Charge through a cross section of a conductor is give by `Q = 5t^(2) - 2t` coulomb .Find the average current through the conduction the interval `t_(1) = 2 s ` to `t_(2) = 4s` .

Options

A. 14A
B. 28A
C. 56A
D. 7A

Explanation

To find the average current through the conductor over the interval from \( t_1 = 2 \) seconds to \( t_2 = 4 \) seconds, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Charge Function**:
The charge \( Q \) through a cross-section of the conductor is given by the equation:
\[ Q(t) = 5t^2 - 2t \]

2. **Calculate Charge at \( t_2 = 4 \) seconds**:
Substitute \( t = 4 \) into the charge function:
\[ Q(4) = 5(4^2) - 2(4) = 5(16) - 8 = 80 - 8 = 72 \text{ coulombs} \]

3. **Calculate Charge at \( t_1 = 2 \) seconds**:
Substitute \( t = 2 \) into the charge function:
\[ Q(2) = 5(2^2) - 2(2) = 5(4) - 4 = 20 - 4 = 16 \text{ coulombs} \]

4. **Determine the Change in Charge (\( \Delta Q \))**:
Calculate the change in charge over the interval:
\[ \Delta Q = Q(4) - Q(2) = 72 - 16 = 56 \text{ coulombs} \]

5. **Determine the Change in Time (\( \Delta t \))**:
Calculate the change in time:
\[ \Delta t = t_2 - t_1 = 4 - 2 = 2 \text{ seconds} \]

6. **Calculate the Average Current (\( I \))**:
The average current is given by the formula:
\[ I = \frac{\Delta Q}{\Delta t} \]
Substitute the values:
\[ I = \frac{56}{2} = 28 \text{ amperes} \]

### Final Answer:
The average current through the conductor from \( t = 2 \) seconds to \( t = 4 \) seconds is \( 28 \) amperes.

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