Chapter 5.3, Problem 114AYU

120. Current in an RC Circuit The equation governing the amount of current I (in amperes) after lime t (in microseconds) in a single RC circuit consisting of a resistance R (in ohms), a capacitance C (in microfarads), and an electromotive force E (in volts) is

$I = \frac{E}{R}{e}^{-t/(RC)}$

(a) If $E = 120$ volts, $R = 2000$ ohms, and $C = 1.0$ microfarad, how much current I₁ is flowing initially ( $t = 0$)? After 1000 microseconds? After 3000 microseconds?

(b) What is the maximum current?

(c) Graph this function $I = I_1$ (t), measuring I along the $y\text{-axis}$ and t along the $x\text{-axis}$.

(d) If $E = 120$ volts, $R=1000$ ohms, and $C = 2.0$ microfarads, how much current I₂ is flowing initially? After 1000 microseconds? After 3000 microseconds?

(e) What is the maximum current?

(f) Graph the function $I = I_2\left(t\right)$ on the same coordinate axes as $I_1(t)$.

To determine

To find:

a. If $E=120$ volts, $R=2000$ ohms and $C=1.0$ microfarads, how much current $I_1$ is flowing initially $t=0$? After $1000$ microseconds? After $3000$ microseconds?

Answer to Problem 114AYU

a. $0.06A$ current $I_1$ is flowing initially $t=0$, $0.0364A$ current $I_1$ is flowing after $1000$ microseconds, $0.0134A$ current $I_1$ is flowing after $3000$ microseconds.

Explanation of Solution

Given:

$I = \frac{E}{R}\left[e^{-\frac{t}{RC}}\right]$

Calculation:

a. $E = 120, R = 2000$ and $C = 1$

How much current $I_1$ is flowing initially $= \frac{120}{2000}\left[e^{-\frac{0}{2000\left(1\right)}}\right]$

$= \frac{12}{200}\left[1\right]$

$= 0.06A$

How much current $I_1$ is flowing after 1000 microseconds $= \frac{120}{2000}\left[e^{-\frac{1000}{2000\left(1\right)}}\right]$

$= \frac{12}{200}\left[e^{-\frac{1}{2}}\right]$

$= \frac{12}{200}\left[0.6065\right]$

$= 0.0364A$

How much current $I_1$ is flowing after 3000 microseconds $= \frac{120}{2000}\left[e^{-\frac{3000}{2000\left(1\right)}}\right]$

$= \frac{12}{200}\left[e^{-\frac{3}{2}}\right]$

$= \frac{12}{200}\left[0.2213\right]$

$= 0.0134A$

To determine

To find:

b. The maximum current.

Answer to Problem 114AYU

b. $0.06A$ maximum current when time approaches zero.

Explanation of Solution

Given:

$I = \frac{E}{R}\left[e^{-\frac{t}{RC}}\right]$

Calculation:

b. We obtain maximum current when time approaches zero

$= \frac{120}{2000}\left[e^{-\frac{0}{2000\left(1\right)}}\right]$

$= \frac{12}{200}\left[1\right]$

$= 0.06A$

To determine

To find:

c. graph the function $I = I_1\left(t\right),$ measuring $I$ along the $y\text{-axis}$ and $t$ along the $x\text{-axis}$.

Answer to Problem 114AYU

c.

Explanation of Solution

Given:

$I = \frac{E}{R}\left[e^{-\frac{t}{RC}}\right]$

Calculation:

c.

To determine

To find:

d. If $E = 120$ volts, $R = 1000$ ohms, and $C = 2.0$ henrys, microfarads, how much current $I_1$ is flowing initially $t=0?$ After $1000$ microseconds? After $3000$ microseconds?

Answer to Problem 114AYU

d. $0.12A$ current $I_1$ is flowing initially $t=0$, $0.0727A$ current $I_1$ is flowing after $1000$ microseconds,$0.0268A$ current $I_1$ is flowing after $3000$ microseconds.

Explanation of Solution

Given:

$I = \frac{E}{R}\left[e^{-\frac{t}{RC}}\right]$

Calculation:

d. $E = 120, R = 1000$ and $C = 2$

How much current $I_1$ is flowing initially $= \frac{120}{1000}\left[e^{-\frac{0}{1000\left(2\right)}}\right]$

$= \frac{12}{100}\left[1\right]$

$= 0.12A$

How much current $I_1$ is flowing after 1000 microseconds $= \frac{120}{1000}\left[e^{-\frac{1000}{1000\left(2\right)}}\right]$

$= \frac{12}{100}\left[e^{-\frac{1}{2}}\right]$

$= \frac{12}{100}\left[0.6065\right]$

$= 0.0727A$

How much current $I_1$ is flowing after 3000 microseconds $= \frac{120}{1000}\left[e^{-\frac{3000}{1000\left(2\right)}}\right]$

$= \frac{12}{100}\left[e^{-\frac{3}{2}}\right]$

$= \frac{12}{100}\left[0.2213\right]$

$= 0.0268A$

To determine

To find:

e. The maximum current.

Answer to Problem 114AYU

e. $0.12 A$ maximum current when time approaches zero.

Explanation of Solution

Given:

$I = \frac{E}{R}\left[e^{-\frac{t}{RC}}\right]$

Calculation:

e. We obtain maximum current when time approaches zero

$= \frac{120}{1000}\left[e^{-\frac{0}{2000\left(1\right)}}\right]$

$= \frac{12}{100}\left[1\right]$

$= 0.12A$

To determine

To find:

f. graph the function $I= I_2\left(t\right),$ measuring $I$ along the $y\text{-axis}$ and $t$ along the $x\text{-axis}$.

Answer to Problem 114AYU

f.

Explanation of Solution

Given:

$I = \frac{E}{R}\left[e^{-\frac{t}{RC}}\right]$

Calculation:

f.