Chapter 8.5, Problem 55AYU

Charging a Capacitor See the illustration. If a charged capacitor is connected to a coil by closing a switch, energy is transferred to the coil and then back to the capacitor in an oscillatory motion. The voltage $V$ (in volts) across the capacitor will gradually diminish to 0 with time $t$ (in seconds).

(a) Graph the function relating $V\text{ and }t$:

$V\left(t\right)=e^{-t/3}\cos \left(\pi t\right)\text{ 0 }\le t\le \text{ 3}$

(b) At what times $t$ will the graph of $V$ touch the graph of $y=e^{-t/3}$? When does the graph of $V$ touch the graph of $y=-e^{-t/3}$?

(c) When will the voltage $V$ be between $-0.4\text{ and }0.4$ volt?

To determine

a. Graph the function relating $\text{V}$ and $\text{t}$:

$\text{V}(\text{t}) = e^{ - \frac{4}{3}}\text{cos}\pi t 0 \le t \le 3$

b. At what times $t$ will the graph of $\text{V}$ touch the graph $of e^{ - \frac{4}{3}}?$

When does the graph of $\text{V}$ touch the graph of $- e^{ - \frac{4}{3}}?$

c. When will the voltage $\text{V}$ be between $–0.4$ and $0.4$ volt?

Answer to Problem 55AYU

a. The graph is plotted.

b. The graph of $v$ touches the graph of $y = e^{ - t/3}$ when $\text{t} = 0$, 2. The touches $y = e^{ - t/3}$ when $t = 1, 3.$

c. $- 0.4 < V < 0.4$ when $0.35 < \text{t} < 0.67, 1.29 < 1.75$ and $2.19 < t \le 3$

Explanation of Solution

Given:

If a charged capacitor is connected to a coil by closing a switch, energy is transferred to the coil and then back to the capacitor in an oscillatory motion. The voltage $\text{V}$ (in volts) across the capacitor will gradually diminish to 0 with time $\text{t}$ (in seconds).

Calculation:

a. Graph of $\text{V}(\text{t}) = e^{ - \frac{4}{3}}\text{cos}\pi t 0 \le t \le 3$

b. On the interval $0 \le t \le 3$, the graph of $v$ touches the graph of $y = e^{ - t/3}$ when $\text{t} = 0$, 2. The touches $y = e^{ - t/3}$ when $t = 1, 3.$

c. From the graph we see that $- 0.4 < V < 0.4$ at various intervals. $- 0.4 < V < 0.4$

When

$0.35 < t < 0.67, 1.29 < t < 1.75$ and $2.19 < t \le 3$