**Problem 5:**A discharged capacitor $ C = 2700F $ is connected in series with the resistor $ R = 1\Omega $.(a) Find the general formula for the voltage drop $ V_{\text{cap}} = f(t) $ across the capacitor if the circuit is connected at time $ t = 0 $ to a battery supplying voltage $ V_{\text{bat}} = 2.7V $.(b) How long does it take for the voltage across the capacitor to reach $ 1.45V $?The differential equation governing the voltage drop on the capacitor in this circuit is:\[\frac{V_{\text{bat}}}{RC} - \frac{dV_{\text{cap}}}{dt} - \frac{V_{\text{cap}}}{RC} = 0.\]

Answered: A discharged capacitor C = 2700F is…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

**Problem 5:**

A discharged capacitor $ C = 2700F $ is connected in series with the resistor $ R = 1\Omega $.

(a) Find the general formula for the voltage drop $ V_{\text{cap}} = f(t) $ across the capacitor if the circuit is connected at time $ t = 0 $ to a battery supplying voltage $ V_{\text{bat}} = 2.7V $.

(b) How long does it take for the voltage across the capacitor to reach $ 1.45V $?

The differential equation governing the voltage drop on the capacitor in this circuit is:

\[
\frac{V_{\text{bat}}}{RC} - \frac{dV_{\text{cap}}}{dt} - \frac{V_{\text{cap}}}{RC} = 0.
\]

Expert Solution

Step 1

The differential equation that governs the voltage drop on the capacitor in the circuit is given by

$\frac{V_{b a t}}{R C} - \frac{d V_{c a p}}{d t} - \frac{V_{c a p}}{R C} = 0$ .

Also, given that $C = 2700 F$ and $R = 1 Ω$

Therefore, $R C = 2700$ .

(a) Given $V_{b a t} = 2.7$

The given differential equation is $\frac{V_{b a t}}{R C} - \frac{d V_{c a p}}{d t} - \frac{V_{c a p}}{R C} = 0$ .

Rewrite,

$\begin{array}{rcl} - \frac{d V_{c a p}}{d t} + \frac{V_{b a t}}{R C} - \frac{V_{c a p}}{R C} & = & 0 \\ - \frac{d V_{c a p}}{d t} + (\frac{V_{b a t}}{R C} - \frac{V_{c a p}}{R C}) & = & 0 \\ (\frac{V_{b a t}}{R C} - \frac{V_{c a p}}{R C}) & = & \frac{d V_{c a p}}{d t} \\ \frac{d V_{c a p}}{d t} & = & \frac{V_{b a t}}{R C} - \frac{V_{c a p}}{R C} \end{array}$