Question 399: Current Electricity

Question

The current in a conductor is expressed as $I=3 t^2+4 t^3$, where $I$ is in Ampere and $t$ is in second. The amount of electric charge that flows through a section of the conductor during $t=1 \mathrm{~s}$ to $t=2 \mathrm{~s}$ is __________ C.

Accepted Answer

To find the amount of electric charge that flows through a section of the conductor, we have to integrate the current over the given time interval. The current $I(t)$ as a function of time $t$ is given by:
$$ I=3t^2+4t^3 $$
The electric charge $Q$ that flows through the conductor from time $t = 1$ s to $t = 2$ s is calculated by integrating the current $I(t)$ with respect to time over this interval:
$$ Q = \int_{t_1}^{t_2} I(t) \, dt $$
Substituting the given limits ($t_1=1$ and $t_2=2$) and the expression for $I(t)$, we get:
$$ Q = \int_{1}^{2} (3t^2+4t^3) \, dt $$
Now we'll integrate the function with respect to $t$:
$$ Q = \left[ \frac{3}{3}t^3 + \frac{4}{4}t^4 ight]_{1}^{2} $$
Simplifying the integrated function:
$$ Q = \left[ t^3 + t^4 ight]_{1}^{2} $$
Substitute the upper and lower limits of the integration:
$$ Q = \left[ (2)^3 + (2)^4 ight] - \left[ (1)^3 + (1)^4 ight] $$
$$ Q = \left[ 8 + 16 ight] - \left[ 1 + 1 ight] $$
$$ Q = 24 - 2 $$
$$ Q = 22 	ext{ C} $$
Therefore, the amount of electric charge that flows through the section of the conductor from $t=1$ s to $t=2$ s is 22 Coulombs.

Current Electricity

The current in a conductor is expressed as I=3 t^2+4 t^3, where I is in Ampere and t is in second. The amount of electric charge that flows through a section of the conductor during t=1 \mathrm{~s} to t=2 \mathrm{~s} is __________ C.

Choose the correct answer: