law, V=IR, relates the three quantities for a resistor. Power dissipated (converted to heat) can be written three equivalent ways depending on which two quantities you already know: P=IV=I²R=V²/R.
2. Kirchhoff's Rules
Simple series/parallel reduction breaks down for circuits with multiple loops and multiple sources. Kirchhoff's two rules always work regardless: the junction rule (charge conservation — current in equals current out at any node) and the loop rule (energy conservation — the sum of voltage changes around any closed loop is zero).
Together, applying the junction rule at enough nodes and the loop rule around enough independent loops gives exactly as many equations as unknown currents, which can then be solved as a system of linear equations.
3. RC Circuits
Once a capacitor is in a loop with a resistor, the current depends on how much charge the capacitor already holds — Kirchhoff's loop rule becomes a differential equation, dQ/dt=(EMF−Q/C)/R, whose solution is an exponential rather than a constant current.
Charging: Q(t)=Q_max(1−e^(−t/RC)), starting at zero and approaching (never quite reaching) the maximum. Discharging: Q(t)=Q₀e^(−t/RC), starting at the initial charge and decaying toward (never quite reaching) zero. Either way, τ=RC sets the timescale — the time to close about 63% of the remaining gap, not a hard finish line.
Key equations
- I = dQ/dt — The definition of current — the instantaneous rate of charge flow through a cross-section.
- V = IR, R = ρL/A — Ohm's law relates V, I, and R for a resistor; the resistivity formula gives R from a conductor's material and geometry.
- P = IV = I²R = V²/R — Three equivalent forms of the same result — pick whichever two of I, V, R you already know.
- ΣI_in = ΣI_out (junction), ΣΔV = 0 (loop) — Kirchhoff's rules: charge conservation at any junction, and energy conservation (net voltage change is zero) around any closed loop — together enough to solve any DC circuit, however tangled.
- Q(t) = Q_max(1−e^(−t/RC)) [charging], Q(t) = Q₀e^(−t/RC) [discharging] — Time-dependence of charge on a capacitor in an RC circuit. τ is the time to reach (1−1/e)≈63% of the way to full charge, or to fall to 1/e≈37% of the initial charge on discharge — not the time to finish, which technically never happens in finite time.