When a capacitor is suddenly connected to a source

MATHEMATICS · Multi-part · AP · Calculus AB · Differential Equations · Differential Equations

Problem

When a capacitor is suddenly connected to a source, the current entering the capacitor decays exponentially with time. In one experiment the current obeys the functional form

I(t) = I_0 e^{-kt}

, where

t

is measured in seconds. • At

t = 0

, the current is measured to be

5.0\,\text{A}

. • At

t = 1.2\,\text{s}

, the current has decreased to

1.84\,\text{A}

. (1) Part 1: (a) Determine the numerical value of the decay constant

k

, including appropriate units. (2) Part 2: (b) Derive an expression for the total charge

Q(t)

that has entered the capacitor between

t = 0

and an arbitrary later time

t

. (3) Part 3: (c) Calculate the total charge that will have entered the capacitor after a very long time has passed (

t \rightarrow \infty

). (4) Part 4: (d) Find the time

t_{90}

at which

90\%

of the total long-time charge found in part (c) has entered the capacitor. (5) Part 5: (e) Discuss whether the capacitor would acquire additional charge after

t_{90}

and, if so, how the rate of charge accumulation compares with earlier times.

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