Question: A charged capacitor C is connected in parallel with a time-varying resistor, R(t), at t = 0.

R(t) = R₀(1 - t / T), for 0 < t < T, and 0 for t > T. Derive a formula for the voltage across C as a function of time.

Answer: v(t) = v₀(1 - t / T) ^{T / (R₀ C)} , for 0 < t < T, and 0 for t > T. If T = R₀C, v(t) itself falls in a straight line.

As T/(R₀C) becomes ≫ 1, or what is the same thing, R₀C/T becomes ≪ 1, v(t)/v₀, shown here in blue, becomes closer to e^{−t / (R₀C)}, shown here in light grey. (Move the slider to change R₀C / T.)