Question

The equation for an instantaneous current in a material is given by i = 5e-t(A). Calculate the amount of total charge flown through the material
during the time interval t1 = 0 s to
t2 = infinity s

Answer 1

Given:

Equation for instantaneous current in a material is given as i = 5e^(-t) Ampere.

To find:

Amount of charge flow in the time interval of t=0 to t = infinite sec.

Calculation:

Current is defined as the instantaneous rate of flow of charge.

$\therefore \: i = \dfrac{dq}{dt}$

$= > dq = i \times dt$

$= > dq = 5 {e}^{ - t} \times dt$

Integrating on both sides:

$\displaystyle = > \int_{0}^{q} dq = \int_{0}^{ \infty} 5 {e}^{ - t} dt$

$\displaystyle = > q = 5 \int_{0}^{ \infty} {e}^{ - t} dt$

$\displaystyle = > q = - 5 \bigg \{ {e}^{ - t} \bigg \}_{0}^{ \infty}$

$\displaystyle = > q = 5 \bigg \{ {e}^{ - t} \bigg \}_{ \infty}^{ 0}$

$\displaystyle = > q = 5 \bigg \{ \dfrac{1}{{e}^{ t}} \bigg \}_{ \infty}^{ 0}$

$\displaystyle = > q = 5 \bigg \{ \dfrac{1}{{e}^{ 0}} - \frac{1}{ \infty} \bigg \}$

$\displaystyle = > q = 5 \bigg \{ \dfrac{1}{{e}^{ 0}} \bigg \}$

$\displaystyle = > q = 5 \bigg \{ \dfrac{1}{1} \bigg \}$

$\displaystyle = > q = 5 \: C$

So, final answer is :

$\boxed{ \red{ \bold{ \huge{ q = 5 \: C }}}}$