Calculation of Electric Charge

To calculate the electric charge flowing through a section of the conductor during the time interval from t = 2 sec to t = 3 sec, we need to find the integral of the current function over this time interval.

Given that the current in the conductor varies with time t as:
I = 2t - 3t^2

Step 1: Finding the Integral
To find the integral of the current function, we need to integrate each term separately.

The integral of 2t with respect to t is:
∫(2t) dt = t^2

The integral of -3t^2 with respect to t is:
∫(-3t^2) dt = -t^3

Therefore, the integral of the current function is:
∫(2t - 3t^2) dt = t^2 - t^3 + C

Where C is the constant of integration.

Step 2: Evaluating the Integral
To evaluate the integral over the given time interval from t = 2 sec to t = 3 sec, we substitute these values into the integral expression:

Q = ∫(2t - 3t^2) dt
Q = [(t^2 - t^3) + C] │from 2 to 3

Substituting t = 3:
Q = [(3^2 - 3^3) + C] - [(2^2 - 2^3) + C]
Q = [9 - 27 + C] - [4 - 8 + C]
Q = -18 + C - (-4 + C)
Q = -18 + C + 4 - C
Q = -14

Therefore, the electric charge flowing through the section of the conductor during the time interval from t = 2 sec to t = 3 sec is -14 Coulombs.

Explanation
The electric charge flowing through a conductor is given by the integral of the current function over a specific time interval. In this case, the current function is given as I = 2t - 3t^2. By finding the integral of this function and evaluating it over the time interval from t = 2 sec to t = 3 sec, we can determine the electric charge.

The integral of the current function is found by integrating each term separately. After evaluating the integral over the given time interval, we obtain the electric charge as -14 Coulombs. The negative sign indicates the direction of the charge flow, which is opposite to the conventional current direction.