In an electrical circuit

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

Problem

In an

RC

electrical circuit, the charge

Q(t)

(in coulombs) on a capacitor at time

t

seconds satisfies

\frac{dQ}{dt}=0.02\bigl(100-Q\bigr).

Initially the capacitor is uncharged, so

Q(0)=0

. (1) Solve the differential equation to obtain

Q(t)

. (2) Determine the time required for the charge on the capacitor to reach

80\text{ C}

. Express your answer to the nearest tenth of a second. (3) Find the current

I(t)=\dfrac{dQ}{dt}

at the time calculated in part (b), and include appropriate units. (4) State the limiting charge as

t\to \infty

and interpret this value in the context of the circuit.