A capacitor discharges through a resistor so that its voltage falls as

PHYSICS · Subjective · Fermi University Physics · Mathematical Methods for Physics · Foundations of mathematical language · Variational and asymptotic methods

Mathematical Methods for Physics, Ch Series, limits and controlled approximation

Problem

A capacitor discharges through a resistor so that its voltage falls as V(t)=V0e−t/τV(t) = V_0 e^{-t/\tau}, where τ=RC\tau = RC. For t/τt/\tau small, the fraction of the initial voltage remaining is sometimes approximated by the two-term truncation V(t)V0≈1−tτ.\frac{V(t)}{V_0} \approx 1 - \frac{t}{\tau}. At t/τ=0.15t/\tau = 0.15, use the Lagrange remainder for e−xe^{-x} (with x=t/τx = t/\tau) to bracket the size of the error made by this truncation, and state upper and lower bounds on the error as a fraction of V0V_0.

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