Capacitor charging
The exponential rise V(t) = V₀(1−e^(−t/τ)) of a capacitor's voltage as it is charged through a resistor. Half the energy delivered by the source ends up in the capacitor; half is dissipated in the resistor, regardless of R.
Definition
When a capacitor is charged through a resistor from a step voltage V₀, its voltage rises exponentially with time constant τ = RC: V_C(t) = V₀(1−e^(−t/τ)). The current into the capacitor starts at V₀/R and decays exponentially to zero on the same timescale. At t = 5τ the process is ≈ 99% complete for practical purposes.
A striking result emerges from the energy bookkeeping. The energy delivered by the source as the capacitor charges from 0 to V₀ is W_source = Q V₀ = CV₀². The energy stored in the capacitor at the end is U_C = ½CV₀². The remaining ½CV₀² is dissipated in the resistor — exactly half of the source's delivered energy, regardless of the value of R. Making R small (fast charging) and making R large (slow charging) both dissipate the same 50% in heat. Only a non-linear or reactive intermediate — a switching regulator, an inductive coupling — can break this even split.
The universality of the 50% loss is why brute-force RC charging is the wrong way to store energy efficiently. Real switched-mode power supplies interrupt the current at high frequency and transfer energy via inductors, achieving 85–95% efficiency. But for simple filtering, timing, and decoupling applications, the 50% loss is negligible because the absolute energy involved is tiny. Anywhere in electronics that a waveform needs to be "softened" on the microsecond-to-millisecond scale, an RC charging transient is the tool that does it.