Current passing through a conductor changes with time as `i = 10e^(-2t)` ampere. Charge passed through a cross-section of conductor in the time-interval t = 0 to `t = oo` is______(in Coulomb).

Explanation

To find the total charge passed through a cross-section of the conductor from time \( t = 0 \) to \( t = \infty \), we start with the relationship between current and charge. The current \( i(t) \) is given by:

\[ i(t) = 10 e^{-2t} \text{ amperes} \]

The current is defined as the rate of flow of charge, which can be expressed mathematically as:

\[ i(t) = \frac{dq}{dt} \]

To find the total charge \( Q \) that has passed through the conductor over the time interval from \( t = 0 \) to \( t = \infty \), we need to integrate the current with respect to time:

\[ Q = \int_{0}^{\infty} i(t) \, dt \]

Substituting the expression for current:

\[ Q = \int_{0}^{\infty} 10 e^{-2t} \, dt \]

Now, we can factor out the constant \( 10 \):

\[ Q = 10 \int_{0}^{\infty} e^{-2t} \, dt \]

Next, we need to evaluate the integral \( \int_{0}^{\infty} e^{-2t} \, dt \). The integral of \( e^{-kt} \) from \( 0 \) to \( \infty \) is given by:

\[ \int_{0}^{\infty} e^{-kt} \, dt = \frac{1}{k} \]

In our case, \( k = 2 \), so:

\[ \int_{0}^{\infty} e^{-2t} \, dt = \frac{1}{2} \]

Now substituting this back into our equation for \( Q \):

\[ Q = 10 \cdot \frac{1}{2} = 5 \text{ coulombs} \]

Thus, the total charge passed through the cross-section of the conductor from \( t = 0 \) to \( t = \infty \) is:

\[ \boxed{5 \text{ coulombs}} \]