RC and RL circuit
/ R-C and R-L /
Flip a switch and a circuit does not respond instantly — a capacitor takes a moment to charge, a coil takes a moment to let current build. That lag, that smooth approach to a final state, is governed by a first-order differential equation, and RC and RL circuits are the textbook examples that engineers reach for constantly.
In an RC circuit (resistor R and capacitor C with applied voltage V), Kirchhoff's voltage law adds the voltage drops around the loop: the resistor's RI plus the capacitor's q/C equals V. Since current I is dq/dt, this becomes R dq/dt + q/C = V — a first-order linear equation for the charge q(t). In an RL circuit (resistor and inductor L), the same loop rule gives L dI/dt + RI = V, a first-order linear equation for the current. Both have the form (rate term) + (decay term) = (source), and both solve into a transient piece that fades exponentially plus a steady-state piece set by the source. For a constant source the charge or current rises or falls toward its final value along a smooth exponential curve.
These circuits are the electrical twins of Newton's cooling and the mixing tank — all first-order linear equations with the same exponential personality. The natural timescale is the time constant: RC for the capacitor circuit, L/R for the inductor circuit, the time for the transient to fade to about 37% of its initial size. After a few time constants the transient is gone and only the steady state remains.
Charging a capacitor through a resistor from a battery V: R dq/dt + q/C = V gives q(t) = CV(1 − e^(-t/(RC))). With R = 1000 ohm and C = 1 microfarad, the time constant RC = 0.001 s, so the capacitor is about 95% charged after roughly 3 milliseconds.
First-order linear: transient fades exponentially, steady state set by the source.
Only single-energy-storage circuits (one capacitor or one inductor) are first-order. Put both R, L, and C in one loop and you get a second-order equation that can oscillate — a different beast entirely (the RLC circuit).