Magnetic fields and forces

RC and RL transients are exponential because the rate depends on what remains

A capacitor’s current is tied to how fast voltage changes; an inductor’s voltage is tied to how fast current changes. Combining either storage relation with resistance makes the change rate proportional to the remaining difference.

Solve the first-order equation

For a charging RC circuit, I = C dV_C/dt and I = (V_s − V_C)/R. Equating gives dV_C/dt = (V_s − V_C)/(RC). Let u = V_s − V_C; then du/dt = −u/(RC), so u = u₀e^{−t/RC} and V_C = V_s(1 − e^{−t/RC}) from zero charge. The time constant is τ = RC.

With R = 10 kΩ and C = 100 μF, τ = 1.0 s. After 2 s, V_C = V_s(1 − e⁻²) = 0.865V_s. At t = τ, the capacitor has reached 63.2% of its final voltage.

The initial and final limits are tests

At t = 0 an uncharged capacitor behaves like a short for voltage continuity; at long time it behaves like an open in a DC circuit. An inductor has the dual current-continuity behavior.

Work it: A 5 V source charges a 2 μF capacitor through 100 kΩ. Calculate τ, initial current, voltage after 0.30 s, and the time to reach 90%.